itfs volume 17 issue 2 april 2009

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 17, NO. 2, APRIL 2009 253

Piecewise Fuzzy Anti-Windup Dynamic OutputFeedback Control of Nonlinear Processes With

Amplitude and Rate Actuator SaturationsTiejun Zhang, Member, IEEE, Gang Feng, Fellow, IEEE, Huaping Liu, Member, IEEE,

and Jianhong Lu

Abstract—In this paper, a novel anti-windup dynamic outputcompensator is developed to deal with the robust H∞ outputfeedback control problem of nonlinear processes with amplitudeand rate actuator saturations and external disturbances. Via fuzzymodeling of nonlinear systems, the proposed piecewise fuzzy anti-windup dynamic output feedback controller is designed based onpiecewise quadratic Lyapunov functions. It is shown that with sec-tor conditions, robust output feedback stabilization of an input-constrained nonlinear process can be formulated as a convex opti-mization problem subject to linear matrix inequalities. Simulationstudy on a strongly nonlinear continuously stirred tank reactor(CSTR) benchmark plant is given to show the performance of theproposed anti-windup dynamic compensator.

Index Terms—Actuator saturation, amplitude and rate con-straints, anti-windup, dynamic output feedback, fuzzy systems,linear matrix inequality (LMI), piecewise quadratic Lyapunovfunction, process control.

I. INTRODUCTION

OUTPUT feedback control plays a major role in processcontrol for industrial plant operation and optimization

[1]–[4] since state variables are not always fully measurable formost of the industrial plants in practice. There have been manyresults on dynamic output feedback stabilization for continu-ous linear systems and discrete-time systems (for example, see[6]–[8]). More recently, the robust H2 and H∞ control problemof discrete-time linear systems with polytopic uncertainties viadynamic output feedback was investigated in [9].

On the other hand, in process industry, actuators are alwayssubject to physical limitations [1], [2], [5], which would lead toperformance deterioration and even instability for large pertur-bations [10], [11]. To deal with these problems, some bounded

Manuscript received February 12, 2007; revised June 9, 2007; acceptedSeptember 3, 2007. First published April 30, 2008; current version publishedApril 1, 2009. The work was supported in part by the Hong Kong ResearchGrant Council under Grant CityU 112806, in part by the National Natural Sci-ence Foundation of China under Grant 50640460116, Grant 60504003, andGrant 60625304, and in part by the National High-Tech R&D Program of Chinaunder Grant 2006AA05A107.

T. J. Zhang and G. Feng are with the Department of Manufacturing Engineer-ing and Engineering Management, City University of Hong Kong, Kowloon,Hong Kong (e-mail: [email protected]; [email protected]).

H. P. Liu is with the Department of Computer Science and Technology,Tsinghua University, Beijing 100084, China (e-mail: [email protected]).

J. H. Lu is with the School of Energy and Environment, Southeast University,Nanjing 210096, China (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TFUZZ.2008.924238

state feedback and output feedback controllers have been de-veloped for linear systems [10]–[16]. In many cases, to avoidthe input saturation, low-gain control laws are designed for aprescribed control constraint and known initial system states.More specifically, the saturation nonlinearity for linear sys-tems is characterized by the convex hull of several linear feed-backs [11], [12]. Alternatively, the saturation term has beentransformed into a deadzone nonlinearity in [14], [15], and ac-cordingly, a sector condition and S-procedure were used toderive some stability conditions. These explicit conditions canbe formulated as linear matrix inequalities (LMIs), and thus,it is very easy to obtain the controller gains by solving theseLMIs. In fact, this LMI-based anti-windup design can also dealwith the tougher actuator rate constraint problem conveniently[16].

However, to our best knowledge, few works have been re-ported in open literature to consider the robust output feedbackcontrol problem for strongly nonlinear industrial processes inthe presence of actuator saturations, external disturbances, andplant uncertainties. Takagi–Sugeno (T–S) fuzzy dynamic mod-els have been widely accepted in the control community dueto its excellent interpretability and capability of representingthe dynamics of nonlinear systems [17]–[22]. There have beenmany controller design methods for T–S fuzzy systems via bothstate feedback and observer-based output feedback [20]–[26]. Ithas been demonstrated that piecewise quadratic Lyapunov func-tions [27], [28] can be used to reduce the conservatism in thefuzzy stabilizing controller design [25], [26]. The controller de-sign for fuzzy systems with actuator saturation has also attractedmuch attention during the last several years, and representativeworks include [29]–[32]. It is noted that only low-gain designapproach was used in [29]. A less conservative approach that isbased on convex hull representations was proposed in [30]. Veryrecently, the saturation function was formulated into a specificnonlinear saturation sector in [31] and a new design approachwas given that requires less number of LMIs. All of [29]–[31]dealt with the case of continuous-time systems. A fuzzy Popovcriterion design approach for both continuous- and discrete-timefuzzy systems was given in [32]. However, all of the aforemen-tioned results are only restricted to state-feedback controllers.To our best knowledge, there is no result available in the open lit-erature on dynamic output feedback controller for fuzzy systemswith actuator saturation. Furthermore, the rate-limited controllerdesign, due to its difficulty, has never been considered for fuzzysystems.

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254 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 17, NO. 2, APRIL 2009

To deal with the aforementioned robust constrained dynamicoutput feedback control problem, an uncertain fuzzy model isused to approximate the uncertain nonlinear process and also todeal with plant modeling errors that are inevitable for approxi-mation. Based on this fuzzy model, we then develop a piecewiserobust fuzzy anti-windup dynamic output compensator for theuncertain nonlinear processes with amplitude and rate satura-tions on manipulated inputs. Motivated by the recent result [16],the design of this anti-windup dynamic compensator is basedon piecewise quadratic Lyapunov functions and can be accom-plished by a convex optimization procedure subject to severalLMI conditions. Both controller parameters and the anti-windupcompensation gain can be obtained simultaneously. This makesjoint optimization possible and may lead to better performance.Finally, the proposed controller design method is applied to thecontrol of the strongly nonlinear chemical continuously stirredtank reactor (CSTR) plant.

The rest of the paper is organized as follows. Section IIintroduces some preliminaries about uncertain fuzzy models,piecewise Lyapunov stability, and sector conditions for inputamplitude and rate saturations. The novel piecewise fuzzy anti-windup dynamic output compensator is proposed in Section III.In Section IV, the case study on a nonlinear CSTR bench-mark is present to demonstrate the effectiveness of the proposedapproach. Finally, some conclusions are drawn in Section V.

Notations: For any two vectors x, y ∈ m , x y means thatx(i) − y(i) ≥ 0 ∀i = 1, . . . ,m, where x(i) and y(i) denotes theith component of x and y, respectively. For any two symmetricmatrices A and B, A > B means that A − B is positive definite.Further, stands for symmetric blocks in matrices and ∗ standsfor blocks in matrices without subsequent use.

II. PRELIMINARIES

A. Uncertain Fuzzy Dynamic Model

T–S fuzzy dynamic models are widely used to represent com-plex systems [17]–[22] with both fuzzy inference rules and localanalytic linear models as follows.

Rule Rl : IF ξ1(k) is Ml1 AND . . . ξr (k) is Ml

r , THEN

x(k + 1) = (Al + ∆Al)x(k) + (Bl + ∆Bl)u(k)+ (Dl + ∆Dl)w(k)

y(k) = Clx(k)

z(k) =

Q1/2y y(k)

R1/2a u(k)

R1/2r ∆u(k)

, l = 1, . . . , L

(1)

where Rl denotes the lth fuzzy inference rule, L the num-ber of fuzzy rules, Ml

j are fuzzy sets, x(k) ∈ n the systemstate variables, u(k) ∈ m the control input variables, ∆u(k)the one-step control difference or also called the control rate,y(k) ∈ p the outputs available for measurement, z(k) ∈ p

the controlled variable that needs to be regulated, w(k) ∈ q

the external disturbance that belongs to L2 [0,∞), ξ(k) ∈ r

some measurable system variables, and (Al,Bl, Cl ,Dl) is thelth nominal local model of the fuzzy system (1). The uncertainterms (∆Al,∆Bl,∆Dl) represent the plant uncertainty and ap-

proximation error between the original nonlinear system and itsapproximating model with the assumption that

[∆Al, ∆Bl, ∆Dl ] = M1l ∆[N

1l , N

2l , N

3l ], ∆T ∆ ≤ In .

(2)Let µl(ξ(k)) be the normalized membership function of theinferred fuzzy set Ml where Ml =

∏rj=1 Ml

j and∑L

i=1 µl = 1.By using a center-average defuzzifier, product inference, andsingleton fuzzifier [17]–[22], the uncertain fuzzy model (1) canbe expressed by the following global model:

x(k + 1) = A(µ)x(k) + B(µ)u(k) + D(µ)w(k)y(k) = C(µ)x(k)

z(k) =

Q1/2y y(k)

R1/2a u(k)

R1/2r ∆u(k)

(3)

where

A(µ) =L∑

l=1

µlAl , B(µ) =L∑

l=1

µlBl

C(µ) =L∑

l=1

µlCl, D(µ) =L∑

l=1

µlDl

µ := µ(ξ) := (µ1 , . . . , µL )

Al = Al + ∆Al, Bl = Bl + ∆Bl, Dl = Dl + ∆Dl.

It should be pointed out that Cl in (1) can be re-garded as one constant matrix C with 0 or 1 ele-ments since y(k) stand for some measurable system states.Without loss of generality, the disturbance w(k) is as-sumed to be with finite energy

∑∞k=0 w(k)T w(k) ≤ wmax <

∞. Note that z(k)T z(k) = y(k)T Qyy(k) + u(k)T Rau(k) +∆u(k)T Rr∆u(k) implies that the H∞ control of (1) also en-forces the transient control performance.

B. Piecewise Quadratic Lyapunov Stability

Similar to [27] and [28], the output space can be parti-tioned into a number of closed polyhedral regions, denoted asSss∈I ⊆ p , where I is the index set of regions. For subse-quent use, a set Ω that represents all possible transitions fromone region to any possible region including itself is defined as

Ω := s, t|y(k) ∈ Ss , y(k + 1) ∈ St. (4)

Define a piecewise Lyapunov function candidate for the regions by

V (x(k)) = x(k)T Psx(k), s ∈ I (5)

then, we have

∆V (x(k)) = V (x(k + 1)) − V (x(k))

= x(k + 1)T Ptx(k + 1) − x(k)T Psx(k). (6)

If the Lyapunov function (5) satisfies that ∆V (k) < 0 for alltrajectories of the system, the system can be guaranteed to bestable in the sense of Lyapunov.


ZHANG et al.: PIECEWISE FUZZY ANTI-WINDUP DYNAMIC OUTPUT FEEDBACK CONTROL OF NONLINEAR PROCESSES 255

Lemma (Piecewise Lyapunov Stability [28]): The au-tonomous piecewise linear system x(k + 1) = Asx(k) isglobally exponentially stable if the following LMIs admit aset of positive definite matrix solutions Ps :

ATs PtAs − Ps < 0, s, t ∈ Ω. (7)

Notice that the piecewise quadratic Lyapunov functions [28] areused here to reduce the conservatism in the stabilization problemof a piecewise linear system, which is quite different from thecase for a linear time-invariant system.

C. Input Saturation and Anti-Windup Condition

Assume now that the manipulated input u is subject to am-plitude limitations defined as follows:

−ρa u(k) ρa (8)

where ρa(i) > 0, i = 1, . . . ,m, denote the component-wise con-trol amplitude bounds. Thus, the actual control input for im-plementation is a saturated one, u(k) = sat(v(k)), with eachcomponent defined by

sat(v)(i) =

+ρa(i) , if ρa(i) v(i)

v(i) , if − ρa(i) v(i) ρa(i)

−ρa(i) , if v(i) −ρa(i).

(9)

Let the function ψa(v) := v − sat(v), then with (9), one has

ψa(v(i)) =

v(i) − ρa(i) , if ρa(i) v(i)

0, if − ρa(i) v(i) ρa(i)

v(i) + ρa(i) , if v(i) −ρa(i)

(10)

which behaves like a deadzone nonlinearity.Moreover, as in [5], the control input in industry might also

be subject to rate constraint

−ρr ∆u(k) ρr (11)

where ρr(i) > 0, i = 1, . . . ,m, denote the component-wise rateconstraint bounds of control inputs. Then, the control variation∆u(k) has the similar saturated components to (9), and thus,we have the corresponding deadzone nonlinearity ψr .

For one polyhedral set Ψ := (α, β)| − ρ(i) ≤ (α(i) −β(i)) ≤ ρ(i) , i = 1, . . . ,m, the so-called sector condition hasbeen described as follows.

Lemma 2 ([14], [15]): If (α, β) ∈ Ψ, then the nonlinearityψ(α) satisfies the inequality

ψ(α)T · W · [ψ(α) − β] ≤ 0 (12)

where W denotes any diagonal positive definite matrix.Note that the sector condition (12) is indeed useful to develop

the convex stability conditions of the constrained linear time-invariant system with anti-windup strategy [14], [15].

Lemma 3 ([16]): Consider the following system composed ofm integrators

v(k + 1) = Im v(k) + σ(k)σ(k) = satr (ω(k))u(k) = sata(v(k))

(13)

where sata(•) and satr (•) denote the amplitude and rate satu-rations, respectively, and ω the input of this integrator system.If |σ(k)(i) | ≤ ρr(i) , i = 1, . . . , m, it follows that

|∆u(i)(k)| = |u(i)(k + 1) − u(i)(k)| ≤ ρr(i) . (14)

Proof: Provided that the Lipschitz constant of the saturationfunction sat(•) equals 1, it follows that

|∆u(i)(k)| = |sata(i)(v(k) + σ(k))(i) − sata(i)(v(k))(i) |≤ |σ(k)(i) | = |satr (ω(i)(k))| ≤ ρr(i) . (15)

III. MAIN RESULTS

Considering the system with some input saturation, the ob-jective here is to design a suitable anti-windup output feedbackcontroller to guarantee the stability of the resulting closed-loopcontrol system with an optimal level of disturbance attenuation,that is, to find a controller such that the upper bound γ > 0 ofinduced L2-norm from the disturbance w(k) to the controlledoutput z(k)

‖z(k)‖‖w(k)‖ ≤ γ (16)

is minimized for all nonzero w(k) ∈ L2 under zero initialconditions.

To achieve this goal, we choose the following piecewise fuzzyanti-windup dynamic output feedback compensator as a candi-date controller.

Rule Rl : IF ξ1(k) is Ml1 AND . . . ξr (k) is Ml

r , THEN

v(k + 1) = Im v(k) + σ(k)xc(k + 1) = Ac

lsxc(k) + Bcs [y(k)T v(k)T ]T

+Ecls(sata(v(k)) − v(k))

+Fcs (satr (yc(k)) − yc(k))

yc(k) = Ccs xc(k) + Dc

s [y(k)T v(k)T ]T

u(k) = sata(v(k)), σ(k) = satr (yc(k)),s ∈ I, l = 1, . . . , L

(17)

where xc(k) ∈ n , yc(k) ∈ m , and v(k) ∈ m . Let

ψr (k) := ψr (yc(k)) = yc(k) − satr (yc(k)) (18)

ψa(k) := ψa(v(k)) = v(k) − sata(v(k)) (19)

it follows that

u(k) = v(k) − ψa(k) (20)

σ(k) = yc(k) − ψr (k) (21)

v(k + 1) = Im v(k) + σ(k) (22)

where yc(k) = Ccs xc(k) + Dc

sC[x(k)T v(k)T ]T , s ∈ I, andC = diagC, Im. Also, by the fuzzy inference mentioned inSection II-A, (17) becomes

xc(k + 1) =L∑

l=1

µl [BcsC[x(k)T v(k)T ]T + Ac

lsxc(k)

− Eclsψa(k) − Fc

s ψr (k)], s ∈ I. (23)



And under the control input (20), the fuzzy system (3) reads

x(k + 1) =L∑

l=1

µl [(Al + ∆Al)x(k) + (Bl + ∆Bl)u(k)

+ (Dl + ∆Dl)w(k)]

=L∑

l=1

µl [Alx(k) + Blv(k) − Blψa(k)

+ Dlw(k)], s ∈ I. (24)

Defining an augmented state vector

x(k) =[

x(k)v(k)

]∈ n+m , x(k) =

[x(k)xc(k)

]∈ 2(n+m )

and with (22)–(24), the closed-loop system can be formulatedas follows:

x(k + 1) =L∑

l=1

µl [Als x(k) − Fsψr (k) − Blsψa(k)

+ Dlw(k)], s ∈ I (25)

Als = Als + ∆Al =[Al + FDc

sC FCcs

BcsC Ac

ls

]+

[∆Al OO O

]Fs =

[FFc

s

], Al =

[Al Bl

O Im

], F =

[OIm

]Bls = Bls + ∆Bl =

[Bl

Ecls

]+

[∆Bl

O

], Bl =

[Bl

O

]Dl = Dl + ∆Dl =

[Dl

O

]+

[∆Dl

O

], Dl =

[Dl

O

].

Moreover, denoting L = [L0 O], L0 = [O Im ], andKs = [ Dc

sC Ccs ] yields v(k) = L · x(k) and yc(k) = Ks ·

x(k); then with (18) and (19), the sector conditions (12) foramplitude and rate saturations become

ψ(Lx(k))T · Wa · [ψ(Lx(k)) − Gax(k)] ≤ 0 (26)

ψ(Ksx(k))T · Wr · [ψ(Ksx(k)) − Gr x(k)] ≤ 0 (27)

for any matrix Ga, Gr ∈ m×2(n+m ) satisfying (Lx, Ga x) ∈Ψ and (Ksx,Gr x) ∈ Ψ, respectively.

Therefore, the control objective becomes equivalent to find-ing an anti-windup dynamic output compensator (17) to ensurethe stability of the closed-loop system (25) and minimize its dis-turbance attenuation performance parameter γ. Now we presentthe corresponding controller design method in the followingtheorem.

Theorem 1: For the system (1), the piecewise fuzzy dy-namic output feedback controller with anti-windup loop (17)can guarantee the stability of the closed-loop system and achievea disturbance rejection level γ, if there exist several sets ofsymmetric positive definite matrices Ps, Vs ∈ (n+m )×(n+m ) ,positive definite diagonal matrices Sa, Sr ∈ m×m , and matri-ces X,Y,W,Us,Als ∈ (n+m )×(n+m ) , Z1

a , Z2a , Z1

r , Z2r , Cs ∈

m×(n+m ) , Qals , Q

rs ∈ (n+m )×m ,Bs ∈ (n+m )×(p+m ) ,Ds ∈

m×(p+m ) , s ∈ I, such that the following optimization prob-lem is solvable

minX

η (28)

subject to the LMIs (29), as shown at the bottom of the page and Ps Us Vs

(L0X)(i) − Z1a(i) (L0)(i) − Z2

a(i) ρ2a(i)

≥ 0,

s ∈ I, i = 1, . . . ,m (30) Ps Us Vs

Cs(i) − Z1r(i) (DsC)(i) − Z2

r(i) ρ2r(i)

≥ 0,

s ∈ I, i = 1, . . . ,m (31)

where X denotes a set of decision variables

η, Ps, Us, Vs, Z1a , Z2

a , Z1r , Z2

r , Sa , Sr ,

X, Y,W, AlsBs, Cs, Ds, Qals, Qr

s

ηIq

O Ps

O Us Vs

O Z1r Z2

r 2Sr

O Z1a Z2

a O 2Sa

−Dl AlX + FCs Al + FDsC FSr Bl Sa X + XT − Pt

−Y T Dl Als Y T Al + BsC Qrs Qa

ls I + W T − Ut Y + Y T − Vt

O Q1/2X Q1/2 O O O O In +m

O O O O O O O O In +m

O R1/2a L0X R

1/2a L0 O R

1/2a Sa O O O O Im

O R1/2r Cs R

1/2r DsC R

1/2r Sr O O O O O O Im

O O O O O (M1l )

T (M1l )

T Y O O O O ε−1In

−N3l N

1l X N

1l O N

2l Sa O O O O O O O εIn

> 0,

s ∈ I, t ∈ Φ(s), l ∈ Υ(s), (29)



and where Φ(s)= t ∈ I: s, t ∈ Ω, Υ(s) = l ∈ 1, . . . , L:∃y ∈ Ss , µl(y) > 0, and

Al =[

Al Bl

O Im

], F =

[O

Im

], Bl =

[Bl

O

]Dl =

[Dl

O

], Q =

[CT QyC O

O O

]where Qy ,Ra , and Rr are given controller design parameters,and γ = η1/2 . Moreover, the controller gains of the piece-wise fuzzy anti-windup dynamic compensator (17) can be ob-tained by

Dcs = Ds , MNT = W − XT Y

Ccs = (Cs − Ds

cCX)(MT )−1

Ecls = N−1(Qa

lsS−1a − Y T Bl)

Fcs = N−1(Qr

sS−1r − Y T F)

Bcs = N−1(Bs − Y T FDc

s)

Acls = N−1(Als − Y T AlX − BsCX − Y T FCc

s MT )M−T .

(32)

Proof: For the closed-loop system (25), consider thepiecewise quadratic Lyapunov function given by V (x(k)) =x(k)T Ps x(k), s ∈ I. If the sector condition (26) exists, andwith Q = diagCT QyC,O, it follows that

V (x(k + 1)) − V (x(k)) + y(k)T Qyy(k)

+ u(k)T Rau(k) + ∆u(k)T Rr∆u(k) − γ2w(k)T w(k)

≤ x(k + 1)T Pt x(k + 1) − x(k)T Ps x(k) + x(k)T Qx(k)

+ u(k)T Rau(k) + ∆u(k)T Rr∆u(k) − γ2w(k)T w(k)

− 2ψa(k)T Wa [ψa(k) − Gax(k)]

− 2ψr (k)T Wr [ψr (k) − Gr x(k)], s, t ∈ Ω. (33)

With the closed-loop system (25), (33) is equivalent to (34), asshown at the bottom of the page, where the first inequality holdsdue to Lemma 3 and 2XT RY ≤ infR>0XT RX + Y T RY as in [20]. Then, one has

V (x(k + 1)) − V (x(k)) + y(k)T Qyy(k) + u(k)T Rau(k)

+ ∆u(k)T Rr∆u(k) − γ2w(k)T w(k) < 0 (35)

if the terms in the bracket of (34) are negative definite; that is,

γ2Iq

O Ps O −WrGr 2Wr O −WaGa O 2Wa

−Dl −Als Fs Bls P−1t

O −Q1/2 O O O I

O −R1/2a L O R

1/2a O O I

O −R1/2r Ks R

1/2r O O O O I

> 0

(36)or equivalently

γ2Iq

O Ps O −WrGr 2Wr O −WaGa O 2Wa


O −Q1/2 O O O I

O −R1/2a L O R

1/2a O O I

O −R1/2r Ks R

1/2r O O O O I

+

O O O O O O O O O O

−∆Dl −∆Als O ∆Bl O O O O O O O O O O O O O O O O O O O O O O

> 0.

(37)

L∑l=1

µl [Als x(k) − Blsψ(k) + Dlw(k)]T · Pt ·L∑

j=1

µj [Ajs x(k) − Bjsψ(k) + Djw(k)] − x(k)T Ps x(k) + x(k)T Qx(k)

+ u(k)T Rau(k) + ∆u(k)T Rr∆u(k) − γ2w(k)T w(k) − 2ψa(k)T Wa [ψa(k) − Gax(k)] − 2ψr (k)T Wr [ψr (k) − Gr x(k)]

≤L∑

l=1

µl [Als x(k) − Blsψ(k) + Dlw(k)]T Pt [Als x(k) − Blsψ(k) + Dlw(k)] − x(k)T Ps x(k) + x(k)T Qx(k)

+ u(k)T Rau(k) + σ(k)T Rrσ(k) − γ2w(k)T w(k) − 2ψa(k)T Wa [ψa(k) − Gax(k)] − 2ψr (k)T Wr [ψr (k) − Gr x(k)]

=L∑

l=1

µl

w(k)x(k)ψr (k)ψa(k)

T

DTl

ATls

−F Ts

−BTls

Pt [ Dl Als −Fs −Bls ] −

γ2Iq

O Ps − Q

O −WrGr 2Wr

O −WaGa O 2Wa

+

O

LT

O

−I

· Ra · [O L O −I ] +

O

KTs

−I

O

· Ra · [ O Ks −I O ]

w(k)x(k)ψr (k)ψa(k)

(34)



Also with the assumption (2) for the uncertain parts of fuzzymodel, let

[∆Al,∆Bl,∆Dl ] = M 1l ∆[N 1

l , N 2l , N 3

l ]

M 1l =

[M

1l

O

], N 1

l =[N

1l O

], N 2

l = N2l , N 3

l =N3l .

Then the second term in (37) is equivalent to XT ∆Y + Y T

∆X ≤ inf‖∆‖≤I εXT X + ε−1 Y T Y , where X = [O O O

O (M 1l )T O O O] and Y = [N 3

l N 1l O −N 2

l O OO O], which is similar to that in [7]. So, (36) holds if thematrix inequality (38), as shown at the bottom of the page,holds. Motivated by the variable change method in [6], now wedefine several matrices to deal with piecewise dynamic outputfeedback

H =[

Y ∗NT ∗

], H−1 =

[X ∗

MT ∗

], Π =

[X I

MT O

].

Thus, multiplying (38) on both sides by

diagI,−Π, Sr , Sa ,HΠ, I, I, I, I, I,Sr = W−1

r , Sa = W−1a

and its transpose yields (39), as shown at the bottom of the page.From the definition of Π and H , it follows that

ΠT HΠ =[

XT W

I Y

], W = XT Y + MNT

ΠT PsΠ =[

Ps UTs

Us Vs

], ΠT PtΠ =

[Pt UT

t

Ut Vt

]ΠT HT AlsΠ =

[AlX + FCs Al + FDc

sC

Als Y T Al + BsC

]Als = Y T AlX +BsCX +Y T FCc

s MT +NAclsM

T

Bs = Y T FDcs + NBc

s , Cs = DcsCX + Cc

s MT ,

Ds = Dcs

ΠT HT BlsSa =[BlSa

Qals

], Qa

ls = (Y T Bl + NEcls)Sa

ΠT HT FsSr =[FSr

Qrs

], Qa

ls = (Y T F + NFcs )Sr

− ΠT HT Dl =[ −Dl

−Y T Dl

], GrΠ = [ Z1

r Z2r ]

Q1/2Π =[Q1/2X Q1/2

O O

], Q =

[CT QyC O

O O

]R1/2

a LΠ =[R

1/2a L0X R

1/2a L0

], GaΠ = [ Z1

a Z2a ]

R1/2r KsΠ =

[R

1/2r Cs R

1/2r DsC

], η = γ2

(M 1l )T HΠ =

[(M

1l )

T (M1l )

T Y], N 2

l Sa = N2l Sa

N 1l Π =

[N

1l X N

1l

], −N 3

l = −N3l .

γ2Iq

O Ps

O −WrGr 2Wr

O −WaGa O 2Wa


O −Q1/2 O O O I

O −R1/2a L O R

1/2a O O I

O −R1/2r Ks R

1/2r O O O O I

O O O O (M 1l )T O O O ε−1I

−N 3l −N 1

l O N 2l O O O O O εI

> 0. (38)

γ2Iq

O ΠT PsΠ

O GrΠ 2Sr

O GaΠ O 2Sa

−ΠT HT Dl ΠT HT AlsΠ ΠT HT F sSr ΠT HT BlsSa ΠT HT P−1t HΠ

O Q1/2Π O O O I

O R1/2a LΠ O R

1/2a Sa O O I

O R1/2r KsΠ R

1/2r Sr O O O O I

O O O O (M 1l )T HΠ O O O ε−1I

−N 3l N 1

l Π O N 2l Sa O O O O O εI

> 0 (39)



With HT P−1t H ≥ H + HT − Pt as in [8], substituting these

variable changes into (39) yields the condition (29) in thetheorem, and from (35), it can be concluded that (29) canguarantee

V (x(k + 1)) − V (x(k)) < −y(k)T Qyy(k) − u(k)T Rau(k)

−∆u(k)T Rr∆u(k) + γ2w(k)T w(k). (40)

It thus follows that when w(k) ≡ 0, the closed-loop systemis asymptotically stable. By summing this inequality fromk = 0 to ∞, with V (x(0)) = 0 from the assumption of zeroinitial condition x(0) = 0 and V (x(∞)) ≥ 0 in the presence ofdisturbance, it implies that

∞∑k=0

[y(k)T Qyy(k) + u(k)T Rau(k)

+ ∆u(k)T Rr∆u(k)] < γ2∞∑

k=0

[w(k)T w(k)]. (41)

Meanwhile, if the condition (30) in the theorem is satisfied,with the change of variables Ps ,L, and Ga , it becomes[

ΠT PsΠ (LΠ − GaΠ)T(i)

(LΠ − GaΠ)(i) ρ2a(i)

]≥ 0. (42)

Pre- and postmultiplying (42), respectively, by diagΠ−T , Iand its transpose, one gets[

Ps (L − Ga)T(i)

(L − Ga)(i) ρ2a(i)

]≥ 0 (43)

or equivalently

|(L − Ga)(i) x(k)|2 ≤ ρ2a(i) .

Obviously, with the condition (30), (Lx, Ga x) ∈ Ψ ensuresthe satisfaction of the sector condition (26). Similarly, if thecondition (31) in the theorem is satisfied, with the change ofvariables Ps ,Ks, and Gr , it becomes[

ΠT PsΠ (KsΠ − GrΠ)T(i)

(KsΠ − GrΠ)(i) ρ2r(i)

]≥ 0. (44)

Pre- and postmultiplying (44), respectively, by diagΠ−T , Iand its transpose, one gets[

Ps (Ks − Gr )T(i)

(Ks − Gr )(i) ρ2r(i)

]≥ 0 (45)

or equivalently

|(Ks − Gr )(i) x(k)|2 ≤ ρ2r(i) .

Obviously, with the condition (31), (Ksx,Gr x) ∈ Ψ ensuresthe satisfaction of the sector condition (27). So, when both theconditions (29), and (30) and (31) are satisfied, the closed-loopsystem (25) can be guaranteed to be stable with H∞ control per-formance. Moreover, less γ in (41) implies better disturbanceattenuation performance. Hence, with the decision variablesX,Y,W, Sr , Sa ,Als ,Bls , Cs ,Ds , Q

als , Q

rs , and η by solving

(28)–(31), the controller gains Acls , B

cs , E

cls , F

cs , Cc

s , and Dcs in

the piecewise fuzzy anti-windup compensator (17) can be ob-tained as in (32), and γ = η1/2 . The proof is thus complete.

Remark 1: As a special case, choosing Gr = Ks,Ga = L,Ksx − Gr x = 0 < ρr , and Lx − Gax = 0 < ρa implies that(Ksx,Gr x) and (Lx, Ga x) always belong to Ψ; thus, the sectorconditions (26) and (27) are verified for any x ∈ n+n . So thecondition (30) and (31) in Theorem 1 can be dropped, and therelated terms Z1

r , Z2r , Z1

a , and Z2a in the condition (29) become

Cs ,DsC, L0X, and L0 , respectively.Remark 2: The piecewise quadratic Lyapunov function (PLF)

has been considered for dynamic output feedback compensatordesign in Theorem 1. Alternatively, if a common Lyapunov func-tion (CLF) is used in the design of fuzzy anti-windup dynamiccompensator like that for linear systems in [16], Ps and Pt inthe conditions (29)–(31) would be replaced by one positive def-inite matrix X , and Vs and Vt by one positive definite block Y ,and all the matrices Us, Ut, and W by In+m ; moreover, the de-cision variables Ps, Us, and Vs would also be eliminated in thetheorem. In fact, the CLF is just one special case of the piece-wise Lyapunov function [27], [28]; thus, PLF provides muchmore freedom of controller design than CLF. And the resultingPLF-based anti-windup controller in Theorem 1 is expected tobe less conservative.

IV. CASE STUDY: ROBUST DYNAMIC OUTPUT FEEDBACK

CONTROL OF CSTR WITH INPUT SATURATIONS

In this section, we consider the case study of a highly nonlin-ear CSTR [2] to evaluate the validity of the proposed piecewisefuzzy H∞ dynamic output feedback control approach.

A. Dynamics of Chemical CSTR Plant

With constant liquid volume, the CSTR for an exothermicirreversible reaction A → B is described by the following dy-namic model based on a component balance for reactant A andon an energy balance

CA =F

V(CAf − CA ) − k0 exp

(− E

RT

)CA

T =F

V(Tf − T ) +

(−∆H)ρCp

k0 exp(− E

RT

)CA

+UA

V ρCp(Tc − T )

y = T, u = Tc (46)

where CA is the concentration of A in the reactor, T is thereactor temperature, and Tc is the temperature of the coolantstream. The objective is to control T by manipulating Tc .Table I contains nominal operating conditions, which corre-spond to an unstable steady state. The resulting steady-stateTc–T curve of CSTR is given in Fig. 1. The open-loop responsein Fig. 2 demonstrates that the reactor exhibits highly nonlinearbehavior in this operating regime. By defining the state vec-tor as x = [CA, T ]T , the manipulated input as u = Tc , and theexternal disturbance as the temperature deviation of feed reac-tant A, w = Tf − Tf s , (46) can be expressed in the form of



TABLE INOMINAL OPERATING CONDITIONS FOR THE CSTR

Fig. 1. Steady-state input–output curve of CSTR.

Fig. 2. Open-loop response for +5 K (a) and −5 K (b) step changes in Tc ofCSTR.

nonlinear state–space model x = f(x) + g(x)u + h(x)w with

f(x) =

[ FV (CAf − x1) − k0 exp

(− E

Rx2

)x1

FV (Tf s − x2) + (−∆H )

ρCpk0 exp

(− E

Rx2

)x1 − U A

V ρCpx2

]

g(x) =

[0

U AV ρCp

], h(x) =

[0FV

].

Fig. 3. Membership functions of fuzzy model and region partition for CSTR.

As shown in Fig. 1, the CSTR (46) has the following threeequilibrium states at steady input u = 300 with w = 0

x1s =[0.875, 324.4]T ,x2

s =[0.5, 350]T ,x3s = [0.2, 370.65]T.

B. H∞ Output Feedback Control of Constrained CSTR viaPiecewise Fuzzy Anti-Windup Dynamic Compensator

Using the fuzzy modeling method described in [22], andchoosing the membership functions for the premise variablex2 or T as

“low”: M 1 =

1, if x2 ≤ 324.41 − (x2 − 324.4)/(350 − 324.4),

if 324.4 < x2 < 3500, if x2 ≥ 350

“middle”: M 2 =

1 − M 1 , if x2 ≤ 3501 − M 3 , if x2 ≥ 350

“high”: M 3 =

0, if x2 ≤ 350(x2 − 350)/(370.65 − 350),

if 350 < x2 < 370.651, if x2 ≥ 370.65

which are shown in Fig. 3, we can easily obtain the followinglocal models of fuzzy system (1) for the nonlinear CSTR (46)under the sampling time Ts = 0.05 min

A1 =[

0.9447 0.00011.3442 0.9043

], B1 =

[0.000010.09951

]A2 =

[0.9055 0.00019.4439 0.9014

], B2 =

[0.000010.09934

]A3 =

[0.7801 0.000135.2608 0.9006

], B3 =

[0.000010.09927

]C1 = C2 = C3 = C = [ 0 1 ]

D1 =[

0.000000.04757

], D2 =

[0.000000.04749

], D3 =

[0.000000.04745

].

The satisfactory approximation ability of fuzzy model is demon-strated as shown in Fig. 4. For the sake of simplicity, we assumethat

M1l = 0.004 · In , N

1l = 0.01 ·

[1 11 1

]N

2l = 0.001 ·

[01

], N

3l = 0.001 ·

[01

], l = 1, 2, 3



Fig. 4. Validation of fuzzy model for CSTR plant (solid: original CSTR;dashed: fuzzy model).

and ε = 0.01 for all the uncertain terms of the fuzzy model. Also,suppose that there exists a finite energy disturbance w(t) =Tf − Tf s = 0.1Tf s exp(−1.2(t)) sin(6t) since the fifth minute.

The parameters for controller design are chosen as Qy =0.1 · I1 , Ra = 0.1 · I1 , Rr = 0.1 · I1 , and the input constraintsare umax = 340, umin = 260,∆umax = 5, that is, the ampli-

tude bound ρa = 40 and the rate bound ρr = 5. The space ofoutput y = x2 is partitioned into three regions, S1 = x2 |x2 <324.4,S2 = x2 |324.4 < x2 < 370.65, and |S3 = x2 |x2 >370.65, which are shown in Fig. 3. Let Ls denote the set offiring fuzzy rules in region s for the piecewise fuzzy controller(17), then one has L1 = 1,L2 = 1, 2, 3, and L3 = 3. Bysolving the problem (28)–(31) in Theorem 1, we can obtain thepiecewise fuzzy anti-windup dynamic output compensator withγ = 0.0669 and the controller gains in (47), as shown at thebottom of this page. The corresponding characteristic matricesfor piecewise Lyapunov functions in controller design are alsogiven in (48), as shown at the bottom of the next page.

In order to compare the performance of the proposed piece-wise fuzzy anti-windup dynamic controller, other two controlapproaches are also applied to the CSTR plant under the samedesign parameters Qy ,Ra , and Rr :

1) CLF-based fuzzy anti-windup dynamic controller de-scribed in Remark 2; and

2) fuzzy dynamic output feedback controller without anti-windup compensator.

When the desired output reference of the closed-loop systemis 350 K, the responses of the closed-loop control system basedon three approaches are shown in Figs. 5–7, respectively. In com-parison with these results, it is easily observed that the proposedpiecewise fuzzy approach can achieve satisfactory output feed-back control performance with closed-loop stability ensured,

Ac11 =

0.25308 −1156.3 4.8748 × 107

1.8069 × 10−7 0.75259 2381.12.5375 × 10−12 8.7628 × 10−5 0.34201

, Bc11 =

2.3728 × 108 −5.8733 × 107

−115.67 22.431−2.5285 −0.45264

,

Ec11 =

0.014411.6198−0.3081

, Cc1 = [ 0.000 0.000 −0.1149 ] , Dc

1 = [−0.5592 −0.7235 ] , F c1 =

−4.2433 × 108

114.91−0.51984

;

Ac12 =

0.09095 −44613 −3.3247 × 105

2.2969 × 10−7 0.76352 −389.841.8868 × 10−11 3.3803 × 10−5 −0.04833

, Bc2 =

1.0035 × 109 4.0061 × 107

−308.33 −0.54291−1.5795 −0.28583

,

Ec12 =

0.009412.1908−0.3062

, Ac22 =

0.09095 −44613 −3.3247 × 105

2.2486 × 10−7 0.73194 −373.613.4676 × 10−11 7.8613 × 10−5 −0.06547

, Ec22 =

0.009410.0010−0.3066

,

Ac32 =

0.09095 −44613 −3.3247 × 105

2.0266 × 10−7 0.63089 −303.851.8868 × 10−11 0.00021814 −0.14878

, Ec32 =

0.00925.8839−0.3031

,

Cc2 = 10−3 [ 0.0000 0.1051 0.7835 ] , Dc

2 = [−2.3648 −1.0396 ] , F c2 =

−4.2433 × 108

115.42−0.51896

;

Ac33 =

0.24685 −20281 −2.2063 × 107

1.3964 × 10−7 0.62458 124457.8232 × 10−11 0.00025 5.445

, Bc3 =

2.3343 × 108 −5.9776 × 107

−109.69 19.709−2.5254 −0.45062

,

Ec3 =

0.01458.7016−0.3043

, Cc3 = [ 0.0000 0.0000 0.0520 ] , Dc

3 = [−0.5501 −0.7211 ] , F c3 =

−4.2433 × 108

110.52−0.5169

. (47)



Fig. 5. Piecewise fuzzy anti-windup dynamic output feedback control ofCSTR (solid: closed-loop response; dashed: output reference).

Fig. 6. Fuzzy anti-windup dynamic output feedback control of CSTR (solid:closed-loop response; dashed: output reference).

Fig. 7. Fuzzy dynamic output feedback control of CSTR without anti-windup(solid: closed-loop response; dashed: output reference).

and the performance shown in Fig. 5 is much better than thosein Figs. 6 and 7 based on the other two approaches.

V. CONCLUSION

This paper has extended the sector condition for constraineddiscrete-time linear systems [15] to the anti-windup controllerdesign of nonlinear processes with amplitude and rate actua-tor saturation. Based on uncertain fuzzy models, the proposedpiecewise fuzzy anti-windup dynamic output feedback compen-sator was formulated as a convex optimization problem, whichcan be solved easily by efficient semidefinite programmingtools. The resulting closed-loop system is guaranteed to be sta-ble with robust H∞ control performance. The simulation resultson a nonlinear CSTR plant have demonstrated the advantagesof this novel dynamic controller. Time delay is commonly metin practical process control, so one interesting future research

P1 =

−0.0027 −0.0380 −0.0076−0.0380 1.2861 −0.9181−0.0076 −0.9181 6.2321

, U1 =

1.4694 −0.1142 1.7611−0.0136 1.0218 −0.0128−0.0029 0.0044 1.0017

,

P2 =

−0.0025 −0.0383 −0.0084−0.0383 1.2866 −0.9179−0.0084 −0.9179 6.2751

, U2 =

0.8519 0.0409 1.74170.0004 0.9999 −0.0157−0.0000 0.0000 1.0016

,

P3 =

−0.0024 −0.0328 −0.0069−0.0328 1.0072 −0.9488−0.0069 −0.9488 6.2076

, U3 =

3.7733 −1.1131 1.42590.0135 0.4275 −0.08180.0025 −0.1156 0.9838

,

V1 = 108 ·

0.0065 −0.0000 0.0000−0.0000 0.0000 0.00000.0000 0.0000 3.2817

, V2 = 108 ·

0.0065 −0.0000 0.0000−0.0000 0.0000 0.00000.0000 0.0000 3.5348

,

V3 = 108 ·

0.0065 −0.0000 0.0000−0.0000 0.0000 0.00000.0000 0.0000 3.2817

. (48)



topic is to investigate novel robust piecewise fuzzy anti-windupcontroller for constrained nonlinear processes with time delay.Another appealing research topic is the study of sampled-dataanti-windup dynamic output feedback control systems when theeffect of the sampling period on digital control performance isconsidered.

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Tiejun Zhang (S’04–M’08) received his B.Eng. andM.Sc. degrees in thermal power engineering from theSoutheast University, Nanjing, China, in 2001 and2004, respectively, and his Ph.D. degree in systemsand control from the City University of Hong Kong,Kowloon, Hong Kong, in 2008.

He is currently a Postdoctoral Research Associatein the Rensselaer Polytechnic Institute, Troy, NY. Hiscurrent research interests focus on modeling, con-trol, and optimization of advanced energy and powersystems.

Gang Feng (S’90–M’92–SM’95–F’09) received theB.Eng. and M.Eng. degrees in automatic control (ofelectrical engineering) from Nanjing Aeronautical In-stitute, Nanjing, China, in 1982 and 1984, respec-tively, and the Ph.D. degree in electrical engineeringfrom the University of Melbourne, Melbourne, Vic.,Australia, in 1992.

During 1997–1998, he was a Lecturer/Senior Lec-turer with the School of Electrical Engineering, Uni-versity of New South Wales, Australia. He was a Vis-iting Fellow at the National University of Singapore

(1997) and Aachen Technology University, Germany (1997–1998). Since 2000,he has been with the City University of Hong Kong, Kowloon, Hong Kong,where he is currently a Professor. His current research interests include robustadaptive control, signal processing, piecewise linear systems, and intelligentsystems and control.

Prof. Feng is an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC

CONTROL and the IEEE TRANSACTIONS ON FUZZY SYSTEMS, and was an Asso-ciate Editor of the IEEE TRANSACTIONS ON SYSTEMS, MAN, & CYBERNETICS,PART C. He is also an Associate Editor of the Journal of Control Theory andApplications. He has also been a member of the Conference Editorial Board ofthe IEEE Control System Society. He was the recipient of the Alexander vonHumboldt Fellowship in 1997–1998, and the 2005 IEEE TRANSACTIONS ON

FUZZY SYSTEMS Outstanding Paper Award in 2007.



Huaping Liu (M’07) was born in Sichuan Province,China, in 1976. He received the Ph.D. degree fromthe Department of Computer Science and Technol-ogy, Tsinghua University, Beijing, China, in 2004.

From 2004 to 2006, he was a Postdoctoral Fellowin the Department of Automation, Tsinghua Univer-sity, where he is currently an Associate Professor inthe Department of Computer Science and Technol-ogy. His current research interests include intelligentcontrol and robotics.

Jianhong Lu received the B.S. degree in power en-gineering and the Ph.D. degree in process controlengineering from the Southeast University, Nanjing,China, in 1985 and 1990, respectively.

He is currently a Professor in the Departmentof Energy Information and Automation, School ofEnergy and Environment, Southeast University. Hiscurrent research interests include advanced controltheory and technology for complex thermal systems.



H∞ Fuzzy Control of Nonlinear Systems UnderUnreliable Communication LinksHuijun Gao, Member, IEEE, Yan Zhao, and Tongwen Chen, Fellow, IEEE

Abstract—This paper investigates the problem of H∞ fuzzy con-trol of nonlinear systems under unreliable communication links.The nonlinear plant is represented by a Takagi–Sugeno (T–S) fuzzymodel, and the control strategy takes the form of parallel dis-tributed compensation. The communication links existing betweenthe plant and controller are assumed to be imperfect (that is, datapacket dropouts occur intermittently, which appear typically ina network environment), and stochastic variables satisfying theBernoulli random binary distribution are utilized to model the un-reliable communication links. Attention is focused on the designof H∞ controllers such that the closed-loop system is stochasti-cally stable and preserves a guaranteed H∞ performance. Twoapproaches are developed to solve this problem, based on thequadratic Lyapunov function and the basis-dependent Lyapunovfunction, respectively. Several examples are provided to illustratethe usefulness and applicability of the developed theoretical results.

Index Terms—Basis-dependent Lyapunov function, H∞ control,nonlinear systems, Takagi–Sugeno (T–S) fuzzy systems, unreliablecommunication links.

I. INTRODUCTION

IN THE REAL WORLD, most physical systems and pro-cesses are not linear, and the inherent nonlinearity makes

their control and regulation a very hard task. This facilitates arapidly growing interest in fuzzy control of nonlinear systems.The main reason is that Takagi–Sugeno (T–S) fuzzy models canbe used to describe nonlinear systems effectively as T–S fuzzysystems with affine terms can approximate any smooth nonlin-ear function to any specified accuracy within any compact set,and the control design can be carried out on the basis of the fuzzymodel. The main feature of T–S fuzzy models is to express thelocal dynamics of each fuzzy rule by a linear system model,and the overall fuzzy model of the system is achieved by fuzzy“blending” of the linear system models. For this local linearityof the T–S fuzzy model, the stability and synthesis analysis canbe simplified based on the well-known framework of Lyapunovfunction method [13]. In recent years, many researchers have

Manuscript received April 10, 2007; revised August 7, 2007. First pub-lished April 30, 2008; current version published April 1, 2009. This work wassupported in part by the Natural Science and Engineering Research Coun-cil of Canada, in part by the National Natural Science Foundation of China(60825303, 60834003), in part by the 973 Project (2009CB320600), in part bythe Research Fund for the Doctoral Programme of Higher Education of China(20070213084), in part by the Heilongjiang Outstanding Youth Science Fund(JC200809), in part by Postdoctoral Science Foundation of China (200801282),and in part by the Fok Ying Tung Education Foundation (111064).

H. Gao and Y. Zhao are with the Space Control and Inertial Technology Re-search Center, Harbin Institute of Technology, Harbin 150001, China (e-mail:[email protected]).

T. Chen is with the Department of Electrical and Computer Engineer-ing, University of Alberta, Edmonton, AB T6G 2R3, Canada (e-mail:[email protected]).


Fig. 1. NCS prototype.

shown their interest in T–S fuzzy models, and a great number ofresults have been reported in the literature. To mention a few, theproblem of stability analysis is investigated in [4], [25] and [38],stabilizing and H∞ control designs are reported in [5], [8], [26],and [50], state estimation is addressed in [1], and reliable con-trol strategies are presented in [44] and [46]. These results areconcerned with various engineering problems, as well as manyclasses of T–S fuzzy systems, including T–S fuzzy systems withparameter uncertainties [26], T–S fuzzy systems with state de-lays [7], [28], [47], T–S fuzzy systems with actuator satura-tion [6], T–S fuzzy systems with singular perturbations [2],[29], [30], T–S sampled-data fuzzy systems [23], and so on.

Most of the aforementioned results concerning T–S fuzzysystems are based on an implicit assumption that the commu-nications between the physical plant and controller are perfect.In many practical situations, however, there may be a nonzeroprobability that some measurements or control inputs will belost during their transmission. This is more obvious and impor-tant for the emerging networked control systems (NCSs) wherefeedback control loops are closed via digital communicationchannels. In a prototype of NCS shown in Fig. 1, several com-ponents communicate over a shared network, and informationflows among the sensor, actuator, and controller according tosome rules. NCSs are becoming increasingly popular becausethey have several advantages over traditional systems, such aslow cost, reduced weight and power requirements, simple instal-lation and maintenance, and high reliability [22]. However, theinsertion of the communication network in the feedback loopsmakes the analysis and synthesis problems much more com-plex. In particular, among a few others, data packet dropout isan important issue to be taken into consideration for the analysisand synthesis of communication-based control systems.

In view of the importance of unreliable communicationsin modern control systems, many researchers have begun to

1063-6706/$25.00 © 2009 IEEE



study how to design control systems with the simultaneousconsideration of communication issues. Concerning the issueof data packet dropout (or data missing), a number of importantresults have been reported. For control problems, readers arereferred to [33], [42], and [49]. In [48], the stability of NCSswith finite data packet dropout is analyzed; in [33], a packet lossmodel for the network is proposed based on which the results fordiscrete-time linear systems with Markovian jumping parame-ters can be applied; in [3], the uncertainty threshold principle isutilized for stability analysis; and in [21], stochastic variablesare utilized to model the data missing phenomenon. The issueof missing measurements has also been investigated for sig-nal processing problems (see, for instance, [32] and [41]). It isworth mentioning that most of these results are concerning lin-ear control systems, and to the best of the authors’ knowledge,up to now, no effort has been made about how to analyze andsynthesize T–S fuzzy systems with unreliable communicationlinks, and control of T–S fuzzy systems in the presence of packetdropouts still remains open and unsolved.

In this paper, we make an attempt to investigate the prob-lem of H∞ fuzzy control of nonlinear systems under unreliablecommunication links. More specifically, the nonlinear plant isrepresented by a T–S fuzzy model, and the control strategytakes the form of parallel distributed compensation. The com-munication links, existing between the plant and controller, areassumed to be imperfect (that is, data packet dropouts occurintermittently), and stochastic variables satisfying the Bernoullirandom binary distribution are utilized to model the unreliablecommunication links. Attention is focused on the design ofstate feedback H∞ controllers such that the closed-loop sys-tem is stochastically stable and preserves a guaranteed H∞performance. Two approaches are developed to solve this prob-lem, based on quadratic Lyapunov function and basis-dependentLyapunov function, respectively. These two approaches providealternatives for designing H∞ controllers with different degreeof conservativeness and computational complexity, and all theconditions are formulated in the framework of linear matrix in-equalities (LMIs). Several examples substantially illustrate theusefulness and applicability of the developed theoretical results.

The remainder of the paper is organized as follows. Section IIformulates the problem under consideration. H∞ analysis andsynthesis results are presented in Section III based on a quadraticapproach, and parallel results are given in Section IV througha basis-dependent approach. Some extensions and illustrativeexamples are given in Sections V and VI, respectively. Finally,concluding remarks are given in Section VII.

Notations: The notation used throughout the paper is fairlystandard. The superscript “T ” stands for matrix transposition;R

n denotes the n-dimensional Euclidean space, and the notationP > 0 (≥0) means that P is real symmetric and positive def-inite (semi-definite). l2 [0,∞) is the space of square-integrablevector functions over [0,∞); the notation | · | refers to theEuclidean vector norm, and ‖ · ‖2 stands for the usual l2 [0,∞)norm. In symmetric block matrices or complex matrix expres-sions, we use an asterisk (∗) to represent a term that is inducedby symmetry, and diag. . . stands for a block diagonal matrix.In addition, Ex and Ex | y will, respectively, mean ex-

Fig. 2. T–S fuzzy system with unreliable communication links.

pectation of x and expectation of x conditional on y. Matrices,if their dimensions are not explicitly stated, are assumed to becompatible for algebraic operations.

II. PROBLEM FORMULATION

Consider a T–S fuzzy system with unreliable communicationlinks shown in Fig. 2, where the physical plant is represented bya T–S fuzzy model, and the unreliable communications exist inboth the downlink (from physical plant to controller) and uplink(from controller to physical plant). Now, we model the physicalplant, controller, and communication links mathematically.

A. Physical Plant

Many nonlinear systems can be expressed as a set of linearsystems in local operating regions. According to fuzzy modelingin [34], a nonlinear discrete-time system can be represented bythe T–S fuzzy model as follows.

Plant Rule i: IF θ1(k) is Mi1 and θ2(k) is Mi2 and · · · andθp(k) is Mip , THEN

xk+1 = Aixk + B2iuk + B1iwk

zk = Cixk + D2iuk + D1iwk (1)

i = 1, . . . , r.

Here, Mij are the fuzzy sets; xk ∈ Rn is the state vector;

uk ∈ Rm is the input vector; wk ∈ R

p is the exogenous distur-bance input which belongs to l2 [0,∞); zk ∈ R

q is the controlledoutput; Ai , B2i , B1i , Ci , D2i , and D1i are known constant ma-trices with appropriate dimensions; r is the number of IF–THENrules; and θk = [ θ1(k), θ2(k), . . . , θp(k) ] is the premise vari-able vector. Throughout the paper, it is assumed that the premisevariables do not depend on the input variable uk explicitly. Givena pair of (xk , uk ), the final output of the fuzzy system is inferredas [34]

xk+1 =r∑

i=1

hi(θk ) [Aixk + B2iuk + B1iwk ]

zk =r∑

i=1

hi(θk ) [Cixk + D2iuk + D1iwk ] (2)

where

hi(θ(k)) =ωi(θ(k))∑ri=1 ωi(θ(k))

ωi(θ(k)) =p∏

j=1

Mij (θj (k))


GAO et al.: H∞ FUZZY CONTROL OF NONLINEAR SYSTEMS UNDER UNRELIABLE COMMUNICATION LINKS 267

with Mij (θj (k)) representing the grade of membership of θj (k)in Mij . Then, it can be seen that

ωi(θk ) ≥ 0, i = 1, 2, . . . , r,

r∑i=1

ωi(θk ) > 0

for all k. In what follows, we will drop the argument of hi(θk )for brevity. Therefore, for all k, we have

hi ≥ 0, i = 1, 2, . . . , r,

r∑i=1

hi = 1.

B. Controller

In this paper, we consider the state feedback control problem,and the control strategy is the parallel distributed compensation,which is described as follows.

Controller Rule i: IF θ1(k) is Mi1 and θ2(k) is Mi2 and · · ·and θp(k) is Mip , THEN

uck = Kixck .

Here, xck ∈ Rn is the input of the controller; uck ∈ R

m is theoutput of the controller; and Ki is the gain matrix of the statefeedback controller. Thus, the controller can be represented bythe following input–output form:

uck =r∑

i=1

hiKixck . (3)

C. Communication Links

Due to the existence of the communication links and the dataloss between the physical plant and controller, the measurementof the plant is no longer equivalent to the input of the con-troller (that is, xk = xck ), and the output of the controller is nolonger equivalent to the input of the plant (that is, uk = uck ). Inthis paper, we model the data loss phenomena via a stochasticapproach. That is, we model them as

uck = α(k)xk , uk = β(k)uck

where α(k) and β(k) are two independent Bernoulli pro-cesses. The process α(k) models the unreliable nature of thelink from the sensor to the controller, and β(k) models that fromthe controller to the actuator. More specifically, α(k) = 0 whenthe link fails (that is, data are lost), and α(k) = 1 means success-ful transmission [the same goes to β(k)]. A natural assumptionon α(k) and β(k) can be made as

Prob α(k) = 1 = E α(k) = α

Prob α(k) = 0 = 1 − α

Prob β(k) = 1 = E β(k) = β

Prob β(k) = 0 = 1 − β.

Based on this, we have

uk =r∑

i=1

hiβ(k)α(k)Kixk . (4)

D. Closed-Loop System

We introduce another Bernoulli process e(k) with

e(k)=α(k)β(k). Thus, we have that e(k) = 1 when both

α(k) = 1 and β(k) = 1, and e(k) = 0 otherwise. Then, we have

Prob e(k) = 1 = E e(k)= e, Prob e(k)=0=1 − e.

Then, from (2) and (4), the closed-loop system is given by

xk+1 =r∑

i=1

r∑j=1

hihj [Aij xk + B1iwk ]

zk =r∑

i=1

r∑j=1

hihj [Cij xk + D1iwk ] (5)

where Aij = Ai + eB2iKj + e(k)B2iKj , Cij =Ci + eD2iKj

+ e(k)D2iKj , and e(k) = e(k) − e. It is clear that Ee(k) =0 and Ee(k)e(k) = e(1 − e).

Before proceeding further, we first introduce the followingdefinition.

Definition 1: The closed-loop system (5) is said to be stochas-tically stable in the mean square if, when w(t) ≡ 0 for any initialcondition x0 , there exists a finite W > 0 such that

E

∞∑k=0

|xk |2∣∣∣∣∣x0

< xT

0 Wx0 .

Then, the problem to be addressed in this paper is expressedas follows.

Problem H∞ fuzzy control under unreliable links (HFCULs):Consider the problem shown in Fig. 2, and suppose the commu-nication link parameter e is given. Given a scalar γ > 0, designa state feedback controller in the form of (3) such that

1) (stochastic stability) the closed-loop system in (5) isstochastically stable in the sense of Definition 1;

2) (H∞ performance) under zero initial conditions, the con-trolled output zk satisfies

‖z‖E ≤ γ‖w‖2 (6)

where ‖z‖E= E

√∑∞k=0 |zk |2. If the previous two

conditions are satisfied, the closed-loop system is calledstochastically stable with a guaranteed H∞ performanceγ.

Remark 1: The stochastic variable e(k) characterizes the pos-sibility of the link from the sensor to controller and then toactuator being successful. It is easy to understand that the largerthe value of e, the higher the chance of successful transmission.This model to characterize data missing phenomenon was firstintroduced in [31].

III. QUADRATIC APPROACH

In this section, Problem HFCUL formulated earlier will besolved via a quadratic approach.



A. H∞ Performance Analysis

In this section, a condition guaranteeing the stochastic sta-bility and H∞ performance is established for the closed-loopsystem in (5).

Theorem 1: Consider the fuzzy system in (2) and suppose thegain matrices Ki , i = 1, . . . , r, of the subsystem controllers in(3) are given. The closed-loop fuzzy system in (5) is stochasti-cally stable with a guaranteed H∞ performance γ, if there existsa matrix P > 0 satisfying

ΠTii PΠii + ΛT

iiΛii − L < 0, i = 1, . . . , r (7)

(Πij + Πj i)T P (Πij + Πj i)

+ (Λij + Λj i)T (Λij + Λj i) − 4L < 0

1 ≤ i < j ≤ r (8)

where

Πij =[

Ai + eB2iKj B1i√e(1 − e)B2iKj 0

], L =

[P 00 γ2I

]Λij =

[Ci + eD2iKj D1i√e(1 − e)D2iKj 0

], P =

[P 00 P

].

(9)

Proof: We first prove the stochastic stability of the closed-loopsystem. To this end, assume wk ≡ 0, and choose a Lyapunovfunction as

V (xk ) = xTk Pxk (10)

where P > 0 is a matrix to be determined. Then, along theclosed-loop fuzzy system in (5), we have

∆Vk = EVk+1 |xk − Vk

= E

r∑

i=1

r∑j=1

hihj Aij xk

T

×P

r∑i=1

r∑j=1

hihj Aij xk

∣∣∣∣∣∣xk

− xTk Pxk

= E

xT

k

r∑i=1

r∑j=1

r∑s=1

r∑t=1

hihjhsht

(Aij + Aji

2

)T

×P

(Ast + Ats

2

)xk

∣∣∣∣xk

− xT

k Pxk

=14xT

k

r∑

i=1

r∑j=1

r∑s=1

r∑t=1

hihjhsht (Ai + Aj )T

×P (As + At) + (Ai + Aj )T P e (B2sKt + B2tKs)

+ e (B2iKj + B2jKi)T P (As + At)

+ e(B2iKj+B2jKi)T P (B2sKt+B2tKs)

xk−xT

k Pxk

≤ 14xT

k

r∑

i=1

r∑j=1

hihj (Ai + Aj )T P (Ai + Aj )

+ 2 (Ai + Aj )T P e (B2iKj + B2jKi)

+ e(B2iKj +B2jKi)T P (B2iKj +B2jKi)

xk

− xTk Pxk

= xTk

r∑i=1

h2i

(ΓT

ii PΓii − P)xk + xT

k

r−1∑i=1

r∑j=i+1

hihj

×[12

(Γij + Γj i)T P (Γij + Γj i) − 2P

]xk

where

Γij =[

Ai + eB2iKj√e(1 − e)B2iKj

]. (11)

By the Schur complement, from inequalities (7) and (8), weknow that

ΓTii PΓii − P < 0, i = 1, . . . , r (12)

12

(Γij + Γj i)T P (Γij + Γj i) − 2P < 0

1 ≤ i < j ≤ r. (13)

Define Φij = (Γij + Γj i)T P (Γij + Γj i) − 4P ; thus

Φ =14

r∑i=1

r∑j=1

hihj [(Γij + Γj i)T P (Γij + Γj i) − 4P ]

=r∑

i=1

h2i

(ΓT

ii PΓii − P)xk +

r−1∑i=1

r∑j=i+1

hihj (14)

×[12

(Γij + Γj i)T P (Γij + Γj i) − 2P

](15)

and we get

E

xTk+1Pxk+1

∣∣xk

− xT

k Pxk ≤ −λmin(−Φ)xTk xk .

Taking mathematical expectation of both sides and for any d ≥1, summing up the inequality on both sides from k = 0, . . . , d,we have

ExT

d+1Pxd+1− xT

0 Px0 ≤ −λmin(−Φ)E

d∑

k=0

|xk |2

which implies

E

d∑

k=0

|xk |2≤(λmin(−Φ))−1(xT

0 Px0 − ExT

d+1Pxd+1)

≤ (λmin(−Φ))−1xT0 Px0

where x0 is the initial condition. When T = 1, . . . ,∞, consid-ering ExT

∞Px∞ ≥ 0, we have

E

∞∑k=0

|xk |2∣∣∣∣∣x0

≤ (λmin(−Φ))−1xT

0 Px0

= xT0 (λmin(−Φ))−1Px0 = xT

0 Wx0



where W= (λmin(−Φ))−1P . By substituting (12) and (13)

into (14), we know Φ < 0, and therefore, λmin(−Φ) > 0.Thus, W = (λmin(−Φ))−1P > 0. According to Definition 1,the closed-loop system is stochastically stable in the meansquare.

Next, the H∞ performance criterion for the closed-loop sys-tem will be established. To this end, assume zero initial condi-tions. An index is introduced as

J=E

xT

k+1Pxk+1∣∣ ξk

− xT

k Pxk

+E

zTk zk

∣∣ ξk

−γ2wT

k wk .

Defining ξk = [xTk wT

k ]T , Gij =[ Aij B1i ] , and Hij =[ Cij D1i ] , we have

J = E

xTk+1Pxk+1

∣∣ ξk

+ E

zTk zk

∣∣ ξk

− ξT

k Lξk

≤ E

r∑

i=1

r∑j=1

hihj

(Gij + Gji

2

)ξk

T

P

×

r∑i=1

r∑j=1

hihj

(Gij + Gji

2

)ξk

∣∣∣∣∣∣ ξk

+ E

r∑

i=1

r∑j=1

hihj

(Hij + Hji

2

)ξk

T

×

r∑i=1

r∑j=1

hihj

(Hij + Hji

2

)ξk

∣∣∣∣∣∣ ξk

− ξTk Lξk

= ξTk

r∑i=1

h2i

(ΠT

ii PΠii + ΛTiiΛii−L

)ξk + ξT

k

r−1∑i=1

r∑j=i+1

hihj

× [(Πij + Πj i)T P (Πij + Πj i)

+ (Λij + Λj i)T (Λij + Λj i) − 4L]ξk .

From inequalities (7) and (8), we know J ≤ 0, that is

E

xTk+1Pxk+1

∣∣ ξk )− xT

k Pxk

+ EzTk zk

∣∣ ξk − γ2wTk wk ≤ 0.

Taking mathematical expectation on both sides, we have

ExTk+1Pxk+1 − ExT

k Pxk + EzTk zk − γ2wT

k wk ≤ 0.

For k = 0, 1, 2, . . . , summing up both sides, under zero initialcondition and ExT

∞Px∞ ≥ 0, we obtain

E

∞∑k=0

zTk zk

−

∞∑k=0

γ2wTk wk ≤ 0.

The proof is completed. Remark 2: Theorem 1 is obtained for e ∈ [0, 1]. When e = 1

(that is, perfect communication links exist between the plant andcontroller), inequalities (7) and (8) reduce to

GTii PGii + MT

ii Mii − L < 0, i = 1, . . . , r

12

(Gij + Gji)T P (Gij + Gji)

+12

(Mij + Mji)T (Mij + Mji) − 2L < 0

1 ≤ i < j ≤ r

where

Gij = [Ai + B2iKj B1i ]

Mij = [Ci + D2iKj D1i ] .

This is the standard H∞ performance criterion for a discrete-time fuzzy closed-loop system reported in [5, Th. 2]. Whene = 0 (that is, all the control signals are lost, and the closed-loop system is nothing but the open-loop system), inequalities(7) and (8) reduce to[

ATi PAi + CT

i Ci − P ATi PB1i + CT

i D1i

∗ BT1iPB1i + DT

1iD1i

]− L < 0

i = 1, . . . , r.

This is the standard H∞ performance criterion for an unforcedsystem reported in [5, Th. 1].

B. Controller Design

In this section, we aim to design a state feedback controllerin the form of (3) such that the closed-loop system in (5) isstochastically stable and the controlled output zk satisfies (6).

Theorem 2: Consider the fuzzy system in (2). There exists astate feedback controller in the form of (4) such that the closed-loop system in (5) is stochastically stable with a guaranteed H∞performance γ, if there exist matrices X > 0 and Mi , i = 1,. . . , r, satisfying L ∆T

ii χTii

∗ X 0

∗ ∗ I

< 0, i = 1, . . . , r (16)

4L ∆Tij + ∆T

ji χTij + χT

ji

∗ X 0

∗ ∗ I

< 0, 1 ≤ i < j ≤ r

(17)

where

∆ij =

[AiX + eB2iMj B1i√e(1 − e)B2iMj 0

]

χij =

[CiX + eD2iMj D1i√e(1 − e)D2iMj 0

]

I =[−I 0

0 −I

], L =

[−X 0

0 −γ2I

]

X =[−X 0

0 −X

]. (18)



Furthermore, if the previous conditions have feasible solutions,the gains Ki of the subsystem controllers in (4) are given by

Ki = MiX−1 .

Proof: According to Theorem 1, if there exists P > 0 satis-fying (7) and (8), then the closed-loop system is stochasticallystable with a guaranteed H∞ performance γ. By the Schur com-plement, the following inequalities are obtained:−L ΠT

ii ΛTii

∗ −P−1 0

∗ ∗ I

< 0, i = 1, . . . , r (19)

−4L ΠTij + ΠT

ji ΛTij + ΛT

ji

∗ −P−1 0

∗ ∗ I

< 0, 1 ≤ i < j ≤ r (20)

where Πij , Λij , P , and L are the same as those in Theorem 1.Performing congruence transformations to inequalities (19) and(20) by diagP−1 , I, I, I, I, I, we have−L ∆T

ii χTii

∗ −P−1 0

∗ ∗ I

< 0, i = 1, . . . , r (21)

−4L ∆Tij + ∆T

ji χTij + χT

ji

∗ −P−1 0

∗ ∗ I

< 0, 1 ≤ i < j ≤ r

(22)

where

∆ij =

[AiP

−1 + eB2iKjP−1 B1i√

e (1 − e)B2iKjP−1 0

]

L =[

P−1 0

0 γ2I

]

χij =

[CiP

−1 + eD2iKjP−1 D1i√

e (1 − e)D2iKjP−1 0

].

Defining X = P−1 , Mi = KiP−1 , we readily obtain (16) and

(17), and the proof is completed.

IV. BASIS-DEPENDENT APPROACH

To reduce the conservativeness in the quadratic framework,in this section, we present another approach based on a basis-dependent Lyapunov function.

A. H∞ Performance Analysis

The following theorem presents a condition guaranteeing theH∞ performance for the closed-loop system in (5).

Theorem 3: Consider the fuzzy system in (2), and suppose thegain matrices Ki, i = 1, . . . , r of the subsystem controllers (3)are given. The closed-loop system in (5) is stochastically stablewith a guaranteed H∞ performance γ, if there exist matrices

Pj > 0, j = 1, . . . , r, satisfying

ΠTii PjΠii + ΛT

iiΛii − Li < 0, i = 1, . . . , r, j = 1, . . . , r

(23)(Πil + Πli

2

)T

Pj (Πil + Πli)

+(

Λil + Λli

2

)T

(Λil + Λli) − Li − Ll < 0

1 ≤ i < l ≤ r, j = 1, . . . , r (24)

where Πij , Λij are given in (9), and

Li =[

Pi 00 γ2I

], Pj =

[Pj 00 Pj

]. (25)

Proof: We first prove the stability of the closed-loop system.To this end, assume wk ≡ 0 and choose a Lyapunov function as

Vk = xTk

[r∑

i=1

hiPi

]xk (26)

where Pi > 0 are to be determined. Define h+j = hj (θk+1).

Then, along the closed-loop system in (5), we have

∆Vk = E Vk+1 |xk − Vk

= E

xTk

r∑j=1

h+j

×[

r∑i=1

r∑l=1

r∑p=1

r∑q=1

hihlhphq ATipPj Alq

]xk

∣∣∣∣∣xk

− xTk

[r∑

i=1

hiPi

]xk

= E

xTk

r∑j=1

h+j

×[

r∑i=1

r∑l=1

r∑p=1

r∑q=1

hihlhphq

(Aip + Api

2

)T

×Pj

(Alq + Aql

2

)]xk

∣∣∣∣xk

− xT

k

[r∑

i=1

hiPi

]xk

=14xT

k

r∑j=1

h+j

r∑

i=1

r∑l=1

r∑p=1

r∑q=1

hihlhphq (Ai + Al)T

× Pj (Ap +Aq )+(Ai + Al)T Pj e (B2pKq + B2qKp)

+ e (B2iKl + B2lKi)T Pj (Ap + Aq )

+e (B2iKl + B2lKi)T Pj (B2pKq + B2qKp)

xk



− xTk

[r∑

i=1

hiPi

]xk

≤ xTk

r∑j=1

h+j

[r∑

i=1

h2i

(ΓT

ii PjΓii − Pi

)+

r−1∑i=1

r∑i<l

×hihl

(Γil + Γli

2

)T

Pj (Γil + Γli) − Pi − Pl

]xk .

From inequalities (23) and (24), we know that

ΓTii PjΓii − Pi < 0, i = 1, . . . , r, j = 1, . . . , r(Γil + Γli

2

)T

Pj (Γil + Γli) − Pi − Pl < 0

1 ≤ i < l ≤ r, j = 1, . . . , r

where Γij is defined in (11). By defining

Φ=

r∑j=1

h+j

[r∑

i=1

h2i

(ΓT

ii PjΓii − Pi

)+

r−1∑i=1

r∑i<l

× hihl

(Γil + Γli

2

)T

Pj (Γil + Γli) − Pi − Pl

]the stochastic stability of the closed-loop system can be provedby following similar lines as the proof of Theorem 1.

Next, the H∞ performance criteria for the closed-loop systemwill be established. To this end, assume zero initial conditions.An index is introduced as

J=E Vk+1 | ξk+E

zTk zk

∣∣ ξk

−γ2wT

k wk − ξTk

r∑i=1

hiPiξk

where ξk is the same as that in the proof of Theorem 1.Since

Vk+1 ≤ ξTk

r∑j=1

r∑i=1

r∑p=1

h+j hihp

×[

ATip

BT1i

]Pj [ Aip B1i ] ξk ,

zTk zk ≤ ξT

k

r∑i=1

r∑p=1

hihp

×[

CTip

DT1i

][ Cip D1i ] ξk ,

we have

J ≤ ξTk

r∑j=1

r∑i=1

r∑p=1

h+j hihp

×[ΠT

ip PjΠip + ΛTipΛip − Li

]ξk

≤ ξTk

r∑j=1

h+j

[r∑

i=1

h2i

(ΠT

ii PjΠii + ΛTiiΛii − Li

)

+r−1∑i=1

r∑l=i+1

hihl

(Πil + Πli

2

)T

Pj (Πil + Πli)

+(

Λil + Λli

2

)T

(Λil + Λli) − Li − Ll

]ξk .

From inequalities (23) and (24), we have J ≤ 0, and thus

E Vk+1 | ξk+EzTk zk

∣∣ ξk

− γ2wT

k wk − ξTk

r∑i=1

hiPiξk ≤ 0.

Taking mathematical expectation on both sides, we obtain

EVk+1 − EVk + EzTk zk − γ2wT

k wk ≤ 0.

For k = 0, 1, 2, . . . , summing up both sides, under zero initialcondition and EV∞ > 0, we obtain

E

∞∑k=0

zTk zk

−

∞∑k=0

γ2wTk wk ≤ 0.

The proof is completed. Remark 3: Theorem 3 is obtained by using the basis-

dependent Lyapunov function defined in (26). It is clear thatwhen Pi = P for any i ∈ 1, . . . , r, (26) becomes the quadraticLyapunov function that is widely used in the literature. WhenPi = P , Theorem 3 reduces to Theorem 1.

B. Controller Design

It is worth mentioning that it is usually difficult to obtainLMI conditions for the controller design in the basis-dependentframework, unless additional overdesign is introduced. In thefollowing, we present a controller design procedure based onthe cone complementarity linearization (CCL) algorithm.

Theorem 4: Consider the fuzzy system in (2). A state feed-back controller in the form of (4) exists, such that the closed-loopsystem in (5) is stochastically stable with a guaranteed H∞ per-formance γ, if there exist matrices Xi > 0, Pi > 0, i = 1, . . . , rsatisfying−Xj 0 Πii

∗ −I Λii

∗ ∗ −Li

< 0, i = 1, . . . , r, j = 1, . . . , r

(27)−2Xj 0 Πil + Πli

∗ −2I Λil + Λli

∗ ∗ −(Li + Ll)

< 0, 1 ≤ i < l ≤ r

j = 1, . . . , r (28)

PiXi = I, i = 1, . . . , r. (29)



where

Xj =[

Xj 00 Xj

]Πij , and Λij are given in (9), and Li is defined in (25).

Proof: According to the Schur complement, (23) and (24) canbe converted to−P−1

j 0 Πii

∗ −I Λii

∗ ∗ −Li

< 0, i = 1, . . . , r, j = 1, . . . , r

−2P−1j 0 Πil + Πli

∗ −2I Λil + Λli

∗ ∗ − (Li + Ll)

< 0, 1 ≤ i < l ≤ r

j = 1, . . . , r.

By defining

Xj =[

Xj 0

0 Xj

]=

[P−1

j 0

0 P−1j

](27) and (28) can be obtained.

It should be noted that the condition in Theorem 4 is not aconvex set due to the existence of the matrix inequalities in (29).Several approaches have been proposed to solve such nonconvexfeasibility problems, among which the CCL method [10] is themost commonly used one (for instance, the CCL algorithm hasbeen used for solving controller design problems [15]). Thebasic idea in the CCL algorithm is that if the LMI [ P I

I L ] ≥ 0is feasible in the n × n matrix variables L > 0 and P > 0, thentr(PL) ≥ n, and tr(PL) = n if and only if PL = I .

Now, using a cone complementarity approach [10], we sug-gest the following nonlinear minimization problem involvingLMI conditions instead of the original nonconvex feasibilityproblem formulated in Theorem 4:

min tr

(r∑

i=1

PiXi

)subject to (27), (28), and

[Pi II Xi

]≥ 0, i = 1, . . . , r. (30)

According to [10], if the solution of the previous minimizationproblem is rn, that is, min tr(

∑ri=1 PiXi) = rn, then the con-

ditions in Theorem 4 are solvable. The following algorithm issuggested to solve the previous problem.

Algorithm 1Step 1: Find a feasible set (P (0)

i , X(0)i ,K

(0)i ) satisfying (27),

(28), and (30). Set t = 0.Step 2: Solve the following LMI problem:

min tr

(r∑

i=1

(PiX

(t)i + P

(t)i Xi

))subject to (27), (28), and (30).

Step 3: Substitute the obtained matrix variables (Pi,Xi,Ki)into (23) and (24). If conditions (23) and (24) are sat-

isfied, with |tr (∑r

i=1 PiXi) − rn| < δ, for some suf-ficiently small scalar δ > 0, then output the feasiblesolutions (Pi,Xi,Ki) . EXIT.

Step 4: If K > N where N is the maximum number of itera-tions allowed, EXIT.

Step 5: Set t = t + 1,(P

(t)i , X

(t)i ,K

(t)i

)= (Pi,Xi,Ki) , and

go to Step 2.

V. EXTENSIONS AND DISCUSSIONS

Parameter uncertainties often exist in practical applications[16], [17], [43]. In this section, the results obtained previouslyare extended to fuzzy systems with time-varying uncertainties.The robust H∞ performance criterion of fuzzy systems with un-certainties is presented and the robust H∞ controller is designed.All these are considered based on the quadratic approach, and theresults based on the basis-dependent approach can be obtainedby similar procedures. Moreover, the advantages and drawbacksof the quadratic approach and the basis-dependent approach arediscussed.

A. Extensions

In this section, the robust H∞ control of fuzzy system withuncertainties is considered. Under control law (3), the closed-loop fuzzy system with parameter uncertainties becomes

xk+1 =r∑

i=1

r∑j=1

hihj [Aij xk + B1iwk ]

zk =r∑

i=1

r∑j=1

hihj [Cij xk + D1iwk ] (31)

where

Aij = (Ai + ∆Ai) + e(B2i + ∆B2i)Kj

+ e(k)(B2i + ∆B2i)Kj , B1i = B1i + ∆B1i

Cij = (Ci + ∆Ci) + e(D2i + ∆D2i)Kj

+ e(k)(D2i + ∆D2i)Kj , D1i = D1i + ∆D1i .

Here, Ai , B2i , B1i , Ci , D2i , and D1i are the same as those in (2),and ∆Ai , ∆B2i , ∆B1i , ∆Ci , ∆D2i , and ∆D1i are real-valuedmatrices with appropriate dimensions satisfying [34][

∆Ai ∆B2i ∆B1i

∆Ci ∆D2i ∆D1i

]=[

E1i

E2i

]Fi(k) [ H1i H2i H3i ]

where E1i , E2i , H1i , H2i , and H3i are known constant matri-ces with compatible dimensions; Fi(k) are unknown nonlineartime-varying matrix functions satisfying FT

i (k)Fi(k) ≤ I . It isassumed that the elements of Fi(k) are Lebesgue measurable.

The following theorems give the results of robust H∞ per-formance analysis and controller design for fuzzy systems withparameter uncertainties, which can be proved by following sim-ilar lines in [14].

1) Robust H∞ Performance Analysis:Theorem 5: Suppose the gain matrices Ki , i = 1, . . . , r of the

subsystem controllers in (3) are given. The closed-loop fuzzysystem in (31) is robustly stochastically stable with a guaranteed



H∞ performance γ, if there exist a matrix P > 0, scalars εi , εij ,satisfying

−P 0 P Πii P E1 i 0

∗ −I Λii E2 i 0

∗ ∗ −L 0 εiTTii

∗ ∗ ∗ εi I 0

∗ ∗ ∗ ∗ εi I

< 0, i = 1, . . . , r (32)

−P 0 P(Πij + Πj i

)P E1 i P E1j 0 0

∗ −I Λij + Λj i E2 i E2j 0 0

∗ ∗ −4L 0 0 εij T Tij εj i T

Tj i

∗ ∗ ∗ εij I 0 0 0

∗ ∗ ∗ ∗ εj i I 0 0

∗ ∗ ∗ ∗ ∗ εij I 0

∗ ∗ ∗ ∗ ∗ ∗ εj i I

< 0

1 ≤ i < j ≤ r

where Πij , L, Λij , and P are defined in (9), and

Tij =

[H1i + eH3iKj H2i√e(1 − e)H3iKj 0

]

E1i =[

E1i 0

0 E1i

], E2i =

[E2i 0

0 E2i

]. (33)

2) Controller Design:Theorem 6: There exists a state feedback controller in the form

of (4) such that the closed-loop fuzzy system in (31) is robustlystochastically stable with a guaranteed H∞ performance γ, ifthere exist matrices X > 0 and Mi , scalars εi , εij , satisfying

L ∆Tii χT

ii 0 εiHT1 ii

∗ X 0 E1 i 0∗ ∗ I E2 i 0∗ ∗ ∗ εi I 0∗ ∗ ∗ ∗ εi I

< 0

i = 1, . . . , r (34)

4L ∆Tij + ∆T

j i χTij + χT

j i 0 0 εij HT1 ij εij HT

1j i

∗ X 0 E1 i E1j 0 0∗ ∗ I E2 i E2j 0 0∗ ∗ ∗ εij I 0 0 0∗ ∗ ∗ ∗ εj i I 0 0∗ ∗ ∗ ∗ ∗ εij I 0∗ ∗ ∗ ∗ ∗ ∗ εj i I

< 0

1 ≤ i < j ≤ r (35)

where ∆ij , χij , X , I , and L are defined in (18), and

H1ij =

[H1iX + eH3iMj H2i√

e(1 − e)H3iMj 0

].

Furthermore, if the previous conditions have feasible solutions,the gains Ki of the subsystem controllers in (4) are given by

Ki = MiX−1 .

B. Discussions

This paper is targeted to solve the problem of H∞ fuzzycontrol with unreliable communication links, which has notbeen solved in the previous literature. Data loss has been rec-ognized as an important issue for the analysis and synthesis ofcommunication-based control systems. Although there are a fewattempts toward this problem, most of the existing results in thisdirection are limited to linear systems. On the other hand, theT–S fuzzy model has been recognized to be an effective meansfor nonlinear systems, while most of the results on fuzzy con-trol are based on the perfect communication assumption. To fillthe gap, the problem of H∞ fuzzy control of nonlinear systemsunder unreliable communication links is investigated in this pa-per, and the results are good complements to the analysis andsynthesis of communication-based control systems.

To solve the problem, we have developed two approachesbased on the quadratic Lyapunov function and the basis-dependent Lyapunov function, respectively. These approachesprovide alternatives for designing H∞ controllers with differentdegrees of conservativeness and computational complexity. It isshown that with the use of basis-dependent Lyapunov function,less conservative results can be obtained than with the use of sin-gle quadratic Lyapunov function, while the computational costof the former is higher than the latter. Most results in this paperare formulated in the form of LMIs, which can be solved eas-ily by numerical software. It is usually difficult to obtain strictLMI conditions for the controller design in the basis-dependentframework, and a CCL algorithm is introduced to solve the non-convex feasibility problem, which provides an alternative forbasis-dependent fuzzy control.

The quadratic Lyapunov function approach has been widelyused in the literature [5], [20], [25]–[27], [36], [38]–[40] sinceit has some advantages: The definition of Lyapunov function issimple, and the computation cost in the design procedure is low.However, the common quadratic Lyapunov functions tend to beconservative and might not exist for some highly nonlinear sys-tems. To reduce the conservatism, basis-dependent Lyapunovfunctions have been adopted in [9], [11], [12], [18], [19], [35],[37], [45], [50], and [51]. It can be seen that the quadraticLyapunov function is a special case of the basis-dependent Lya-punov function, and thus, the latter is more general and lessconservative. Meanwhile, the basis-dependent Lyapunov func-tion also has some disadvantages; the design procedures becomemore complex, and the computational requirement is usuallydemanding.

VI. ILLUSTRATIVE EXAMPLE

In this section, we present three examples to illustrate thetheoretical results developed earlier.

Example 1: In this example, our aim is to compare Theo-rems 1 and 3 to show that the basis-dependent approach is lessconservative. Consider the following system:

Model Rule 1 : IF xk (1) is h1(xk (1))

THEN

xk+1 = A1xk + B21uk + B11wk

zk = C1xk + D21uk + D11wk



TABLE ICOMPARISON OF THE MINIMUM H∞ PERFORMANCE γ∗ BY DIFFERENT

APPROACHES (THEOREMS 1 AND 3)


THEN

xk+1 = A2xk + B22uk + B12wk

zk = C2xk + D22uk + D12wk

(36)

where

A1 =[

1.5 −0.51 0

], B21 =

[11

]B11 =

[0.20.3

], C1 = [ 1 0.5 ]

A2 =[−1 −0.51 0

], B22 =

[−21

]B12 =

[0.5−0.1

], C2 = [ 0.5 1 ]

D11 = 0.4, D12 = 0.2, D21 = 1, D22 = 0.5.

The fuzzy controller is given by


THEN uck = K1xck


THEN uck = K2xck

where K1 = [−0.65 0.5 ] , and K2 = [−0.87 0.11 ] .The aim is to compare the H∞ performances of the closed-

loop system obtained by Theorems 1 and 3. The minimum H∞performances obtained by using Theorems 1 and 3 are listed inTable I, where the communication link parameter e is selectedfrom 0.7 to 0.9.

Table I shows that the minimum H∞ performance γ∗ obtainedby Theorem 3 is much smaller than that by Theorem 1, showingthat Theorem 3 is less conservative. It is interesting to point outthat when e = 0.6833, Theorem 1 fails to find solutions, whileTheorem 3 still yields feasible solutions.

Example 2: In this example, we aim to motivate the problemconsidered in this paper, and to show the usefulness of ourresults. Consider the following fuzzy system:


THEN xk+1 = A1xk + B21uk


THEN xk+1 = A2xk + B22uk (37)

Fig. 3. Closed-loop state variables when e = 1.

where

A1 =[

1 −0.51 0

], B21 =

[11

]A2 =

[−1 −0.51 0

], B22 =

[−21

].

The controller is given by


THEN uck = K1xck


THEN uck = K2xck (38)

where K1 = [0.7460 0.1240], and K2 = [−0.9988 0.8724] .We assume the membership function to be

h1(xk (2)) =

1, xk (2) ≤ −1

0.5 − 0.5xk (2), −1 ≤ xk (2) ≤ 1

0, xk (2) ≥ 1

h2(xk (2)) = 1 − h1

and assume the initial condition to be x0 = [ 0.9 −0.7 ]T .First, when e = 1 (corresponding to perfect communication

links), Fig. 3 shows the state response of the closed-loop system,from which we can see that all the state variables converge tozero. However, when e = 0.8 (corresponding to the case that20% data are lost during the communication), the state variablesno longer converge to zero, as shown in Fig. 4. The data missingis randomly generated according the probability shown in Fig. 5.

Now, for e = 0.8, we apply Theorem 2, and the followingfeasible solutions are obtained:

X =[

2.0920 0.00890.0089 5.5432

]K1 = [−0.8093 −0.0075 ] , K2 = [−0.6742 −0.0128 ] .



Fig. 4. Closed-loop state variables when e = 0.8 with controller gain matricesin (38).

Fig. 5. Random data loss.

Fig. 6 shows that the controller designed by Theorem 2 makesthe closed-loop state variables converge to zero. This well mo-tivates the necessity to consider the communication issue in thecontrol design.

Example 3: Consider an inverted pendulum controlled by adc motor via a gear train shown in Fig. 7, whose fuzzy modelingwas done in [24]. With a sampling time T = 0.02, we have thediscrete-time model as

xk+1 =2∑

i=1

hi(x1k ) (Aixk + B2iuk )

yk =2∑

i=1

hi(x1k )Fixk

Fig. 6. Closed-loop state variables when e = 0.8.

Fig. 7. Inverted pendulum controlled by a dc motor.

where

A1 =

1.0002 0.02 0.02

0.196 1.0001 0.0181

−0.0184 −0.1813 0.8170

A2 =

1 0.02 0.0002

0 0.9981 0.0181

0 −0.1811 0.8170

B21 =

0

0.0019

0.1811

, B22 =

0

0.0019

0.1811

F1 = [ 1 0 0 ] , F2 = [ 1 0 0 ] .

For simulation, we add some disturbance terms and a controlledoutput

xk+1 =2∑

i=1

hi(x1k ) (Aixk + B2iuk + B1iwk )

zk =2∑

i=1

hi(x1k ) (Cixk + D2iuk + D1iwk ) (39)

where

B12 =

−0.054−0.094

0

, B11 =

0.0540.094

0



Fig. 8. Closed-loop state variables when w(t) ≡ 0.

C1 =

0.0540.0050.1

T

, C2 =

0.0540.0050.1

T

D11 = 0.1, D12 = 0.1.

Our aim is to design a state feedback paralleled controller inthe form of (4) such that the system (39) is stochastically stablewith a guaranteed H∞ performance γ.

Without loss of generality, the condition e = 0.8 is discussed.By solving LMIs (16) and (17), the minimum H∞ performanceγ∗ = 0.9332 is obtained, and the feasible solutions are

X =

0.4021 −0.3768 −1.0765−0.3768 0.5258 −0.3634−1.0765 −0.3634 24.7426

K1 = [−75.3905 −52.8595 −6.8464 ]

K2 = [−60.9557 −49.9968 −4.8232 ] . (40)

To show the effectiveness of the obtained results, we assume themembership function to be

h1 =

1, x0(1) = 0sin(x0(1))

x0(1), else

h2 = 1 − h1 .

First, we assume the initial condition to be x0 =[ 0.8 −0.4 −0.8 ]T . Fig. 8 shows the state responses of theclosed-loop system when the external disturbance w(k) ≡ 0,from which we can see that the two states converge to zero.

Next, to illustrate the disturbance attenuation performance,assume zero initial condition, and the external disturbance w(k)is assumed to be

w(k) =

0.2, 20 ≤ k ≤ 30

−0.2, 40 ≤ k ≤ 50

0, else.

(41)

Fig. 9. Closed-loop state variables with w(k) in (41).

Fig. 10. Signals w(k) and z(k).

Fig. 9 shows the changing curves of the state variables, andFig. 10 shows the signals w(k) and z(k). It is found that ‖z‖2 =0.4164 and ‖w‖2 = 0.9381, which yields γ = 0.4439 (belowthe minimum γ∗ = 0.9332), showing the effectiveness of thecontroller design.

VII. CONCLUDING REMARKS

In this paper, the problem of H∞ fuzzy control of nonlinearsystems under unreliable communication links has been inves-tigated. The T–S fuzzy model is utilized to model the nonlin-ear plant, and the communication link failure is modeled via astochastic variable satisfying the Bernoulli random binary dis-tribution. Two approaches have been developed to design H∞controllers such that the closed-loop system is stochasticallystable and preserves a guaranteed H∞ performance. Three ex-amples have been given to illustrate the developed theoreticalresults.



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Huijun Gao (M’06) was born in Heilongjiang,China, in 1976. He received the M.S. degree in elec-trical engineering from Shenyang University of Tech-nology, Shenyang, China, in 2001, and the Ph.D. de-gree in control science and engineering from HarbinInstitute of Technology, Harbin, China, in 2005.

From November 2003 to August 2004, he was aResearch Associate in the Department of Mechan-ical Engineering, University of Hong Kong, HongKong. In November 2004, he joined Harbin Insti-tute of Technology, where he is currently a Professor.

From October 2005 to October 2007, he was a Postdoctoral Researcher in theDepartment of Electrical and Computer Engineering, University of Alberta,Edmonton, AB, Canada. He is an Associate Editor of the Journal of Intelligentand Robotics Systems, Circuits, System and Signal Processing, etc. His currentresearch interests include network-based control, robust control, and time-delaysystems and their industrial applications.

Prof. Gao is an Associate Editor of the IEEE TRANSACTIONS ON SYS-TEMS, MAN AND CYBERNETICS PART B: CYBERNETICS and the IEEE TRANS-ACTIONS ON INDUSTRIAL ELECTRONICS. He was an outstanding reviewer for theIEEE TRANSACTIONS ON AUTOMATIC CONTROL AND AUTOMICA in 2008 and2007, respectively, and an appreciated reviewer for the IEEE TRANSACTIONS ON

SIGNAL PROCESSING in 2006. He was the recipient of the University of AlbertaDorothy J. Killam Memorial Postdoctoral Fellow Prize in 2005, the NationalOutstanding Youth Science Fund in 2008, and the National Outstanding Doc-toral Thesis Award in 2007. He was the corecipient of the National NaturalScience Award of China in 2008.

Yan Zhao received the B.S. degree in chemical engi-neering and equipment control and the M.S. degreein mechanical engineering from Inner Mongolia Uni-versity of Technology, Hohhot, China, in 2002 and2005, respectively. She is currently working towardthe Ph.D. degree in control science and engineer-ing with the Harbin Institute of Technology, Harbin,China.

Her current research interests include fuzzy con-trol systems, robust control, and networked controlsystems.

Tongwen Chen (S’86–M’91–SM’97–F’06) receivedthe B.Eng. degree in automation and instrumentationfrom Tsinghua University, Beijing, China, in 1984and the M.A.Sc. and Ph.D. degrees in electrical engi-neering from the University of Toronto, Toronto, ON,Canada, in 1988 and 1991, respectively.

From 1991 to 1997, he was an Assistant/AssociateProfessor with the Department of Electrical and Com-puter Engineering, University of Calgary, Calgary,AB, Canada. Since 1997, he has been with the De-partment of Electrical and Computer Engineering,

University of Alberta, Edmonton, AB, where he is currently a Professor. Heheld visiting positions at the Hong Kong University of Science and Technology,Tsinghua University, and Kumamoto University. He was the coauthor of thebook Optimal Sampled-Data Control Systems (Springer-Verlag, 1995). He wasa McCalla Professor for 2000–2001 and a Killam Professor for 2006–2007 atthe University of Alberta. He was an Associate Editor for several internationaljournals, including Automatica and Systems and Control Letters. His current re-search interests include computer- and network-based control systems, processcontrol, multirate digital signal processing, and their applications to industrialproblems.

Prof. Chen was an Associate Editor for the IEEE TRANSACTIONS ON

AUTOMATIC CONTROL. He is a registered Professional Engineer in Alberta.He received the Fellowship from the Japan Society for the Promotion of Sci-ence for 2004.



An Adaptive Consensus Support Model for GroupDecision-Making Problems in a Multigranular Fuzzy

Linguistic ContextFrancisco Mata, Luis Martınez, and Enrique Herrera-Viedma

Abstract—Different consensus models for group decision-making (GDM) problems have been proposed in the literature.However, all of them consider the consensus reaching process arigid or inflexible one because its behavior remains fixed in allrounds of the consensus process. The aim of this paper is to im-prove the consensus reaching process in GDM problems defined inmultigranular linguistic contexts, i.e., by using linguistic term setswith different cardinality to represent experts’ preferences. To dothat, we propose an adaptive consensus support system model forthis type of decision-making problem, i.e., a process that adaptsits behavior to the agreement achieved in each round. This adap-tive model increases the convergence toward the consensus and,therefore, reduces the number of rounds to reach it.

Index Terms—Consensus, fuzzy preference relation, group deci-sion making (GDM), linguistic modeling.

I. INTRODUCTION

ONE of the reasons why decision-making processes havebeen widely studied in the literature is the increasing com-

plexity of the social–economic environment [12], [19]. Manyorganizations have moved from a single decision maker to agroup of experts to accomplish this task successfully. A groupdecision-making (GDM) problem may be defined as a deci-sion problem with several alternatives and experts that try toachieve a common solution taking into account their opinionsor preferences.

Our interest is focused on GDM problems in which the expertshave to express their preferences on qualitative aspects thatcannot be assessed by means of quantitative values. In thesecases, the use of linguistic terms instead of precise numericalvalues seems to be more appropriate. For example, to evaluatethe “comfort” of a car, linguistic terms like “good,” “fair,” or“poor” could be preferred by the experts instead of numericalvalues [48].

The use of the fuzzy linguistic approach [60]–[62] to assessqualitative aspects by using linguistic variables, i.e., variableswhose values are not numbers but words or sentences in a naturalor an artificial language, has proven successful in decision-

Manuscript received April 27, 2007; revised November 14, 2007, April 20,2008, and December 2, 2008; accepted December 15, 2008. First publishedJanuary 27, 2009; Current version published April 1, 2009. This work wassupported in part by the Research Projects TIN2006-02121 and TIN2007-61079.

F. Mata and L. Martınez are with the Department of Computer Science, Uni-versity of Jaen, Jaen 23700, Spain (e-mail: [email protected]; [email protected]).

E. Herrera-Viedma is with the Department of Computer Science andArtificial Intelligence, University of Granada, Granada 18071, Spain (e-mail:[email protected]).


Fig. 1. Resolution process of a GDM problem.

making problems [1], [3], [5], [20], [27], [38], [42], [49], [58],[59].

In GDM problems, there are cases where experts have differ-ent levels of knowledge about the alternatives, and as a conse-quence, they might use linguistic term sets with different car-dinality to express their preferences. In such cases, we say thatthe GDM problem is defined in a multigranular fuzzy linguisticcontext [11], [13], [21], [28], [32], [35], [50], [52], [57].

Usually, GDM problems are solved by carrying out selectionprocesses to obtain a solution set of alternatives from the prefer-ences given by the experts [19], [23], [25], [53]. However, it mayhappen that, after the selection process, some experts considerthat their preferences have not been taken into account properlyto obtain the solution, and hence, they might reject it. One wayto avoid this situation would be the application of a consensusprocess (see Fig. 1) so that the experts discuss and modify theirpreferences in order to reach a sufficient agreement, before ap-plying the selection process [5], [7], [9], [26], [31], [33], [41].Selection processes for GDM problems defined in multigranularlinguistic contexts were introduced in [21] and [28]; hence, herewe focus on the consensus process.

Consensus modeling is an important area of research in deci-sion analysis [5], [7]–[9], [14], [16]–[18], [26], [31], [33], [35],[37], [39], [40], [46], [47], [54], [55]. Consensus is defined as astate of mutual agreement among members of a group where allopinions have been heard and addressed to the satisfaction ofthe group [54]. A consensus reaching process is a dynamic anditerative process composed by several rounds where the expertsexpress, discuss, and modify their preferences. Normally, thisprocess is guided by the figure of a moderator, who helps theexperts to make their preferences closer to each other [40], [54].

1063-6706/$25.00 © 2009 IEEE



In each consensus round, the moderator evaluates the currentagreement among the experts’ preferences. If the agreement isnot acceptable, i.e., it is lower than a specified threshold, themoderator would then recommend to modify the furthest pref-erences from the collective ones in order to make them closer.Otherwise, when the agreement is acceptable, the moderatorwould apply the selection process in order to obtain the finalsolution for the GDM problem.

The main interest in consensus research has been the devel-opment of new processes with different structures and method-ologies to achieve its aim [7], [10], [33], [35], [44]. However,the enhancement of these processes has not been the focus inthis research field yet. For instance, it is easy to check thatif the agreement is “very low” (initial rounds), then the num-ber of changes of preferences should be greater than when theagreement is “high” (final rounds). Thus, adapting the consen-sus reaching process to the level of agreement achieved in eachdiscussion round could significantly improve its performance.

The aim of this paper is to propose an adaptive consensussupport system (ACSS) model to support consensus processesin GDM problems with multigranular linguistic information,which improves the consensus reaching process by adaptingthe search for preferences in disagreement to the current levelof consensus at each round. To do so, three different methodsto identify the preferences that each expert should modify, inorder to increase the agreement in the next consensus round, aredefined. The result is a model that improves the convergencerate toward the consensus, and therefore decreases the numberof rounds to achieve it.

The rest of the paper is set out as follows. In Section II, pre-liminaries about the multigranular fuzzy linguistic GDM prob-lems and the consensus process are presented. The proposedACSS model is described in detail in Section III. In Section IV,the application of the proposed ACSS model is given, whilein Section V, we draw our conclusions. Finally, the Appendixintroduces the meaning and features of the measurements usedto evaluate the agreement.

II. PRELIMINARIES

In order for this paper to be as self-contained as possible,we include in this section a brief review of the fuzzy linguisticapproach, focusing on GDM problems defined in multigranularfuzzy linguistic contexts, and the main elements and features ofthe consensus processes.

A. Multigranular Fuzzy Linguistic GDM Problems

The fuzzy linguistic approach assesses qualitative attributesby using linguistic assessments by means of linguistic vari-ables [60]–[62]. This approach has been successfully applied todifferent problems [2], [3], [6], [20], [22], [29], [30], [34], [36],[43], [56], [63].

In this approach, assessments of the preferences on pairs ofalternatives are provided in the form of linguistic terms or labelsof a linguistic term set S = s0 , s1 , . . . , sg,#(S) = g + 1. Animportant issue to analyze is the “granularity of uncertainty,”i.e., the cardinality of the linguistic term set. The granularity of

S should be small enough so as not to impose useless precisionlevels on the users but large enough to allow a discrimination ofthe assessments in a limited number of degrees. Additionally,the following properties are assumed.

1) The set S is ordered: si ≥ sj , if i ≥ j.2) There is the negation operator: Neg(si) = sj such that

j = g − i.The semantics of S can be given by fuzzy numbers defined on

the unit interval [0,1]. One way to characterize a fuzzy numberis by using a representation based on parameters of its member-ship function [4]. For example, the following semantics can beassigned to a set of seven terms via triangular fuzzy numbers:

N = None = (0, 0, 0.17)

V L = Very Low = (0, 0.17, 0.33)

L = Low = (0.17, 0.33, 0.5)

M = Medium = (0.33, 0.5, 0.67)

H = High = (0.5, 0.67, 0.83)

V H = Very High = (0.67, 0.83, 1)

P = Perfect = (0.83, 1, 1).

A GDM problem is classically defined as a decision situ-ation where a set of experts, E = e1 , e2 , . . . , em (m ≥ 2),express their preferences about a set of feasible alternatives,X = x1 , x2 , . . . , xn (n ≥ 2). In many decision situations, itis assumed that each expert ei provides his or her preferencesby means of a fuzzy preference relation [19] Pei = [plk

i ], l, k ∈1, . . . , n, with plk

i = µPei(xl, xk ) assessed in the unit inter-

val [0,1] and being interpreted as the preference degree of thealternative xl over xk according to the expert ei. In this paper,we use linguistic preference relations to represent the experts’preferences as in [23] and [24], i.e., with plk

i = µPei(xl, xk )

assessed in a linguistic term set S = s0 , s1 , . . . , sg.The ideal situation for GDM problems defined in linguistic

contexts would be that all the experts use the same linguisticterm set S to express their preferences about the alternatives.However, in some cases, experts may belong, e.g., to distinctresearch areas, and therefore could have different backgroundand levels of knowledge. A consequence of this is that they needto express their preferences by using linguistic term sets withdifferent granularity Si = si

0 , . . . , sigi, i ∈ 1, 2, . . . ,m. In

these cases, the GDM problem is defined in a multigranularfuzzy linguistic context [11], [13], [21], [28], [32], [35], [50],[52], [57].

B. Consensus Process

A consensus reaching process in a GDM problem is an iter-ative process composed by several discussion rounds, in whichexperts are expected to modify their preferences according tothe advice given by the moderator (see Fig. 1). The moderatorplays a key role in this process. Normally, the moderator is aperson who does not participate in the discussion but knows thepreferences of each expert and the level of agreement duringthe consensus process. He/she is in charge of supervising and


MATA et al.: ADAPTIVE CONSENSUS SUPPORT MODEL FOR GDM PROBLEMS IN A MULTIGRANULAR FUZZY LINGUISTIC CONTEXT 281

Fig. 2. Phases of the consensus process supervised by the moderator.

driving the consensus process toward success, i.e., to achieve themaximum possible agreement and reduce the number of expertsoutside of the consensus in each new consensus round.

An overall scheme of the different phases carried out in aconsensus process guided by a moderator is shown in Fig. 2.

1) Computing level of agreement: The moderator com-putes the current agreement among all experts from theirpreferences.

2) Checking level of agreement: The moderator comparesthe current level of agreement with a consensus thresholdfixed previously. If the consensus threshold is achieved,the selection process will be applied to obtain the finalsolution. Otherwise, the consensus process will continueits execution.

3) Search for preferences: The moderator searches for theexperts’ preferences furthest from the collective ones andsuggests how to change them in order to improve theagreement in the next round.

In order to evaluate the agreement, it is required to com-pute similarity measures among the experts [7], [26], [27], [31],[35], [39]. Two types of measurements to guide the consensusreaching process were proposed in [26]:

a) consensus degrees to evaluate the level of agreementamong all the experts. They will be used to identify thepreference values where the agreement is not sufficient;

b) proximity measures to evaluate the distance between theexperts’ individual preferences and the group or collectiveones. They will be used to identify the experts who shouldchange their preferences in the next rounds.

These measurements are computed at the three different levelsof representation of information of a preference relation: pairsof alternatives, alternatives, and relation.

Level 1: Pairs of alternatives. Given a pair of alternatives(xl, xk ):

1) cplk is the agreement among all experts on the pair ofalternatives (xl, xk );

2) pplki is the proximity between the preference value of

expert ei, plki , and the collective one on the pair of alter-

natives (xl, xk ).Level 2: Alternatives. Given the alternative xl ∈ X:1) cal is the agreement among all experts on xl ;

2) pali is the proximity between the preference values of

expert ei and the collective ones on xl .Level 3: Preference relation.1) cr is the global agreement among all experts on all the

pairs of alternatives of a preference relation.2) pri is the global proximity between the preferences given

by ei and the collective ones of a preference relation.A further detailed description of the meaning as well as a

description of the computation of these measurements can befound in the Appendix.

III. ADAPTIVE CONSENSUS SUPPORT SYSTEM MODEL FOR

GDM IN A MULTIGRANULAR FUZZY LINGUISTIC CONTEXT

Several authors [7], [31], [35], [44] have proposed differentmodels to carry out consensus processes where the human mod-erator’s role is assumed by the own model. In all of them, theconsensus reaching process is considered as a rigid or inflexi-ble one because its behavior remains fixed in all rounds of theconsensus process. However, it is obvious that when the levelof agreement between the experts is “high”, a few number ofchanges of opinions from some of the experts might lead toconsensus in a few discussion rounds. On the contrary, whenthe level of agreement among the experts is “low,” a high num-ber of changes of opinions and many group discussion roundsmight be necessary for consensus to be achieved. In this sec-ond case, it seems reasonable that many experts’ preferencesshould be changed if they try to achieve a common solution.As the level of agreement increases, less and less experts mightneed to change their opinions. In fact, in these cases, it mightbe expected that only those experts whose preference valuesare furthest from the group ones should change them. In otherwords, the number of changes in different stages of a consensusprocess is clearly related to the actual level of agreement. A con-sensus model that implements this idea will improve the GDMprocesses.

In this section, following the aforementioned idea, we presentan ACSS model for multigranular fuzzy linguistic GDM prob-lems that improves the convergence rate toward the consensus,and therefore decreases the number of rounds to achieve it. Itconsists of four phases (see Fig. 3).

1) Making the linguistic information uniform: In this phase,all experts’ multigranular linguistic preferences are unifiedinto a single linguistic domain.

2) Computing the consensus degree and control of the con-sensus process: The consensus degree among all experts iscalculated. If the consensus degree is high enough, the se-lection process is then applied. Otherwise, the consensusprocess keeps going.

3) Adaptive search for preferences: Different policies or pro-cedures for searching the preferences to be changed ineach consensus round are applied based on a broad clas-sification of the global consensus level as very low, low,and medium. Each preference search procedure (PSp) willreturn the set of preferences each expert should change in



Fig. 3. ACSS model in a multigranular fuzzy linguistic context.

order to make his/her preferences closer to the collectiveopinion.

4) Production of advice: Once the sets of preferences in dis-agreement have been identified, an advice system sug-gests the direction of the changes to be recommended tothe experts in order to improve the agreement in the nextconsensus round.

In the following sections, the aforementioned phases are de-scribed in detail.

A. Making the Linguistic Information Uniform

To manage multigranular fuzzy linguistic information, weneed to make it uniform, i.e., experts’ preferences have to betransformed (using a transformation function) into a single do-main or linguistic term set that we call the basic linguistic termset (BLTS), denoted by ST [21]. To do this, it seems reasonableto impose a cardinality high enough to maintain the uncertaintydegrees associated with each one of the possible domains to beunified. This means that the cardinality of the BLTS has to beas high as possible. Therefore, in a general multigranular fuzzylinguistic context, to select ST , we proceed as it was proposedin [21].

1) If there is only one linguistic term set, from the set of dif-ferent domains to be unified, with maximum cardinality,then we choose that one as the BLTS, ST .

2) If there are two or more linguistic term sets with maximumcardinality, then the selection of ST will depend on thesemantics associated with them.

a) If all of them have the same semantics, i.e., thesame fuzzy membership functions associated withthe linguistic terms but with different syntax, thenany one of them could be selected as ST .

b) If two or more of them have different semantics, thenST is defined as a generic linguistic term set with anumber of terms greater than the number of terms aperson is able to discriminate, which is normally 7or 9 [51].

Once ST has been selected, the following multigranular trans-formation function is applied to transform every linguistic valueinto a fuzzy set defined on ST .

Fig. 4. Transforming l1 ∈ S into a fuzzy set on ST .

Definition 1 [21]: If S = l0 , . . . , lp and ST = c0 , . . . , cgare two linguistic term sets, with g ≥ p, then a multigranu-lar transformation function τSST

: S → F (ST ) is defined asfollows:

τSST(li) = (ch , αh)

∣∣∣αh = maxy minµli (y), µch(y),

h = 0, . . . , g

where F (ST ) is the set of fuzzy sets defined on ST , and µli (y)and µch

(y) are the membership functions of the fuzzy sets as-sociated with the linguistic terms li and ch , respectively.

Example 1: Let S = l0 , l1 , . . . , l4 and ST =c0 , c1 , . . . , c6 be two term sets with the following semantics:

l0 = (0, 0, 0.25) c0 = (0, 0, 0.16)

l1 = (0, 0.25, 0.5) c1 = (0, 0.16, 0.34)

l2 = (0.25, 0.5, 0.75) c2 = (0.16, 0.34, 0.5)

l3 = (0.5, 0.75, 1) c3 = (0.34, 0.5, 0.66)

l4 = (0.75, 1, 1) c4 = (0.5, 0.66, 0.84)

c5 = (0.66, 0.84, 1)

c6 = (0.84, 1, 1).

The fuzzy set obtained when applying τSSTto l1 is (see Fig. 4)

τSST(l1) = (c0 , 0.39), (c1 , 0.85), (c2 , 0.85),

(c3 , 0.39), (c4 , 0), (c5 , 0), (c6 , 0).

In order to unify all the experts’ preferences, different multi-granular transformation functions τSi ST

are defined. Each lin-guistic preference value plk

i ∈ Si will be transformed in afuzzy set plk

i = τSi ST(plk

i ) = (ch , αlkh )|h = 0, . . . , g on ST .

To simplify, we will use the membership degrees (αlk0 , . . . , αlk

g )



to denote each fuzzy set plki

Pei =

p11

i =(α110 , . . . , α11

g ) · · · p1ni = (α1n

0 , . . . , α1ng )

.... . .

...

pn1i =(αn1

0 , . . . , αn1g ) · · · pnn

i = (αnn0 , . . . , αnn

g )

.

B. Computing the Consensus Degree and Control of theConsensus Process

Once all the linguistic preferences have been unified by meansof fuzzy sets on the BLTS, the following two steps are applied.

1) Computing the consensus degree: The level of agreementachieved in the current round is obtained. To do so, a globalconsensus degree, called consensus on relation cr ∈ [0, 1],is computed (see the Appendix).

2) Control of the consensus process: In this phase, both theglobal consensus degree cr and the consensus threshold γare compared such that:

a) if cr ≥ γ, the level of agreement is sufficient, theconsensus process will stop, and the selection pro-cess will be applied;

b) if cr < γ, a new consensus round is applied.Note that γ is fixed in advance and represents the necessary

level of agreement for a solution to be accepted by the group.A γ value too high may cause that the first condition will neverbe satisfied, and in consequence, we have a never-ending con-sensus process. In order to avoid this situation, we define aparameter Max rounds that limits the maximum number of con-sensus rounds. This parameter has already been used by otherauthors in the control of consensus processes [7], [33].

The value of γ will obviously depend on the particular prob-lem we are dealing with. When the consequences of the decisionto be made are of utmost importance, the minimum level of con-sensus required to make that decision should be logically as highas possible, and it is not unusual if a minimum value of 0.8 orhigher is imposed. At the other extreme, we have cases wherethe consequences are not so serious (but are still important),and it is urgent to obtain a solution to the problem, and thus, aminimum consensus value as close as possible to 0.5 could berequired.

C. Adaptive Search for Preferences

If the agreement among all experts is low, then there exist a lotof experts’ preferences in disagreement. In such a case, in orderto bring the preferences closer to each other and so to improvethe consensus situation, the number of changes in the experts’preferences should be high. However, if the agreement is high,the majority of preferences is close and only a low number ofexperts’ preferences are in disagreement; it seems reasonable tochange only these particular preferences. We distinguish threelevels of consensus: very low, low, and medium consensus. Eachlevel implies a different search policy to identify the preferenceswith low agreement degree. When the level of consensus is verylow, all experts will be advised to modify all the preferencesvalues identified in disagreement, while if the level of consensusis greater, the search will be limited to the preference values

Fig. 5. Adaptive search for preferences.

in disagreement of those experts furthest from the group. Todo so, the system establishes three different PSps: “PSp forvery low consensus,” “PSp for low consensus,” and “PSp formedium consensus.” Each PSp will identify the preferences indisagreement in a different way. This fact defines the adaptivecharacter of our model.

The adaptive search for preferences consists of two processes(see Fig. 5).

1) Choose the most suitable PSp: Two parameters θ1 andθ2 , whose values depend on the particular problem dealtwith, are fixed at the beginning of the consensus process todifferentiate the three consensus situations: very low, low,and medium consensus. Depending on both parameters,we choose the most appropriate PSp to apply to eachparticular consensus round: a) PSp for very low consensusif cr ≤ θ1 ; b) PSp for low consensus if cr ≤ θ2 ; and c)PSp for medium consensus otherwise.

2) Apply the PSp: Each PSp finds out a set of pref-erences, PREFECH i = (l, k), l, k ∈ 1, 2, . . . , n, l =k, to be changed by each expert ei in order to improve theagreement in the next round. In each PSp, the agreementis analyzed in a different preference representation level.

a) In PSp for very low consensus, the level of pairs ofalternatives is considered.

b) In PSp for low consensus, the level of alternative isconsidered.

c) In PSp for medium consensus, the level of preferencerelation is considered.

The three PSps are described in detail below:1) PSp for Very Low Consensus (PSpV L ): Usually, at the

beginning of the consensus process, experts’ preferences arequite far from each other, and therefore, the agreement will bevery low. In these situations, it seems reasonable to require manychanges in order to make the preferences closer to one another.To do this, the procedure suggests modifying the preference val-ues on all the pairs of alternatives where the agreement is nothigh enough. These changes may be carried out either by someexperts, for example, the experts furthest from the group as pro-posed in [35], or by all experts. We consider the second optionmore appropriate because it prevents some experts imposingtheir preferences in the first rounds, and as a consequence, the



Fig. 6. Chosen preferences by P SpV L .

consensus process could be guided toward their own opinions,which is known as “tyranny of the majority,” and is a problemthat should be avoided in consensus reaching processes [54].Also, with the second option, all experts would be willing toshare the final solution because their preferences were takeninto account to obtain the solution.

PSpV L finds out the set of preferences to be changed byei,PREFECH V L

i , as follows.1) First, the pairs of alternatives with a consensus degree

smaller than a threshold ρ defined at level of pairs of alter-natives, P = (l, k)|cplk < ρ, are identified. The valueof ρ may be static and fixed before starting the consen-sus process or dynamic with respect to the level of con-sensus reached in each round. The selection of such athreshold plays a very important role in the identifica-tion process because a static value too high may implymany changes (leading to all experts having to change al-most all their preference values), while a value too lowmay imply very few changes. We consider that a dy-namic value that changes during the consensus processis better than static one fixed in advance. In this pa-per, we have assigned to ρ the average of the consen-sus degree at level of all pairs of alternatives, ρ = cp,such that cp =

∑nl=1(

∑nk=1,l =k cplk )/(n2 − n). Then,

P = (l, k)|cplk < cp, l, k = 1, . . . , n.2) The set of preference values PREFECH V L

i to be changedby each expert ei will be

PREFECH V Li = P.

In Fig. 6, the characteristics and the behavior of this procedureare graphically described.

2) PSp for Low Consensus (PSpL ): After several discussionrounds, the agreement among all experts should be greater thanat the beginning with θ1 < cr ≤ θ2 . In this situation, it seemslogical to reduce the number of changes and modify the pointof view for the analysis of the agreement. While in the PSpV L

we focused on all the pairs of alternatives in disagreement, inthe PSpL the agreement is analyzed from the point of view of

the alternatives and only the preference values in disagreementof those alternatives where agreement is not sufficient will beconsidered.

Another important difference with respect to the PSpV L isthe number of experts involved in the change of preferences.While in the PSpV L , all experts are required to modify theidentified preference values, in the PSpL , the following re-striction is added: the experts required to modify the identi-fied preference values will be those with proximity value atlevel of alternatives, for those identified alternatives in dis-agreement, smaller than an alternative proximity threshold β,i.e., ei |pal

i < β, β ∈ [0, 1], i ∈ 1, . . . , m. As in the previ-ous case, the value of β may be static or dynamic. Again, weconsider the second option more appropriate because it meansthat the restriction adapts to the proximity values obtained ineach consensus round. A possible dynamic value in this casecould be the arithmetic mean of all proximity on alternativesβ = pal =

∑mi=1 pal

i/m.PSpL finds out the set of preferences to be changed by each

ei,PREFECH Li , as follows.

1) The consensus degrees at level of alternatives are obtained(see the Appendix): cal |l = 1, . . . , n.

2) Alternatives to be changed Xch are identified. A dynamicconsensus threshold at level of alternatives is proposed inthis case, such as the average of the consensus degreesat level of alternative ca =

∑nl=1 cal/n, and then, Xch =

l|cal < ca.3) Pairs of alternatives to be changed are identified: P =

(l, k)|l ∈ Xch ∧ cplk < cp.4) The proximity of the alternatives that should be changed

is computed for all experts (see the Appendix): pali |l ∈

Xch∀ei ∈ E.5) The proximity threshold β = pal used to identify the ex-

perts that will be required to modify the identified pairs ofalternatives is computed.

6) Then, the sets of preference values that are required to bemodified are

PREFECH Li = (l, k) ∈ P | pal

i < pal.

Clearly, the new restriction reduces the number of preferencesand experts required to make changes. Consequently, we have

#

(⋃i

PREFECH Li

)≤ #

(⋃i

PREFECH V Li

).

Graphically, the behavior of this procedure is shown in Fig. 7.By comparing this figure with the previous one, we can checkthat, indeed, the number of changes required is reduced.

3) PSp for Medium Consensus (PSpM ): In the last con-sensus rounds, the agreement will be close to the desired con-sensus threshold, θ2 < cr < γ. Therefore, the agreement canbe improved by suggesting fewer changes than in the previ-ous two cases. Consequently, a new restriction is added to thePSpM , which will reduce the number of experts required tomodify their opinions: only those experts who have proximityvalues on the pairs of alternatives identified in disagreementsmaller than a specific proximity threshold at level of pairs of



Fig. 7. Chosen preferences by P SpL .

Fig. 8. Chosen preferences by P SpM .

alternatives will have to change their opinions. This is illustratedin Fig. 8.

The computation of the set of preferences to be changed byei,PREFECH M

i in PSpM , is as follows.1) Operations enumerated from 1 to 5 in PSpL are carried

out.2) The proximity threshold to be used in identifying the ex-

perts required to modify the identified pairs of alternativesin disagreement is computed:

pplk =

∑mi=1 pplk

i /m |(l, k) ∈ P

.

3) The sets of preference values that are required to be mod-ified are

PREFECH Mi = (l, k) ∈ P | pal

i < pal ∧ pplki < pplk.

Clearly, we have #(⋃

i PREFECH Mi ) ≤

#(⋃

i PREFECH Li ).

Therefore, this adaptive search of preferences in disagreementreduces the number of changes as the consensus increases.

The main features of the PSps are shown in the Table I.

TABLE ISUMMARY TABLE OF PSPS

D. Production of Advice

Once the preferences to be changed have been identified, themodel shows the right direction of the changes in order to im-prove the agreement. For each preference value to be changed,the model will suggest increasing or decreasing the current as-sessment.

A guidance advice system based on several direction param-eters was proposed in [35] to increase the agreement. However,this system presented some difficulties. In this paper, we presenta new mechanism based on a set of direction rules to identify andsuggest the changes. These rules compare the central values ofthe fuzzy sets on ST of the individual and collective preferenceassessments cv(plk

i ) and cv(plkc ). The central value represents

the center of gravity of the information contained in the fuzzyset (see the Appendix). The new direction rules, DR, in our caseare as follows.

DR.1: If (cv(plki ) − cv(plk

c )) < 0, then the expert ei should in-crease the assessment associated with the pair of alternatives(xl, xk ).


c )) > 0, then the expert ei should de-crease the assessment associated with the pair of alternatives(xl, xk ).


c )) = 0, then the expert ei should notmodify the assessment associated with the pair of alternatives(xl, xk ).

IV. APPLICATION OF THE ACSS MODEL

In this section, we apply the ACSS model presented in Sec-tion III to a GDM problem with multigranular fuzzy linguisticinformation.

A. GDM Framework

Let us suppose that a supermarket wants to buy 10 000bottles of Spanish wine from among four possible brandsof wine or alternatives: x1 = Marques de Caceres, x2 =Los Molinos, x3 = Somontano, x4 = Rene Barbier.

The manager decided to inquire eight experts about theiropinions E = e1 , . . . , e8. The experts have to reach a highlevel of agreement before choosing the best brand of wine. Dueto the fact that the experts involved in the problem have differentlevels of knowledge about wine, three linguistic term sets withdifferent cardinalities may be used to provide their preferences.



1) Experts e3 and e7 use label set A

a0 = (0, 0, 0.13) a1 = (0, 0.13, 0.25)

a2 = (0.13, 0.25, 0.38) a3 = (0.25, 0.38, 0.5)

a4 = (0.38, 0.5, 0.63) a5 = (0.5, 0.63, 0.75)

a6 = (0.63, 0.75, 0.88) a7 = (0.75, 0.88, 1)

a8 = (0.88, 1, 1).

2) Experts e4 , e5 , and e8 use label set B

b0 = (0, 0, 0.17) b1 = (0, 0.17, 0.33)

b2 = (0.17, 0.33, 0.5) b3 = (0.33, 0.5, 0.67)

b4 = (0.5, 0.67, 0.83) b5 = (0.67, 0.83, 1)

b6 = (0.83, 1, 1).

3) Experts e1 , e2 , and e6 use label set C

c0 = (0, 0, 0.25) c1 = (0, 0.25, 0.5)

c2 = (0.25, 0.5, 0.75) c3 = (0.5, 0.75, 1)

c4 = (0.75, 1, 1).

Initially, the experts provide the following linguistic prefer-ence relations:

Pe1 =

− c0 c0 c2c4 − c3 c4c3 c0 − c1c2 c1 c3 −

Pe2 =

− c2 c0 c4c1 − c1 c1c3 c3 − c1c0 c4 c3 −

Pe3 =

− a1 a4 a3a5 − a8 a4a4 a1 − a2a5 a5 a7 −

Pe4 =

− b0 b4 b5b6 − b1 b6b3 b4 − b2b0 b1 b4 −

Pe5 =

− b4 b1 b6b2 − b3 b2b4 b3 − b2b0 b5 b3 −

Pe6 =

− c2 c3 c1c2 − c0 c1c0 c4 − c4c4 c4 c0 −

Pe7 =

− a0 a3 a7a8 − a0 a4a4 a8 − a5a1 a4 a3 −

Pe8 =

− b6 b1 b3b0 − b0 b5b6 b6 − b5b4 b1 b0 −

.

The parameters applied to the ACSS model are1) γ = 0.75;2) θ1 = 0.65 and θ2 = 0.72;3) Max rounds = 10.

B. First Round

1) Making the Linguistic Information Uniform: Accordingto conditions set out in Section III-A, ST = A. To unify thedifferent linguistic term sets, three multigranular transformationfunctions τAST

, τBST, τC ST

are used (see Tables II and III).2) Computing the Consensus Degree and Control of the Con-

sensus Process:a) Computing consensus degree: The consensus degree

obtained at the different levels is as follows (see the Appendix).

TABLE IIFUZZY SETS OBTAINED FOR τA ST

AND τB ST

TABLE IIIFUZZY SETS OBTAINED FOR τC ST

i) Pairs of alternatives

CM =

− 0.6 0.69 0.68

0.58 − 0.58 0.660.71 0.58 − 0.690.61 0.61 0.63 −

.

ii) Alternatives(ca1 = 0.642, ca2 = 0.6, ca3 = 0.645, ca4 = 0.646

).

iii) Relation: cr = 0.633.b) Control of the consensus process: Because cr =

0.63 < γ = 0.75, the adaptive search of preference values indisagreement is activated.

3) Adaptive Search for Preferences:a) Choose the most suitable PSp: Given that cr =

0.633 ≤ θ1 = 0.65, the level of agreement is very low, andtherefore, PSpV L is applied.

b) Apply the PSpV L :i) Identification of pairs of alternatives in disagree-

ment

P = (1, 2), (2, 1), (2, 3), (3, 2), (4, 1), (4, 2), (4, 3).ii) Set of preferences to be changed by all experts

PREFECH V Li = P, i = 1, . . . , 8.

4) Production of Advice:a) According to rule DR1, the experts are required to in-

crease the following preference assessments:

p121 = c0 → c1 p12

3 = a4 → a5

p415 = b0 → b1 p23

7 = a0 → a1

p321 = c0 → c1 p32

3 = a1 → a2

p435 = b3 → b4 p41

7 = a1 → a2

p421 = c1 → c2 p12

4 = b0 → b1

p216 = c2 → c3 p42

7 = a4 → a5

p212 = c1 → c2 p23

4 = b1 → b2

p236 = c0 → c1 p43

7 = a3 → a4



p232 = c1 → c2 p41

4 = b0 → b1

p436 = c0 → c1 p21

8 = b0 → b1

p412 = c0 → c1 p42

4 = b1 → b2

p127 = a0 → a1 p23

8 = b0 → b1

p215 = b2 → b3 p42

8 = b1 → b2

p325 = b3 → b4 p43

8 = b0 → b1 .

b) According to rule DR2, the experts are required to de-crease the following preference assessments:

p211 = c4 → c3 p43

2 = c3 → c2

p324 = b4 → b3 p41

6 = c4 → c3

p231 = c4 → c3 p23

3 = a8 → a7

p434 = b4 → b3 p42

6 = c4 → c3

p411 = c2 → c1 p42

3 = a5 → a4

p125 = b4 → b3 p21

7 = a8 → a7

p431 = c3 → c2 p21

3 = a5 → a4

p235 = b3 → b2 p32

7 = a8 → a7

p122 = c2 → c1 p41

3 = a5 → a4

p425 = b5 → b4 p12

8 = b6 → b5

p322 = c3 → c2 p43

3 = a7 → a6

p126 = c2 → c1 p32

8 = b6 → b5

p422 = c4 → c3 p21

4 = b6 → b5

p326 = c4 → c3 p41

8 = b4 → b3 .

C. Second Round

1) Gathering Information (New Preferences): According tothe previous advices, the experts implemented all suggestedchanges, and the new provided preferences are

Pe1 =

− c1 c0 c2c3 − c2 c4c3 c1 − c1c1 c2 c2 −

Pe2 =

− c1 c0 c4c2 − c2 c1c2 c3 − c1c1 c3 c2 −

Pe3 =

− a2 a4 a3a4 − a7 a4a4 a2 − a2a4 a4 a6 −

Pe4 =

− b1 b4 b5b5 − b2 b6b3 b3 − b2b1 b2 b3 −

Pe5 =

− b3 b1 b6b3 − b2 b2b4 b4 − b2b1 b4 b4 −

Pe6 =

− c1 c3 c1c3 − c1 c1c0 c3 − c4c3 c3 c1 −

Pe7 =

− a1 a3 a7a7 − a1 a4a4 a7 − a5a2 a5 a4 −

Pe8 =

− b5 b1 b3b1 − b1 b5b6 b5 − b5b3 b2 b1 −

.

TABLE IVPROXIMITY AT LEVEL OF ALTERNATIVES

Note: In the remaining rounds, we shall only show the mostrelevant information, the evolution of the consensus degrees andthe operation of PSps.

2) Computing the Consensus Degree and Control of the Con-sensus Process:

a) Consensus degree:i) Pairs of alternatives

CM =

− 0.77 0.69 0.68

0.73 − 0.73 0.660.72 0.71 − 0.690.77 0.8 0.78 −

.


).


0.726 < γ = 0.75, the adaptive search of preference values indisagreement is activated.

3) Adaptive Search for Preferences:a) Choose the most suitable PSp: Given that θ2 = 0.72 <

cr = 0.726 < γ = 0.75, the level of agreement is medium, andtherefore, PSpM is applied.

b) Apply the PSpM :i) Identifying the alternatives with consensus degree

not high enough

Xch = l|cal < 0.726 = x1 , x3.

ii) For each one of the aforementioned alternatives, thepreference values in disagreement are identified

P = (l, k)|l ∈ Xch, cplk < 0.726= (1, 3), (1, 4), (3, 1), (3, 2), (3, 4).

iii) Computing the proximity values at level of alterna-tives for these elements in Xch (Table IV).

iv) Computing the proximity thresholds used to iden-tify the experts required to modify their preferences

pa1 = 0.8 pa3 = 0.81.

v) Computing the proximity thresholds used to selectthe experts required to modify the identified pairs



of alternatives in disagreement

pp12 = 0.83 pp13 = 0.78 pp14 = 0.76pp21 = 0.81 pp23 = 0.82 pp24 = 0.76pp31 = 0.82 pp32 = 0.79 pp34 = 0.77pp41 = 0.83 pp42 = 0.86 pp43 = 0.85.

vi) Sets of preferences to be changed

PREFECH M1 = (3, 1), (3, 2)

PREFECH M6 = (1, 3), (1, 4), (3, 1), (3, 4)

PREFECH M3 = (3, 2)

PREFECH M8 = (3, 1), (3, 2), (3, 4).

4) Production of Advice:a) According to rule DR1, the experts are required to

increase the following preference assessments:

p321 = c1 → c2 p32

3 = a2 → a3

p316 = c0 → c1 p14

6 = c1 → c2 .

b) According to rule DR2, the experts are required todecrease the following preference assessments:

p311 = c3 → c2 p13

6 = c3 → c2

p346 = c4 → c3 p31

8 = b6 → b5

p328 = b5 → b4 p34

8 = b5 → b4 .

D. Third Round

1) Gathering Information (New Preferences):

Pe1 =

− c1 c0 c2c3 − c2 c4c2 c2 − c1c1 c2 c2 −

Pe3 =

− a2 a4 a3a4 − a7 a4a4 a3 − a2a4 a4 a6 −

Pe6 =

− c1 c2 c2

c3 − c1 c1c1 c3 − c3

c3 c3 c1 −

Pe8 =

− b5 b1 b3b1 − b1 b5b5 b4 − b4

b3 b2 b1 −

.

2) Computing the Consensus Degree and Control of the Con-sensus Process:

a) Consensus degree:i) Pairs of alternatives

CM =

− 0.77 0.8 0.77

0.73 − 0.73 0.730.82 0.8 − 0.760.77 0.8 0.78 −

.


).


0.756 > γ = 0.75, the desired level of consensus is achieved,and the selection process is applied.

Fig. 9. Experts’ preferences behavior during the consensus reaching process.

E. Graphic Description of the Experts’ Preferences Behavior

Fig. 9 illustrates graphically the experts’ preferences in thefirst and third rounds. Each point represents the preference valuegiven by each expert (in a different color) on a pair of alterna-tives. In this figure, the movements of the experts’ preferenceson the pairs (1, 3) and (3, 2) are highlighted by means of abox. We note that the preferences come closer each other, form-ing a group (some experts’ preferences cannot be seen becausethey are hidden by some other equal assessment), and thereforeincreases the level of agreement.

V. CONCLUSION

In this paper, we have analyzed the consensus processes inGDM problems under multigranular linguistic information, andhave proposed an ACSS model to guide it and reduce the numberof consensus rounds. To do so, different procedures to find outthe experts’ preference values furthest from the collective oneshave been defined. These procedures are applied according tothe level of agreement achieved in each consensus round. Withthis proposal, we overcome the convergence problems detectedin other consensus approaches existing in the literature [3], [7],[26], [33], [35], [44].

APPENDIX

This appendix contains the measurements to evaluate theagreement, i.e., consensus degrees and proximity measures.

Experts might use linguistic term sets with different car-dinality and semantics. To unify this information, each ex-pert’s linguistic preference plk

i is transformed in a fuzzy setplk

i = (αlki0 , . . . , α

lkig ). Some drawbacks related to the use of tra-

ditional distance measurements were pointed out in [35], and analternative similarity function s(·) was proposed to overcomethem. The similarity function takes as its arguments the cen-tral values of the fuzzy sets to compare. Given a fuzzy setplk

i = (αlki0 , . . . , α

lkig ), its central value defined as [15], [45],

cv(plki ) =

∑gh=0 hαlk

ih/∑g

h=0 αlkih represents the center of grav-

ity of the information contained in the fuzzy set.The similarity between two preference values, s(plk

i , plkj ) ∈

[0, 1], is defined as s(plki , plk

j ) = 1 −∣∣(cv(plk

i ) − cv(plkj ))/g

∣∣.The closer s(plk

i , plkj ) is to 1, the more similar plk

i and plkj are,

while the closer s(plki , plk

j ) is to 0, the more distant plki and plk

j

are.Consensus degrees measure the agreement between ex-

perts’ preferences. For each pair of experts i and j (i <j), a similarity matrix SMij =

(smlk

ij

)is calculated with



smlkij = s(plk

i , plkj ), l, k = 1, . . . , n ∧ l = k. The consen-

sus matrix, CM = (cmlk ), is obtained by aggregating all thesimilarity matrices. This aggregation is carried out at level ofpairs of alternatives: cmlk = φ(smlk

ij ), i, j = 1, . . . ,m, l, k =1, . . . , n ∧ i < j. In this paper, we use the arithmetic meanas aggregation function φ, although different aggregation op-erators could be used according to the particular properties toimplement.

Consensus degrees are obtained from the consensus matrixCM and in each one of the three different levels of the prefer-ence relation Pei .

Level 1 (Consensus on pairs of alternatives): The consensusdegree on a pair of alternatives (xl, xk ), called cplk , measuresthe agreement among all experts on that pair of alternatives:cplk = cmlk∀l, k = 1, . . . , n ∧ l = k.

Level 2 (Consensus on alternatives): The consensus de-gree on an alternative xl , called cal , measures theagreement among all experts on that alternative: cal =(∑n

k=1,l =k (cplk + cpkl))/2(n − 1).Level 3 (Consensus on the relation): The consensus degree on

the relation, called cr, measures the global agreement amongthe experts’ preferences: cr =

∑nl=1 cal/n.

Proximity measurements evaluate the proximity betweenthe individual experts’ preferences and the collective ones.The collective preference Pec = (plk

c ) is calculated by aggre-gating the set of (uniformed) individual preference relationsPe1 , . . . , Pem : plk

c = ψ(plk1 , . . . , plk

m ), with ψ, an “aggrega-tion operator.” Then, for each expert ei , we calculate a proxim-ity matrix PMi = (pmlk

i ), pmlki = s(plk

i , plkc ), which measures

the distance between Peiand Pec

.The proximity measurements are computed in each one of

the three levels of the relation, too.

Level 1 (Proximity on pairs of alternatives): Given an expert ei ,his/her proximity measure on a pair of alternatives (xl, xk ),called pplk

i , measures the proximity between his/her pref-erence and the collective one on that pair of alternatives:pplk

i = pmlki ∀l, k = 1, . . . , n ∧ l = k.

Level 2 (Proximity on alternatives): Given an expert ei ,his/her proximity measure on an alternative xl , calledpal , measures the proximity between his/her prefer-ence and the collective one on that alternative: pal

i =(∑n

k=1,k = l(pplki + ppkl

i ))/2(n − 1).Level 3 (Proximity on the relation): Given an expert ei , his/her

proximity measure on the relation, called pri , measures theglobal proximity between his/her individual preferences andthe collective one: pri = (

∑nl=1 pal

i)/n.

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Francisco Mata was born in 1971. He received theM.Sc. degree in computer sciences from the Univer-sity of Granada, Granada, Spain, in 1994, and thePh.D. degree in computer sciences from the Univer-sity of Jaen, Jaen, Spain, in 2006.

He is currently a Lecturer in the Department ofComputer Science, University of Jaen. His current re-search interests include linguistic preference model-ing, consensus reaching process, group decision mak-ing, and multiagent systems.

Luis Martınez was born in 1970. He received theM.Sc. and Ph.D. degrees in computer sciences, fromthe University of Granada, Granada, Spain, in 1993and 1999, respectively.

He is currently a Senior Lecturer in the Depart-ment of Computer Science, University of Jaen, Jaen,Spain. His current research interests include linguis-tic preference modeling, decision making, fuzzy-logic-based systems, computer-aided learning, sen-sory evaluation, recommender systems, and elec-tronic commerce. He was a coeditor of several journal

special issues on fuzzy preference modeling and fuzzy sets theory.

Enrique Herrera-Viedma was born in 1969. Hereceived the M.Sc. and Ph.D. degrees in computersciences from the University of Granada, Granada,Spain, in 1993 and 1996, respectively.

He is currently a Professor in the Department ofComputer Science and Artificial Intelligence, Uni-versity of Granada. He has authored or coauthoredmore than 50 papers published in international jour-nals. He has coedited one international book and 11special issues in international journals on topics suchas computing with words and preference modeling,

soft computing in information retrieval, and aggregation operators. He is a mem-ber of the Editorial Board of the Fuzzy Sets and Systems, Soft Computing, andInternational Journal of Information Technology and Decision Making. His cur-rent research interests include group decision making, decision support systems,consensus models, linguistic modeling, aggregation of information, informationretrieval, genetic algorithms, digital libraries, Web quality evaluation, and rec-ommender systems.



H∞ Fuzzy Filtering of Nonlinear SystemsWith Intermittent Measurements

Huijun Gao, Member, IEEE, Yan Zhao, James Lam, Senior Member, IEEE, and Ke Chen

Abstract—This paper is concerned with the problem of H∞fuzzy filtering of nonlinear systems with intermittent measure-ments. The nonlinear plant is represented by a Takagi–Sugeno(T–S) fuzzy model. The measurements transmission from the plantto the filter is assumed to be imperfect, and a stochastic variablesatisfying the Bernoulli random binary distribution is utilized tomodel the phenomenon of the missing measurements. Attentionis focused on the design of an H∞ filter such that the filter er-ror system is stochastically stable and preserves a guaranteed H∞performance. A basis-dependent Lyapunov function approach isdeveloped to design the H∞ filter. By introducing some slack ma-trix variables, the coupling between the Lyapunov matrix and thesystem matrices is eliminated, which greatly facilitates the filter-design procedure. The developed theoretical results are in the formof linear matrix inequalities (LMIs). Finally, an illustrative exam-ple is provided to show the effectiveness of the proposed approach.

Index Terms—Basis-dependent Lyapunov function, H∞ filterdesign, intermittent measurements, nonlinear systems, Takagi–Sugeno (T–S) fuzzy systems.

I. INTRODUCTION

IN RECENT years, there has been a growing interest in theTakagi–Sugeno (T–S) fuzzy model since it is a powerful so-

lution that bridges the gap between linear control and complexnonlinear systems [4], [36], [37]. The important advantage of theT–S fuzzy model is its universal approximation of any smoothnonlinear function by a “blending” of some local linear systemmodels. Based on that local linearity, many complex nonlinearproblems can be simplified by employing the Lyapunov functionapproach [9]. The earlier approach employs quadratic Lyapunovfunctions, which has shown great effectiveness and has beenwidely used up until now [5]–[7], [27], [30]. This approach at-tempts to find a common positive definite matrix to satisfy a set

Manuscript received May 8, 2007; revised July 16, 2007. First publishedApril 30, 2008; current version published April 1, 2009. This work was sup-ported in part by the National Natural Science Foundation of China (60825303,60834003), in part by the 973 Project (2009CB320600), in part by the ResearchFund for the Doctoral Programme of Higher Education of China (20070213084),in part by the Heilongjiang Outstanding Youth Science Fund (JC200809), in partby the Postdoctoral Science Foundation of China (200801282), in part by theFok Ying Tung Education Foundation (111064), and in part by HKU CRCG200707176077.

H. Gao is with the Space Control and Inertial Technology ResearchCenter, Harbin Institute of Technology, Harbin 150001, China (e-mail:[email protected]).

Y. Zhao is with the Space Control and Inertial Technology Research Center,Harbin Institute of Technology, Harbin, Heilongjiang 150001, China (e-mail:[email protected]).

J. Lam is with the Department of Mechanical Engineering, University ofHong Kong, Hong Kong (e-mail: [email protected]).

K. Chen is with the Department of Electrical and Computer Engineering,University of Alberta, Edmonton, AB T6G 2R3, Canada (e-mail: [email protected]).


of linear matrix inequalities (LMIs), which is recognized to beconservative, and for some highly nonlinear complex systems,the common Lyapunov matrix even does not exist [45]. Thishas motivated the development of the more recent approach,which employs basis-dependent Lyapunov functions. Results inmany papers have shown that this approach is less conserva-tive because the basis-dependent Lyapunov function is also a“blending” of some piecewise Lyapunov functions [15], [44].

Since the state variables in control systems are not alwaysavailable, state estimation is another important problem thathas been attracting attention from researchers around the world,and a great number of important results have been reported.To mention a few, the filtering problem has been solved forlinear systems for uncertain systems [35], Markovian jumpingsystems [3], [17], [25], [38], sample-data systems [21], [23],[33], systems with singular perturbation [18], [20], and systemswith time delays [11], [38]. Different norms have been used tomeasure the filtering performance (see, for instance, the H∞norm [12], [16], the L1 norm [1], and the L2–L∞ norm [19]).There are also some results investigating the filtering problemsfor nonlinear systems [10], [20], [24].

Among the aforementioned references, H∞ filtering is one ofthe most important strategies [14], [38], [41]. The advantage ofH∞ filtering lies in that no statistical assumption on the noisesignals is needed, and thus, it is more general than classicalKalman filtering [12]. Due to the powerful approximation prop-erty of T–S fuzzy model, recently, there have been a number ofresults on H∞ filtering for T–S fuzzy systems [8], [39], [45].A robust H∞ filter design for continuous T–S fuzzy modelsbased on the notion of quadratic stability proposed in [8], [13],and [45] are concerned with the H∞ filtering problem for a classof discrete-time fuzzy systems using basis-dependent Lyapunovfunctions with reduced conservatism. It is worth noting that allthese results are based on the implicit assumption that the com-munication between the physical plant and filter is perfect, thatis, the signals transmitted from the plant will arrive at the filtersimultaneously and perfectly.

On another research front, networked control systems havedrawn much attention due to their great advantages over tra-ditional systems such as low cost, reduced weight and powerrequirements, simple installation and maintenance, and high re-liability. However, the utilization of networks as communicationchannels brings us new challenges, and the analysis and syn-thesis problems become more difficult and complicated due totheir limited transmission capacity. Among a few other impor-tant problems, data packet dropout is an important issue to beaddressed. So far, there have been a number of results focusingon stability analysis of networked systems [29], [43]. Recently,

1063-6706/$25.00 © 2009 IEEE



increasing attention has been paid to the synthesis problems.For example, state-feedback control is investigated in [40], andH∞ control is developed in [22] and [42]. It is noted that mostof these results focus on the control-related problems. Morerecently, there have been a few results on the filtering problemfor networked systems: References [31] and [32] consider thefiltering problem for stochastic systems with missing measure-ments, and [26] investigates the problem of performing Kalmanfiltering with intermittent observations, while [11] and [34] dis-cuss that for stochastic systems with time delays. To the bestof the author’s knowledge, up until now, there has been no re-search on the filter design for T–S fuzzy systems in the presenceof intermittent measurements, which still remains important andchallenging. This motivates the present study.

In this paper, we investigate the problem of H∞ filter designfor nonlinear systems with intermittent measurements. The non-linear plant is represented by a T–S fuzzy model. The measure-ments transmitted between the plant and the filter are assumed tobe imperfect, and the phenomenon of the missing measurementsis assumed to satisfy the Bernoulli random binary distribution.Given a T–S fuzzy system, our objective is to design an H∞filter such that the filter error system is stochastically stable andpreserves a guaranteed H∞ performance. A basis-dependentLyapunov function approach is developed to design a desiredH∞ filter. The introduction of some slack matrix variables elim-inates the coupling between the system matrices and Lyapunovmatrix, which simplifies the filter design. The theoretical resultsare in the form of LMIs, which can be solved by standard nu-merical software. An example shows the effectiveness of theproposed approach.

The rest of this paper is organized as follows. Section IIformulates the problem under consideration. The stability con-dition and H∞ performance of the filter error system are givenin Section III. The filter design problem is solved in Section IV.An illustrative example is given in Section V, and we concludethe paper in Section VI.

The notation used in the paper is standard. The super-script “T ” stands for matrix transposition; R

n denotes the n-dimensional Euclidean space, and the notation P > 0 (≥ 0)means that P is real symmetric and positive definite (semidefi-nite). l2 [0,∞) is the space of square-integrable vector functionsover [0,∞); the notation | · | refers to the Euclidean vectornorm, and ‖ · ‖2 stands for the usual l2 [0,∞) norm. In sym-metric block matrices or complex matrix expressions, we usean asterisk (∗) to represent a term that is induced by symmetry,and diag. . . stands for a block-diagonal matrix. In addition,Ex and Ex| y will, respectively, mean expectation of xand expectation of x conditional on y. Matrices, if their dimen-sions are not explicitly stated, are assumed to be compatible foralgebraic operations.


The filtering problem with intermittent measurements isshown in Fig. 1, where the physical plant is represented bya T–S fuzzy model, and the data missing phenomenon occursintermittently from the plant to the filter. In the following, wemodel the whole problem mathematically.

Fig. 1. Filtering problem with intermittent measurements.

A. Physical Plant

The plant under consideration is a nonlinear discrete-timesystem that is represented by the T–S fuzzy model as follows:

1) Plant Rule i: IF θ1(k) is Mi1 and θ2(k) is Mi2 and · · ·and θp(k) is Mip , THEN

xk+1 = Aixk + Biwk

yk = Cixk + Diwk

zk = Lixk

i = 1, . . . , r. (1)

Here, Mij are the fuzzy sets, xk ∈ Rn is the state vector,

wk ∈ Rp is the noise signal that is assumed to be arbitrary sig-

nal in l2 [0,∞), zk ∈ Rq is the signal to be estimated, yk ∈ R

m

is the measurement output, Ai , Bi , Ci , Di , and Li are knownconstant matrices with appropriate dimensions, r is the num-ber of IF–THEN rules, and θ(k) = [ θ1(k), θ2(k), . . . , θp(k) ]is the premise variable vector and measurable. The fuzzy basisfunctions are given by

hi(θ(k)) =

∏pj=1 Mij (θj (k))∑r

i=1∏p

j=1 Mij (θj (k))

with Mij (θj (k)) representing the grade of membership of θj (k)in Mij . In what follows, we will drop the argument of hi(θk )for brevity. Therefore, for all k, we have

hi ≥ 0, i = 1, 2, . . . , r

r∑i=1

hi = 1. (2)

Let ρ be a set of basis functions satisfying (2). A more compactpresentation of the T–S discrete-time fuzzy model is given by

xk+1 = A(h)xk + B(h)wk

yk = C(h)xk + D(h)wk

zk = L(h)xk (3)

where

A(h) =r∑

i=1

hiAi, B(h) =r∑

i=1

hiBi, C(h) =r∑

i=1

hiCi

D(h) =r∑

i=1

hiDi, L(h) =r∑

i=1

hiLi (4)

and h= (h1 , h2 , . . . , hr ) ∈ ρ.


GAO et al.: H∞ FUZZY FILTERING OF NONLINEAR SYSTEMS WITH INTERMITTENT MEASUREMENTS 293

B. Filter

In this paper, we consider the following fuzzy filter to estimatezk .

1) Filter Rule i: IF θ1(k) is Mi1 and θ2(k) is Mi2 and · · ·and θp(k) is Mip , THEN

xk+1 = Af ixk + Bf iyf k

zk = Lf ixk

i = 1, . . . , r. (5)

Here, xk ∈ Rn , and zk ∈ Rq, and Af i , Bf i , and Lf i are to be

determined. Thus, the filter can be represented by the followinginput–output form:

xk+1 = Af (h)xk + Bf (h)yf k

zk = Lf (h)xk . (6)

C. Communication Link

It is assumed that measurements are intermittent, that is, thedata may be lost during their transmission. In this case, the inputyf k of the filter is no longer equivalent to the output yk of theplant (that is, yk = yf k ). In this paper, the data loss phenomenonis modeled via a stochastic approach:

yf k = e(k)yk

where e(k) is Bernoulli process. e(k) models the intermit-tent nature of the link from the plant to the filter. More specif-ically, e(k) = 0 when the link fails (that is, data is lost), ande(k) = 1 means successful transmission. A natural assumptionon e(k) can be made as

Probe(k) = 1= Ee(k)= e, Probe(k) = 0= 1 − e.


xk+1 = Af (h)xk + Bf (h)e(k)yk

zk = Lf (h)xk . (7)

D. Filter Error System

From (3) and (7), the filter error system is given by

xk+1 = A1(h)xk + e(k)A2(h)xk +B1(h)wk + e(k)B2(h)wk

zk = L(h)xk (8)

where

xk =[

xk

xk

], zk = zk − zk (9)

A1(h) =[

A(h) 0

eBf (h)C(h) Af (h)

], B1(h)=

[B(h)

eBf (h)D(h)

]

A2(h) =[

0 0

Bf (h)C(h) 0

], B2(h) =

[0

Bf (h)D(h)

]L(h) = [ L(h) −Lf (h) ] (10)

and e(k) = e(k) − e. It is clear that Ee(k) = 0 and thatEe(k)e(k) = e(1 − e).


Definition 1: The filter error system in (8) is said to be stochas-tically stable in the mean square when w(k) ≡ 0 for any initialcondition x0 if there exists a finite W > 0 such that

E

∞∑k=0

|xk |2∣∣∣∣∣x0

< xT

0 Wx0 .


Problem H∞ filtering with intermittent measurements(HFIM): Consider the filtering problem shown in Fig. 1, andsuppose the intermittent transmission parameter e is known.Given a scalar γ > 0, design a fuzzy filter in the form of (7)such that

1) (stochastic stability) the filter error system in (8) isstochastically stable in the sense of Definition 1;

2) (H∞ performance) under zero initial condition, the erroroutput zk satisfies

||z||E ≤ γ||w||2 (11)

where

||z||E= E

√√√√ ∞∑

k=0

zTk zk

.

If the previous two conditions are satisfied, the filter errorsystem is called stochastically stable with a guaranteed H∞performance γ.

III. FILTERING PERFORMANCE ANALYSIS

In this section, the filtering analysis problem is concerned.More specifically, we assume that the filter matrices in (6) areknown, and we will study the condition under which the filtererror system in (8) is stochastically stable in the mean squarewith a given H∞ performance γ. The following theorem showsthat the H∞ performance of the filter error system can be guar-anteed if there exist some fuzzy-basis-dependent matrices andadditional matrices satisfying a certain linear matrix inequality(LMI).

Theorem 1: Consider the fuzzy system in (3), and supposethat the filter in (6) is given. The filter error system in (8)is stochastically stable with a given H∞ performance γ, ifthere exist fuzzy-basis-dependent matrices P (h) > 0, Ω(h),

for any h ∈ ρ, h+ = (h1(θk+1), h2(θk+1), . . . , hr (θk+1)) ∈ ρ,

satisfying

Θ 0 0 ΩT (h+)A1(h) ΩT (h+)B1(h)

∗ Θ 0 fΩT (h+)A2(h) fΩT (h+)B2(h)

∗ ∗ −I L(h) 0

∗ ∗ ∗ −P (h) 0

∗ ∗ ∗ ∗ −γ2I

< 0

(12)



where

Θ = P (h+) − Ω(h+) − ΩT (h+)

f =√

e(1 − e).Proof: We first prove the stochastic stability of the filter er-

ror system in (8). To this end, assume wk ≡ 0, and choose aLyapunov function as

Vk = xTk [P (h)] xk . (13)

When wk ≡ 0, (8) becomes

xk+1 = A1(h)xk + e(k)A2(h)xk

zk = L(h)xk .

Then, we have

∆Vk = E Vk+1 | xk − Vk

= ExT

k

(AT

1 (h) + e(k)AT2 (h)

)P (h+)

× (A1(h) + e(k)A2(h))xk |xk

− xT

k P (h)xk

= xTk

(AT

1 (h)P (h+)A1(h) + f 2AT2 (h)P (h+)A2(h)

− P (h))xk .

Note that the inequality

[P (h+) − Ω(h+)]T P−1(h+)[P (h+) − Ω(h+)] ≥ 0

implies that

P (h+) − (Ω(h+) + ΩT (h+)) ≥ −ΩT (h+)P−1(h+)Ω(h+)

which together with (12) yields

Θ 0 0 ΩT (h+)A1(h) ΩT (h+)B1(h)

∗ Θ 0 fΩT (h+)A2(h) fΩT (h+)B2(h)

∗ ∗ −I L(h) 0

∗ ∗ ∗ −P (h) 0

∗ ∗ ∗ ∗ −γ2I

< 0

(14)where Θ = −ΩT (h+)P−1(h+)Ω(h+). Clearly, Ω(h+) is in-vertible. Diag Ω−T (h+),Ω−T (h+), I, I, I and postmultiply-ing diag Ω−1(h+),Ω−1(h+), I, I, I on the left and right sidesof (14), we obtained the following inequality:

−P−1(h+) 0 0 A1(h) B1(h)

∗ −P−1(h+) 0 fA2(h) fB2(h)

∗ ∗ −I L(h) 0

∗ ∗ ∗ −P (h) 0

∗ ∗ ∗ ∗ −γ2I

< 0

by Schur complement, which leads to[AT

1 (h) fAT2 (h) LT (h)

BT1 (h) fBT

2 (h) 0

]P (h+) 0 0

0 P (h+) 0

0 0 I

×

A1(h) B1(h)

fA2(h) fB2(h)

L(h) 0

−[

P (h) 0

0 γ2I

]< 0. (15)

Here, (15) implies

AT1 (h)P (h+)A1(h) + f 2AT

2 (h)P (h+)A2(h) − P (h) < 0

and thus, we have

∆Vk < 0.

Define

Φ= AT

1 (h)P (h+)A1(h) + f 2AT2 (h)P (h+)A2(h) − P (h)

and we get

E Vk+1 | xk − Vk ≤ −λmin(−Φ)xTk xk .

Taking mathematical expectation of both sides, for any T ≥ 1,and summing up the inequality on both sides from k = 0, . . . , T ,we have

E VT +1 − V0 ≤ −λmin(−Φ)E|xk |2

which implies

E|xk |2 ≤ (λmin(−Φ))−1(V0 − E VT +1).

Considering EV (k) ≥ 0 for all k ≥ 0, we have

E

∞∑k=0

|xk |2∣∣∣∣∣ x0

≤ (λmin(−Φ))−1 xT

0 max(P (h))x0

= xT0 (λmin(−Φ))−1 max(P (h))x0

= xT0 Wx0

where x0 is the initial condition, and W= (λmin(−Φ))−1

max(P (h)). According to Definition 1, the filter error systemis stochastically stable in the mean square.

Next, the H∞ performance criteria for the filter error sys-tem in (8) will be established. To this end, assume zero initialconditions. An index is introduced as

J = E Vk+1 | ξk + zTk zk − γ2wT

k wk − xTk P (h)xk

where

ξk =[

xk

wk

].

Since

E Vk+1 |ξk

= E

ξTk

[AT

1 (h) + e(k)AT2 (h)

BT1 (h) + e(k)BT

2 (h)

] [P (h+) 0

0 P (h+)

]× [ A1(h) + e(k)A2(h) B1(h) + e(k)B2(h) ] ξk | ξk

= E

ξTk

([AT

1 (h)

BT1 (h)

]P (h+) [ A1(h) B1(h) ]

+f 2[

AT2 (h)

BT2 (h)

]P (h+) [ A2(h) B2(h) ]

)ξk

∣∣∣∣ ξk

and

zTk zk = ξT

k

[LT (h)

0

][ L(h) 0 ] ξk



we have

J = ξTk

([AT

1 (h)

BT1 (h)

]P (h+) [ A1(h) B1(h) ]

+ f 2[

AT2 (h)

BT2 (h)

]P (h+) [ A2(h) B2(h) ]

+[

LT (h)

0

][ L(h) 0 ] −

[P (h) 0

0 γ2I

])ξk .

From inequality (15), we know that

J ≤ 0

that is

E Vk+1 | ξk + zTk zk − γ2wT

k wk − xTk P (h)xk ≤ 0.

Take mathematical expectation on both sides, we have

EVk+1 − EVk + EzTk zk

− γ2wT

k wk ≤ 0.

For k = 0, 1, 2, . . . , summing up both sides, consideringEV (k) ≥ 0 for all k ≥ 0, under zero initial condition, weobtain

E

∞∑k=0

zTk zk

−

∞∑k=0

γ2wTk wk ≤ 0

which is equivalent to (11). The proof is completed. Remark 1: If there is no data dropout in the channel between

the physical plant and the filter, that is, perfect communicationlinks exist between the plant and the filter, then we have thefollowing corollary, which can be proved by following similararguments, as in the proof of Theorem 1.

Corollary 1: Consider the fuzzy system in (3) and supposethat the filter in (6) is given. When e = 1, the filter error systemin (8) is stochastically stable with a given H∞ performance γ,if there exist fuzzy-basis-dependent matrices P (h) > 0, Ω(h),for any h, h+ ∈ ρ, satisfying

Θ 0 ΩT (h+)A1(h) ΩT (h+)B1(h)

∗ −I L(h) 0

∗ ∗ −P (h) 0

∗ ∗ ∗ −γ2I

< 0 (16)

where h+ and Θ are defined in (12).

IV. FILTER DESIGN

In this section, we will design a fuzzy filter in the form of (6)based on Theorem 1, that is, to determine the filter matrices in(6) such that the filter error system in (8) is stochastically stablewith a guaranteed H∞ performance. Since the condition in (12)cannot be utilized to obtain the filter directly, we introduce someslack matrices, which will simplify the filter design procedure.

Theorem 2: Consider the fuzzy system in (3). For a given pos-itive constant γ, if there exist fuzzy-basis-dependent matrices

Q(h) =[

Q1(h) Q2(h)

QT2 (h) Q3(h)

]> 0

and R,S,W, Af (h), Bf (h), Lf (h), for any h, h+ ∈ ρ satisfying[ϕ11 ϕ12

∗ ϕ22

]< 0 (17)

where

Ξ =[

Q1(h+) Q2(h+)

QT2 (h+) Q3(h+)

]−[

R + RT S + W

WT + ST W + WT

](18)

ϕ11 =

Ξ 0 0

∗ Ξ 0

∗ ∗ −I

, ϕ12 =

ϕ

(11)12 ϕ

(12)12

ϕ(21)12 ϕ

(22)12

ϕ(31)12 0

ϕ22 =

[ −Q1(h) −Q2(h)

−QT2 (h) −Q3(h)

]0

0 −γ2I

(19)

and

ϕ(11)12 =

[RT A(h) + eBf (h)C(h) Af (h)

ST A(h) + eBf (h)C(h) Af (h)

]

ϕ(12)12 =

[RT B(h) + eBf (h)D(h)

ST B(h) + eBf (h)D(h)

]

ϕ(21)12 = f

[Bf (h)C(h) 0

Bf (h)C(h) 0

], ϕ

(22)12 = f

[Bf (h)D(h)

Bf (h)D(h)

]ϕ

(31)12 = [L(h) −Lf (h) ] (20)

then there exists a fuzzy filter in the form of (6) such thatthe filtering error system in (8) is stochastically stable with aprescribed H∞ norm bound γ. Moreover, if the aforementionedcondition is satisfied, the matrices for the filter in (6) are givenby [

Af (h) Bf (h)

Lf (h) 0

]=[

Ω−T4 0

0 I

] [Af (h) Bf (h)

Lf (h) 0

]

×[

Ω−14 Ω3 0

0 I

](21)

where Ω3 and Ω4 can be obtained by the decomposition on W .Proof: Suppose that there exist matrices Q(h) > 0, R, S,

W, Af (h), Bf (h), and Lf (h) satisfying (17). From (17), weknow that W > 0. One can always find square and nonsingularmatrices Ω3 and Ω4 that W = ΩT

4 Ω−13 Ω4 . Let

R = Ω1 , S = Ω2Ω−13 Ω4

Ω =[

Ω1 Ω2

Ω4 Ω3

], T =

[I 0

0 Ω−13 Ω4

](22)

and

T−T

[Q1(h+) Q2(h+)

QT2 (h+) Q3(h+)

]T−1 =

[P1(h+) P2(h+)

PT2 (h+) P3(h+)

][

Af (h) Bf (h)

Lf (h) 0

]



=[

Ω−T4 0

0 I

] [Af (h) Bf (h)

Lf (h) 0

] [Ω−1

4 Ω3 0

0 I

]. (23)

By (22) and (23), one has

T−T ΞT−1 =[

P1(h+) P2(h+)

PT2 (h+) P3(h+)

]− Ω − ΩT . (24)

With the support of (10), (22), and (23), it can be verified that

TT ΩT A1(h)T

=[

ΩT1 A(h)+ eΩT

4 Bf (h)C(h) ΩT4 Af (h)Ω−1

3 Ω4

ΩT4 Ω−T

3 ΩT2 A(h)+ eΩT

4 Bf (h)C(h) ΩT4 Af (h)Ω−1

3 Ω4

]

=[

RT A(h) + eBf (h)C(h) Af (h)

ST A(h) + eBf (h)C(h) Af (h)

]fTT ΩT A2(h)T

= f

[ΩT

4 Bf (h)C(h) 0

ΩT4 Bf (h)C(h) 0

]= f

[Bf (h)C(h) 0

Bf (h)C(h) 0

]TT ΩT B1(h)

=[

ΩT1 B(h) + eΩT

4 Bf (h)D(h)

ΩT4 Ω−T

3 ΩT2 B(h) + eΩT

4 Bf (h)D(h)

]

=[

RT B(h) + eBf (h)D(h)

ST B(h) + eBf (h)D(h)

]fTT ΩT B2(h)

= f

[ΩT

4 Bf (h)D(h)

ΩT4 Bf (h)D(h)

]= f

[Bf (h)D(h)

Bf (h)D(h)

]L(h)T

= [L(h) −Lf (h)Ω−13 Ω4 ] = [L(h) −Lf (h) ] . (25)

Letting

Ω(h) = Ω, P (h) = T−T

[Q1(h) Q2(h)

QT2 (h) Q3(h)

]T−1 (26)

one can readily obtain from (24)–(26) that (17) is equivalent to

TT 0 0 0 0

0 TT 0 0 0

0 0 I 0 0

0 0 0 TT 0

0 0 0 0 I

×

Θ 0 0 ΩT (h+)A1(h) ΩT (h+)B1(h)

∗ Θ 0 fΩT (h+)A2(h) fΩT (h+)B2(h)

∗ ∗ −I L(h) 0

∗ ∗ ∗ −P (h) 0

∗ ∗ ∗ ∗ −γ2I

×

T 0 0 0 0

0 T 0 0 0

0 0 I 0 0

0 0 0 T 0

0 0 0 0 I

< 0

where Θ is defined in (12), which together with (17) impliesthat, for any h, h+ ∈ ρ, (12) holds.

The proof is completed. The condition in (17) cannot be directly employed for filter

design. One way to facilitate Theorem 2 for the construction of afuzzy filter is to convert (17) into a finite set of LMI constraints.To this end, one must further restrict the choice of the fuzzy-basis-dependent Lyapunov functions. The following theoremgives a possible way to achieve this.

Theorem 3: Consider the fuzzy system in (3). For a given

positive constant γ, if there exist matrices Qi =[

Q1i Q2i

QT2i Q3i

]>

0, and R, S, W, Af i , Bf i , Lf i , for all i, j, l ∈ 1, . . . , rsatisfying

[ψ11 ψ12∗ ψ22

]< 0 (27)

where

Ξ =[

Q1l Q2l

QT2l Q3l

]−[

R + RT S + W

WT + ST W + WT

](28)

ψ11 =

Ξ 0 0

∗ Ξ 0

∗ ∗ −I

,

ψ22 =

[−Q1i −Q2i

−QT2i −Q3i

]0

0 −γ2I

ψ12 =

[RT Ai + eBf iCj Af i

ST Ai + eBf iCj Af i

] [RT Bi + eBf iDj

ST Bi + eBf iDj

]f

[Bf iCj 0

Bf iCj 0

]f

[Bf iDj

Bf iDj

][ Li −Lf i ] 0

(29)

then there exists a fuzzy filter in (6) such that the filter er-ror system in (8) is stochastically stable with a prescribed H∞norm bound γ. Moreover, if the earlier condition is satisfied, thematrices for the filter in (6) are given by

[Af (h) Bf (h)

Lf (h) 0

]=

r∑i=1

hi

[Ω−T

4 0

0 I

] [Af i Bf i

Lf i 0

]

×[

Ω−14 Ω3 0

0 I

](30)

where Ω3 and Ω4 can be obtained by the decomposition on W .



Proof: Suppose that there exist matrices R, S, W, Af i , Bf i ,

Lf i , and Qi =[

Q1i Q2i

QT2i Q3i

]> 0, for all i, j, l ∈ 1, . . . , r

satisfying (17). Then, we use these matrices and the fuzzy basisfunction h ∈ ρ to define the following functions:

Q(h) =r∑

l=1

hl

[Q1l Q2l

QT2l Q3l

], Af (h) =

r∑i=1

hiAf i

Bf (h) =r∑

i=1

hiBf i , Lf (h) =r∑

i=1

hiLf i

which together with (4) imply that[ϕ11 ∗ϕ21 ϕ22

]=

r∑i=1

r∑j=1

r∑l=1

hihjh+l

[ψ11 ∗ψ21 ψ22

]

and (17) is clearly verified, where h+l is hl(θk+1), as is defined

in (12), and ϕ11 , ϕ21 , ϕ22 , ψ11 , ψ21 , and ψ22 are defined as in(17), (19), (27), and (29).

Corollary 2: Consider the fuzzy system in (3). When e = 1,for a given positive constant γ, if there exist matrices Qi =[

Q1i Q2i

QT2i Q3i

]> 0, and R, S, W, Af i , Bf i , and Lf i, for all i, j,

l ∈ 1, . . . , r satisfying[Π11 Π12

∗ Π22

]< 0 (31)

where

Π11 =[

Ξ 0

0 −I

], Π22 =

[−Q1i −Q2i

−QT2i −Q3i

]0

0 −γ2I

Π12 =

[RT Ai + Bf iCj Af i

ST Ai + Bf iCj Af i

] [RT Bi + Bf iDj

ST Bi + Bf iDj

][ Li −Lf i ] 0

(32)

and Ξ is defined in (28), then there exists a fuzzy filter in theform of (6), and the filter error system in (8) is stochasticallystable with a prescribed H∞ norm bound γ. Moreover, if theprevious condition is satisfied, the matrices for the filter in (6)are given by (30).

Remark 2: Theorem 3 is obtained by using the basis-dependent Lyapunov function. It is clear that when Qi = Qfor any i ∈ 1, . . . , r, (13) becomes the quadratic Lyapunovfunction that has been widely used in the literature. Then,the following corollary based on the quadratic approach isobtained.

Corollary 3: Consider the fuzzy system in (3). For a given

positive constant γ, if there exist matrices Q =[

Q1 Q2QT

2 Q3

]>

0, R, S,W, Af i , Bf i , and Lf i , for all i, j ∈ 1, . . . , r satisfying[ψ11 ψ12∗ ψ22

]< 0 (33)

Fig. 2. Tunnel diode circuit.

where

Ξ =[

Q1 Q2

QT2 Q3

]−[

R + RT S + W

WT + ST W + WT

]

ψ11 =

Ξ 0 0

∗ Ξ 0∗ ∗ −I

, ψ22 =

[ −Q1 −Q2

−QT2 −Q3

]0

0 −γ2I

and ψ12 is defined in (29), then there exists a fuzzy filter in theform of (6) such that the filter error system in (8) is stochasticallystable with a prescribed H∞ norm bound γ. Moreover, if theearlier condition is satisfied, the matrices for the filter in (6) aregiven by (30).

Remark 3: Theorem 3 is obtained by restricting the fuzzy-basis-dependent Lyapunov functions. The expression of fuzzy-basis-dependent Lyapunov functions adopted here is consistentwith the compact presentation of system matrices in (3), which isadopted by mostc of the literature. This fuzzy-basis-dependentLyapunov approach has been recognized to be less conservative.However, in this basis-dependent framework, the restriction onthe Lyapunov function still introduces some overdesign. How tofurther reduce this conservatism still needs further investigation.

Remark 4: The number of inequalities in Theorem 3 willincrease with the number of fuzzy rules of the model, thus;a computational problem might arise for high-order nonlinearsystems. One effective way to solve this problem is to try toreduce the number of fuzzy rules when modeling the nonlinearsystem based on fuzzy logic, which can be found in [28].

V. ILLUSTRATIVE EXAMPLE

In this section, we use an example to illustrate the effective-ness of the theoretical results developed before.

Consider a tunnel diode circuit shown in Fig. 2, whose fuzzymodeling was done in [2], where x1(t) = vC (t), x2(t) = iL (t),w(t) is the disturbance noise input, y(t) is the measurementoutput, and z(t) is the controlled output. With a sampling timeT = 0.02, the discrete-time model is obtained as

xk+1 = A(h)xk + B(h)wk

yk = C(h)xk + D(h)wk

zk = L(h)xk (34)



where

A1 =[

0.9887 0.9024

−0.0180 0.8100

], B1 =

[0.0093

0.0181

]

A2 =[

0.90337 0.8617

−0.0172 0.8103

], B2 =

[0.0091

0.0181

]C1 = [ 1 0 ] , C2 = [ 1 0 ]

D1 = 1, D2 = 1, L1 = [ 1 0 ] , L2 = [ 1 0 ] .

To show the effectiveness of the obtained results, we assume themembership function to be

h1 =

x(1)k + 3

3, −3 ≤ x

(1)k ≤ 0

0, x(1)k < −3

3 − x(1)k

3, 0 ≤ x

(1)k ≤ 3

0, x(1)k > 3

h2 = 1 − h1 . (35)

The purpose here is to design a filter in the form of (5) such thatthe system in (34) is stochastically stable with a guaranteed H∞norm bound γ.

Suppose e = 0.8. By solving LMI (27), the minimum H∞performance γ∗ = 0.1463 is obtained, and the filter matricesare obtained:

Af 1 =[

4.9722 14.2575

9.1080 69.4648

], Bf 1 =

[−0.3451

−1.4883

]

Af 2 =[

4.6054 13.9920

8.4237 67.9560

], Bf 2 =

[−0.2005

−0.8318

](36)

Lf 1 = [−1.0000 −0.0002 ] (37)

Lf 2 = [−0.9955 0.0152 ] (38)

W =[

5.5335 11.5195

11.5162 71.9180

]. (39)

By (30), we have

Af 1 =[

0.9524 0.8488

−0.0259 0.8300

], Bf 1 =

[−0.0289

−0.0161

]

Af 2 =[

0.8827 0.8423

−0.0242 0.8100

], Bf 2 =

[−0.0182

−0.0086

]Lf 1 = [−1.0000 −0.0002 ]

Lf 2 = [−0.9955 0.0152 ] .

First, we assume that wk ≡ 0 and the that initial conditionx0 = [ 0.2 −0.8 ] , x0 = [ 0 0 ] . Fig. 3 shows that the esti-mation error response converges to zero.

Fig. 3. Estimation error when wk ≡ 0.

Fig. 4. Data packet dropout.

To illustrate the performance of the designed filter, we assumethe initial conditions and the external disturbance w(k) to be

w(k) =

2, 30 ≤ k ≤ 50

−2, 70 ≤ k ≤ 100

0, elsewhere.

(40)

In the simulation, the data packet dropouts are generated ran-domly according to e = 0.8, which is shown in Fig. 4. Fig. 5shows the response of signal z(k). Fig. 6 shows the simu-lation results of zk and zk . By calculation, we obtain that||z||22 = 1.0881 and ||w||22 = 208, which yields γ = 0.0723 (be-low the minimum γ∗ = 0.1463), showing the effectiveness ofthe filter design.



Fig. 5. Estimation error.

Fig. 6. Estimation signals.

VI. CONCLUDING REMARKS

In this paper, the problem of H∞ fuzzy filtering of non-linear systems under unreliable communication links has beeninvestigated. The T–S fuzzy system is utilized to model thenonlinear plant, and the communication link failure is modeledvia a stochastic variable satisfying the Bernoulli random binarydistribution. The basis-dependent Lyapunov function has beenused to design an H∞ filter such that the filter error systemis stochastically stable and preserves a guaranteed H∞ perfor-mance. Some slack matrices have been introduced to facilitatethe H∞ filter design. An example has been given to illustratethe effectiveness of the proposed approach.

It should be noted that in practical networked control systems,in addition to data missing, the phenomenon of transmissiondelay often occurs. It can also degrade the performance of thesystems and even cause system instability. In this paper, wehave only considered data missing, but the study of networked

control of fuzzy systems with simultaneous consideration ofpacket dropout and signal delay deserves further investigation.

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From November 2003 to August 2004, he wasa Research Associate in the Department of Mechan-ical Engineering, University of Hong Kong, HongKong. In November 2004, he joined Harbin Insti-tute of Technology, where he is currently a Professor.

From October 2005 to October 2007, he was a Postdoctoral Researcher in theDepartment of Electrical and Computer Engineering, University of Alberta,Edmonton, AB, Canada. He is an Associate Editor of the Journal of Intelligent

and Robotics Systems, Circuits, System and Signal Processing, etc. He was anoutstanding reviewer for the Automica in 2007. His current research interestsinclude network-based control, robust control, and time-delay systems and theirindustrial applications.

Prof. Gao is an Associate Editor of the IEEE TRANSACTIONS ON SYSTEMS,MAN AND CYBERNETICS PART B: CYBERNETICS and the IEEE TRANSACTIONS

ON INDUSTRIAL ELECTRONICS. He was an outstanding reviewer for the IEEETRANSACTIONS ON AUTOMATIC CONTROL in 2008, and an appreciated reviewerfor the IEEE TRANSACTIONS ON SIGNAL PROCESSING in 2006. He was the re-cipient of the University of Alberta Dorothy J. Killam Memorial PostdoctoralFellow Prize in 2005, the National Outstanding Youth Science Fund in 2008,and the National Outstanding Doctoral Thesis Award in 2007. He was the core-cipient of the National Natural Science Award of China in 2008.

Yan Zhao received the B.S. degree in chemical engi-neering and equipment control and the M.S. degreein mechanical engineering from the Inner MongoliaUniversity of Technology, Hohhot, China, in 2002and 2005, respectively. She is currently working to-ward the Ph.D. degree in control science and engi-neering with Harbin Institute of Technology, Harbin,China.


James Lam (S’86–M’87–SM’99) received the B.Sc.degree (with first class) in mechanical engineer-ing from the University of Manchester, Manchester,U.K., in 1983, and the M.Phil. and Ph.D. degreesin control engineering from the University of Cam-bridge, Cambridge, U.K., in 1985 and 1988, respec-tively. He received the Ashbury Scholarship, the A.H.Gibson Prize, and the H. Wright Baker Prize for hisacademic performance.

He is currently a Professor in the Department ofMechanical Engineering, University of Hong Kong,

Hong Kong. He is an Associate Editor of the Asian Journal of Control, Interna-tional Journal of Systems Science, Journal of Sound and Vibration, InternationalJournal of Applied Mathematics and Computer Science, Journal of the FranklinInstitute, Dynamics of Continuous, Discrete and Impulsive Systems (Series B:Applications and Algorithms), and Automatica. He is also a member of theEditorial Board of the Institute of Engineering and Technology (IET) ControlTheory and Applications, Open Electrical and Electronic Engineering Journal,Research Letters in Signal Processing, International Journal of Systems, Controland Communications, and Journal of Electrical and Computer Engineering. Hiscurrent research interests include reduced-order modeling, delay systems, de-scriptor systems, stochastic systems, multidimensional systems, robust control,and filtering. He was an Editor-in-Chief of the Institute of Electrical Engineers(IEE) Proceedings Control Theory and Applications.

Prof. Lam is a Chartered Mathematician and a Chartered Scientist. He is aFellow of the Institute of Mathematics and Its Applications, and the IET. He isa Scholar and a Fellow of the Croucher Foundation. He is an Associate Editorof the IEEE TRANSACTIONS ON SIGNAL PROCESSING.

Ke Chen received the B.S. degree in mathematics in2002 and the M.S. degree in bioinformatics in 2005from Nankai University, Tianjin, China. He is cur-rently working toward the Ph.D. degree in electricaland computer engineering with the University of Al-berta, Edmonton, AB, Canada.

His current research interests include the applica-tion of mathematical models in biological sciences.



An Interval Type-2 Fuzzy Rough Set Modelfor Attribute Reduction

Haoyang Wu, Yuyuan Wu, and Jinping Luo

Abstract—Rough set theory is a very useful tool for describingand modeling vagueness in ill-defined environments. Traditionalrough set theory is restricted to crisp environments. However,nowadays, it has been extended to fuzzy environments, resulting inthe development of the so-called fuzzy rough sets. Type-2 fuzzy setspossess many advantages over type-1 fuzzy sets, but for the gen-eral type-2 fuzzy sets, the computational complexity is severe. Onthe other hand, set-theoretic and arithmetic computations for theinterval type-2 fuzzy sets are very simple. Motivated by the afore-mentioned accomplishments, in this paper, the concept of fuzzyrough sets is generalized to interval type-2 fuzzy environments.Subsequently, a method of attribute reduction within the intervaltype-2 fuzzy rough set framework is proposed. Lastly, the proper-ties of the interval type-2 fuzzy rough sets are presented.

Index Terms—Fuzzy rough sets, interval type-2 fuzzy sets, roughapproximation, rough sets, type-2 fuzzy sets.

I. INTRODUCTION

ROUGH set theory was proposed by Pawlak in 1982 [1]. Itis a very useful tool to detect the data dependencies and re-

duce the number of attributes contained in a data set. Since then,the rough set theory has attracted a lot of researchers around theworld, and has been applied to many domains [3]–[14]. Its suc-cess is partially owing to the following properties: 1) only thefacts hidden in the data are analyzed; 2) no additional infor-mation about the data is required, such as thresholds or expertknowledge; and 3) given a data set with discretized attributevalues, it is possible to find a subset (termed as reduct) withinthe original attributes that are the most informative [4]. Fuzzyset theory was proposed by Zadeh in 1965 [2]. It extended theclassical notion of set and permitted the gradual assessment ofthe membership of elements in a set. To date, fuzzy set the-ory has shown enormous promise in many disciplines, such asartificial intelligence, control systems and so on [15], [16].

It is generally accepted that theories of fuzzy and rough setsare related but distinct, and they are complementary [10]. Tra-ditional rough set theory is based on an equivalent relation.However, the values of attributes may be both symbolic andreal-valued. In other words, the traditional rough set theory will

Manuscript received September 6, 2007; revised June 29, 2008, September17, 2008, and December 10, 2008; accepted December 17, 2008. First publishedJanuary 27, 2009; current version published April 1, 2009.

H. Wu and Y. Wu are with the State Key Laboratory of MultiphaseFlow in Power Engineering, School of Energy and Power Engineering, Xi’anJiaotong University, Xi’an 710049, China (e-mail: [email protected];[email protected]).

J. Luo is with the Kunlun Technology Industry Corporation, Hangzhou310012, China (e-mail: [email protected]).



have difficulties in handling such values mentioned earlier [4].For example, two close real values may only differ as a resultof noise; however, in traditional rough set theory, one of themmay be considered different from the other in a different orderof magnitude. A possible solution is to discretize the data setbeforehand and obtain a new crisp data set, but this method alsogenerates information loss [4].

To overcome this shortcoming, some methods that can ex-tend the traditional rough set theory to deal with real-valueddomains have been proposed [3], [17], [18]. Among thesemethods, hybridization of fuzzy and rough sets can lead oneto consider not only rough approximations of fuzzy sets, butalso approximations of sets by means of similarity relationsor fuzzy partitions. Dubois and Prade [3] claimed that thesehybrid notions came up in a natural way when a linguistic cat-egory, denoting a set of objects, had to be approximated interms of already existing labels, or when the indiscernibilityrelation between the objects no longer obeyed ideal laws ofequivalence and was a matter of degree. According to Jensenand Shen [6], fuzzy rough sets offer a high degree of flexi-bility, providing robust solutions and advanced tools for dataanalysis.

On the other hand, there exist some disadvantages in usingfuzzy sets for reasoning from the data, and the hybridization offuzzy and rough sets may help to deal with them. According toLiu et al. [11], when a fuzzy implication inference rule and apremise are given, only one fixed value that acts as the final rea-soning consequence can be obtained. However, in many cases,there exist some special constraints that must be satisfied. Liuet al. [11] studied the fuzzy reasoning based on fuzzy rough sets.According to their research, the fuzzy reasoning consequencecan be expressed by a fuzzy interval, and a suitable value canbe selected from the fuzzy interval as a final reasoning conse-quence. Thus, it is possible to satisfy some special constraints.

So far, many studies have been carried out in the field offuzzy rough sets. For example, Dubois and Prade [3] combinedfuzzy and rough sets by proposing a definition for rough fuzzysets and fuzzy rough sets. Shen and Jensen [8] proposed an ap-proach that integrated a fuzzy rule induction algorithm with afuzzy rough method for feature selection. Tsang et al. [9] es-tablished solid mathematical foundations for attribute reductionwith fuzzy rough sets. Furthermore, Yeung et al. [10] inves-tigated the generalization of fuzzy rough sets, while Wu andZhang [12] proposed constructive and axiomatic approaches offuzzy approximation operators. Lingras and Jensen [13] pre-sented a review on the fuzzy and rough hybridization in su-pervised learning, information retrieval, feature selection, andneural and evolutionary computing.

1063-6706/$25.00 © 2009 IEEE



To date, most of the research works in the field of fuzzyrough sets are restricted to ordinary (type-1) fuzzy environ-ments. Recently, Wu et al. [14] proposed a measure of rough-ness of a type-2 fuzzy set based on similarity relation. Jensenand Shen [7] proposed an interval-valued approach for the fuzzyrough feature selection, which could not only handle the missingvalues effectively, but also handle the uncertainty that could notbe modeled by a type-1 approach. According to Mendel [20],there exist at least four sources of uncertainties in type-1 fuzzylogic systems (T1 FLS), which are as follows: 1) meanings ofthe words that are used in the antecedents and consequents ofrules can be uncertain (words mean different things to differentpeople); 2) consequents may have a histogram of values associ-ated with them, especially when knowledge is extracted from agroup of experts, all of whom do not collectively agree; 3) mea-surements that activate a T1 FLS may be noisy and thereforeuncertain; and 4) the data that are used to tune the parametersof a T1 FLS may also be noisy. All these uncertainties lead touncertain fuzzy-set membership functions.

Ordinary type-1 fuzzy sets cannot model such uncertaintiesdirectly, because they are characterized by crisp membershipfunctions. However, type-2 fuzzy sets that are characterized bytype-1 fuzzy membership functions are capable of modelingthe four uncertainties, because their membership functions arethemselves fuzzy. In the past decades, many researchers inves-tigated type-2 fuzzy sets and their properties, e.g., Dubois andPrade [19], Mendel [20], Karnik and Mendel [21], Mendel andJohn [22], and so on. According to Liang and Mendel [23] andMendel et al. [24], most people only use interval type-2 fuzzysets in practical fields because of the computational complexityin using general type-2 fuzzy sets.

Motivated by the aforementioned accomplishments, in thispaper, the concept of fuzzy rough sets proposed by Dubois andPrade [3] is generalized to interval type-2 (IT2) fuzzy envi-ronments. Section II of this paper recalls the preliminaries oftype-2 fuzzy sets and fuzzy rough sets, while Section III givesthe definitions of IT2 fuzzy rough sets. Section IV proposesa method of attribution reduction based on the concept of IT2fuzzy rough sets. Lastly, Section V draws the conclusions. Fur-thermore, the properties of IT2 fuzzy rough sets are given in theAppendix.

II. PRELIMINARIES

A. Type-2 Fuzzy Sets

In this section, the preliminaries of type-2 fuzzy sets given byMendel [20], Karnik and Mendel [21], and Mendel and John [22]are briefly recalled, while some notations are modified.

1) Definitions of Type-2 Fuzzy Sets:Definition 1: Let U be a finite and nonempty set, which

is referred to as the universe, I = [0, 1]. A type-2 fuzzy set,denoted as A, is characterized by a type-2 membership functionµA (x, u) : U × I → I , where x ∈ U and u ∈ Jx ⊆ I , i.e.

A ≡ ((x, u), µA (x, u)) | x ∈ U, u ∈ Jx ⊆ I (1)

where 0 ≤ µA (x, u) ≤ 1. A can also be expressed as

A ≡∫

x∈U

∫u∈Jx

µA (x, u)/(x, u)

≡∫

x∈U

[∫u∈Jx

fx(u)/u

] /x, Jx ⊆ [0, 1] (2)

where fx(u) = µA (x, u).The class of all type-2 fuzzy sets of the universe U is denoted

as F (U) in the following sections.Definition 2: At each value of x, say x = x′, the 2-D plane

whose axes are u and µA (x′, u) is called the vertical slice ofµA (x, u). A secondary membership function is a vertical sliceof µA (x, u). It is µA (x = x′, u) for x′ ∈ U and ∀u ∈ Jx ′ ⊆ I ,i.e.

µA (x = x′, u) ≡ µA (x′) ≡∫

u∈Jx ′

fx ′(u)/u, Jx ′ ⊆ I (3)

where 0 ≤ fx ′(u) ≤ 1. The amplitude of a secondary member-ship function is called a secondary grade. In formula (2), fx(u)and µA (x, u) are all secondary grades.

Definition 3: The domain of a secondary membership func-tion is called the primary membership of x. In formula (3), Jx ′

is the primary membership of x′. Uncertainty in the primarymemberships of a type-2 fuzzy set A consists of a bounded re-gion that is called the footprint of uncertainty (FOU), which isthe union of all primary memberships, i.e.

FOU(A) ≡⋃x∈U

Jx. (4)

In the following sections, FOU(A) is also denoted as DA.Let DA(x) ≡ Jx ∀x ∈ U . Therefore, formula (2) can be re-expressed as

A =∫ ∫

(x,u)∈DA

µA (x, u)/(x, u). (5)

For any A ∈ F (U), a lower and an upper membership func-tion are the two type-1 membership functions that are the boundsfor DA. The lower membership function (denoted as DA) is as-sociated with the lower bound of DA, and the upper membershipfunction (denoted as DA) is associated with the upper bound ofDA.

Definition 4: If all the secondary grades of a type-2 fuzzy setA are equal to 1, i.e. µA (x, u) = 1 ∀x ∈ U,∀u ∈ Jx ⊆ I , thenA is defined as an interval type-2 fuzzy set. In the followingsections, the class of all interval type-2 (IT2) fuzzy sets of theuniverse U is denoted as FIT2(U).

Fig. 1(a) illustrates two IT2 fuzzy sets N and Z. The shadedregions in Fig. 1(a) are FOU . Fig. 1(b) illustrates an example ofsecondary membership function µN (−0.3), whose correspond-ing primary membership is [0.4, 0.6].

For any X ∈ FIT2(U) and x ∈ U, µX (x) is an interval type-1(IT1) fuzzy set on I . Therefore, DX(x) can be expressed as arange [lX (x), rX (x)] ⊆ I and DX =

⋃x∈U DX(x). Accord-

ing to formula in [20, B-9], type-2 fuzzy set inclusion is definedas follows.


WU et al.: INTERVAL TYPE-2 FUZZY ROUGH SET MODEL FOR ATTRIBUTE REDUCTION 303

Fig. 1. IT2 fuzzy sets corresponding to the information system given in Table I.(a) Two IT2 fuzzy sets. (b) Secondary membership function µN (−0.3).

Definition 5: For X, Y ∈ F (U), X ⊆ Y ⇔ µX (x) ≤ µY (x)∀x ∈ U .

Proposition 1: For X, Y ∈ FIT2(U), X ⊆ Y ⇔ DX ⊆DY ⇔ µDX (x) ≥ µDY (x) and µDX (x) ≤ µDY (x), x ∈ U .

Proof: For X, Y ∈ FIT2(U) and x ∈ U, µX (x) and µY (x)are IT1 fuzzy sets on I . Thus, µX (x) ≤ µY (x) is equiv-alent to DX(x) ⊆ DY (x). Therefore, X ⊆ Y ⇔ µX (x) ≤µY (x), x ∈ U ⇔ DX(x) ⊆ DY (x), x ∈ U ⇔ DX ⊆ DY .

On the other hand, DX ⊆ DY ⇔ µDX (x) ≥ µDY (x) andµDX (x) ≤ µDY (x), x ∈ U .

Therefore, X ⊆ Y ⇔ µDX (x) ≥ µDY (x) and µDX (x) ≤µDY (x), x ∈ U .

2) Operations of Type-2 Fuzzy Sets: Let U be a nonemptyuniverse, A, B ∈ F (U)

A =∫

x∈U

µA (x)/x =∫

x∈U

[∫u∈J u

x

fx(u)/u

]/x, Ju

x ⊆ I

B =∫

x∈U

µB (x)/x =∫

x∈U

[∫w∈J w

x

gx(w)/w

]/x, Jw

x ⊆I.

The union and intersection for type-2 fuzzy sets are definedas follows:

a) Union of two type-2 fuzzy sets A ∪ B

µA∪B

(x) ≡∫

u∈J ux

∫w∈J w

x

[fx(u)gx(w)]/(u ∨ w)

≡ µA (x) µB (x), x ∈ U (6)

where is the minimum or algebraic product t-normfor fuzzy intersection, ∨ is the maximum operation, iscalled the join operation, µ

A∪B(x), µA (x), and µB (x) are

the secondary membership functions, and all are type-1fuzzy sets.

b) Intersection of two type-2 fuzzy sets A ∩ B

µA∩B

(x) ≡∫

u∈J ux

∫w∈J w

x

[fx(u)gx(w)]/(u ∧ w)

≡ µA (x) µB (x), x ∈ U (7)

where ∧ is the minimum operation, is called the meet oper-ation, µ

A∩B(x), µA (x), and µB (x) are the secondary member-

ship functions, and all are type-1 fuzzy sets.3) Operations of IT2 Fuzzy Sets: Let U be a

nonempty universe, X1 , . . . , Xn ∈ FIT2(U). For anyx ∈ U, µX 1

(x), . . . , µXn(x) are vertical slices and all are

IT1 fuzzy sets. For simplicity, let µX 1(x), . . . , µXn

(x) bedenoted as F1 , . . . , Fn . Correspondingly, their domains are[l1 , r1 ], . . . , [ln , rn ] ⊆ I .

a) Union of IT2 fuzzy sets X1 ∪ . . . ∪ Xn : The join ni=1Fi

of n IT1 fuzzy sets F1 , . . . , Fn is an IT1 fuzzy set with do-main [(l1 ∨ . . . ∨ ln ), (r1 ∨ . . . ∨ rn ], where ∨ is the max-imum operation

µX 1 ∪...∪Xn(x) ≡ F1 . . . Fn

=∫

q∈[(l1 ∨...∨ln ),(r1 ∨...∨rn )]1/q, x ∈ U. (8)

b) Intersection of IT2 fuzzy set X1 ∩ . . . ∩ Xn : The meetn

i=1Fi of n IT1 fuzzy sets F1 , . . . , Fn is an IT1 fuzzyset with domain [(l1 . . . ln ), (r1 . . . rn )], where denotes either the minimum or algebraic product t-norm.In the following sections, is specified as the minimumt-norm

µX 1 ∩...∩Xn(x) ≡ F1 . . . Fn

=∫

q∈[(l1 ∧...∧ln ),(r1 ∧...∧rn )]1/q, x ∈ U. (9)

B. Fuzzy Rough Sets

The concepts of vagueness (for fuzzy sets) and indiscerni-bility (for rough sets) are related but distinct concepts, bothof which are caused by uncertainties in knowledge or data.Fuzzy rough sets encapsulate these two concepts, and there-fore, are adequate to deal with the uncertainties. Dubois andPrade [3] pioneered the concept of fuzzy rough sets. Theyconstructed a pair of lower and upper approximation oper-ators of fuzzy sets with respect to a fuzzy similarity rela-tion by using the t-norm Min and its dual conorm Max.Here, the preliminaries of fuzzy rough sets proposed byDubois and Prade [3] are briefly recalled with some modifiednotations.

Definition 6: Let U be a nonempty universe, R be a fuzzybinary relation on U , and F(U) be the fuzzy power set of U .For any A ∈ F(U), a fuzzy rough set is a pair (R∗(A), R∗(A))



such that for every x ∈ U

R∗(A)(x) ≡ infy∈U

max1 − R(x, y), µA (y)

= ∧y∈U

[(1 − R(x, y)) ∨ µA (y)] (10)

R∗(A)(x) ≡ supy∈U

minR(x, y), µA (y)

= ∨y∈U

[R(x, y) ∧ µA (y)]. (11)

According to Dubois and Prade [3], a fuzzy rough set repre-sents the distortion of a fuzzy set A owing to the indiscernibilityrelation on U , as if looking at A with blurring glasses. For anyA,B ∈ F(U) and x, y ∈ U , the approximation operators R∗and R∗ have the following properties:

(FR1) R∗Ac = (R∗A)c , R∗Ac = (R∗A)c

(FR2) R∗(A ∩ B) = R∗A∩R∗B,R∗(A∪B) = R∗A ∪ R∗B(FR3) R∗A ⊆ A,A ⊆ R∗A(FR4) R∗(U − y)(x) = R∗(U − x)(y),

R∗x1(y) = R∗y1(x)(FR5) R∗U = U,R∗∅ = ∅(FR6) R∗A ⊆ R∗(R∗A), R∗(R∗A) ⊆ R∗A.

Here, x1(y) = 1 for y = x and x1(y) = 0 otherwise.

III. INTERVAL TYPE-2 FUZZY ROUGH SETS

As mentioned in Section I, although fuzzy rough sets havebeen investigated fruitfully, so far, most of the studies in this fieldare based on the concept of type-1 fuzzy sets, i.e., membershipfunctions are crisp. When problems to be solved contain uncer-tain membership functions, type-2 fuzzy sets should be used;hence, conventional fuzzy rough sets are infeasible. In this sec-tion, the concept of fuzzy rough sets proposed by Dubois andPrade [3] will be generalized to interval type-2 fuzzy environ-ments. The properties of IT2 fuzzy rough sets are provided inthe Appendix.

A. Definitions of IT2 Fuzzy Relation

Definition 7: Let U and W be two nonempty universes. Atype-2 fuzzy set R ∈ F (U × W ) is defined as a type-2 fuzzybinary relation from U to W

R ≡ (((x, y), t), µR ((x, y), t))|(x, y) ∈ U × W,

t ∈ J(x,y ) ⊆ I (12)

where 0 ≤ µR ((x, y), t) ≤ 1. R can also be expressed as

R ≡∫

(x,y )∈U×W

∫t∈J(x , y )

µR ((x, y), t)/((x, y), t)

≡∫

(x,y )∈U×W

[∫t∈J(x , y )

f(x,y )(t)/t

]/(x, y) (13)

where f(x,y )(t) = µR ((x, y), t).At each value of (x, y), say (x, y) = (x′, y′), the 2-D plane,

whose axes are t and µR ((x′, y′), t), is a vertical slice of

µR ((x, y), t) denoted as µR (x′, y′)

µR (x′, y′) ≡∫

t∈J(x ′ , y ′)

f(x ′,y ′)(t)/t. (14)

The FOU of R is denoted as

DR(x, y) ≡ J(x,y )

DR ≡ FOU(R) ≡⋃

(x,y )∈U×W

DR(x, y). (15)

Definition 8: Let U and W be two nonempty universes. AnIT2 fuzzy set R ∈ FIT2(U × W ) is defined as an IT2 fuzzyrelation from U to W

R ≡∫ ∫

((x,y ),t)∈DR

1/((x, y), t)

where

DR ≡((x, y), t) | (x, y) ∈ U×W, t ∈ I, µR ((x, y), t)=1.(16)

Definition 9: Let A, B ∈ FIT2(W ), define A B if DA ⊆DB and DA ⊆ DB. If A B and B A, then A = B.

B. Definitions of IT2 Fuzzy Rough Sets

Definition 10: Let R ∈ FIT2(U × W ), define a mapping f :U → FIT2(W ) by

µf (x)(y, v) ≡ µR ((x, y), v), x ∈ U, y ∈ W, v ∈ I. (17)

Hence, Df(x) = (y, v) ∈ W × I|µf (x)(y, v) = 1. The

lower and upper membership functions of Df(x) are denotedas Df(x),Df(x) ∈ F(W ), respectively.

Since R ∈ FIT2(U × W ), for any x ∈ U and y ∈W,µR (x, y) is an IT1 fuzzy set on I . Hence, DR(x, y) canbe expressed as a range [lR (x, y), rR (x, y)] ⊆ I . Therefore, forany x ∈ U, y ∈ W

µDf (x)(y) = lR (x, y) µDf (x)(y) = rR (x, y). (18)

Definition 11: Let R ∈ FIT2(U × W ), A ∈ FIT2(W ), define

lower and upper IT2 fuzzy rough approximation operators f , f :FIT2(W ) → FIT2(U) by

f(A) ≡∫

x∈U

∫u∈Df (A)(x)

1/(x, u) (19)

f(A) ≡∫

x∈U

∫u∈Df (A)(x)

1/(x, u) (20)

where for any x ∈ U,Df(A)(x) ≡ [µf (DA)(x), µ

f (DA)(x)],

Df(A)(x) ≡ [µf (DA)

(x), µf (DA)

(x)], and

µf (DA)(x) ≡ inf

y∈Wmax1 − µDf (x)(y), µ

DA(y)

= ∧y∈W

[(1 − µDf (x)(y)) ∨ µDA

(y)] (21)



µf (DA)(x) ≡ inf

y∈Wmax1 − µDf (x)(y), µ

DA(y)

= ∧y∈W

[(1 − µDf (x)(y)) ∨ µDA

(y)] (22)

µf (DA)

(x) ≡ supy∈W

minµDf (x)(y), µDA

(y)

= ∨y∈W

[µDf (x)(y) ∧ µDA

(y)] (23)

µf (DA)

(x) ≡ supy∈W

minµDf (x)(y), µDA

(y)

= ∨y∈W


(y)]. (24)

The pair (f(A), f(A)) is defined as an interval type-2 fuzzyrough set.

Obviously, Df(A) = f(DA), Df(A) = f(DA), and Df

(A) = f(DA), Df(A) = f(DA).Definition 12: Let U and W be two nonempty universes and

R ∈ FIT2(U × W ), then:1) if for any x ∈ U , there always exists y ∈ W such that

µR ((x, y), 1) = 1, and µR ((x, y), u) = 0 for 0 ≤ u < 1,then R is defined as a serial IT2 fuzzy relation from U toW ;

2) if U = W , then R is defined as an IT2 fuzzy relation onU .

Definition 13: Let U be a nonempty universe and R be an IT2fuzzy relation on U , then:

1) if for any x ∈ U, µR ((x, x), 1) = 1, and µR ((x, x), u) =0 for 0 ≤ u < 1, then R is defined as a reflexive IT2 fuzzyrelation on U .

2) if for any x, y ∈ U, J(x,y ) = J(y ,x) , and µR ((x, y), u) =µR ((y, x), u) for all u ∈ J(x,y ) , then R is defined as asymmetric IT2 fuzzy relation on U .

3) if for any x, y ∈ U, lR (x, y) ≥ ∨z∈U [lR (x, z) ∧ lR (z, y)]and rR (x, y) ≥ ∨z∈U [rR (x, z) ∧ rR (z, y)], then R is de-fined as a transitive IT2 fuzzy relation on U .

C. Example

Let U = a1 , a2 , a3,W = b1 , b2 , b3 , b4, R ∈ FIT2(U ×W ), A ∈ FIT2(W ). Let DR(ai, bj ) = [lR (i, j), rR (i, j)],DA(bj ) = [lA (j), rA (j)], i = 1 ∼ 3, j = 1 ∼ 4. lR , rR , lA , rA aregiven as follows:

lR =

0.9 0.3 0.5 0.60.8 0.4 0.4 0.70.8 0.4 0.5 0.6

rR =

1.0 0.5 0.7 0.81.0 0.6 0.6 0.80.9 0.5 0.7 0.8

lA = [ 0.8 0.4 0.5 0.7 ] rA = [ 0.9 0.5 0.6 0.8 ] .

Hence, for every x ∈ U,Df(x),Df(x) ∈ F(W ) are describedas follows:

Df(a1) = 0.9/b1 + 0.3/b2 + 0.5/b3 + 0.6/b4

Df(a2) = 0.8/b1 + 0.4/b2 + 0.4/b3 + 0.7/b4

Df(a3) = 0.8/b1 + 0.4/b2 + 0.5/b3 + 0.6/b4

Df(a1) = 1.0/b1 + 0.5/b2 + 0.7/b3 + 0.8/b4

Df(a2) = 1.0/b1 + 0.6/b2 + 0.6/b3 + 0.8/b4

Df(a3) = 0.9/b1 + 0.5/b2 + 0.7/b3 + 0.8/b4 .

DA,DA ∈ F(W ) are described as follows:

DA = 0.8/b1 + 0.4/b2 + 0.5/b3 + 0.7/b4

DA = 0.9/b1 + 0.5/b2 + 0.6/b3 + 0.8/b4 .

For every x ∈ U,Df(A)(x),Df(A)(x) can be calculated byusing Definition 11. For simplicity, at this juncture, only x = a1will be considered

µf (DA)(a1) = ∧

y∈W[(1 − µDf (a1 )(y)) ∨ µ

DA(y)] = 0.5

µf (DA)(a1) = ∧

y∈W[(1 − µDf (a1 )(y)) ∨ µ

DA(y)] = 0.6

µf (DA)

(a1) = ∨y∈W

[µDf (a1 )(y) ∧ µDA

(y)] = 0.8

µf (DA)

(a1) = ∨y∈W

[µDf (a1 )(y) ∧ µDA

(y)] = 0.9.

Hence, Df(A)(a1) = [0.5, 0.6],Df(A)(a1) = [0.8, 0.9].For all x ∈ U , the results are

Df(A)(a1) = [0.5, 0.6] Df(A)(a2) = 0.6

Df(A)(a3) = [0.5, 0.6]

Df(A)(a1) = Df(A)(a2) = Df(A)(a3) = [0.8, 0.9].

This example is for the case of IT2 fuzzy rough sets. As acomparison, the aforementioned IT2 fuzzy sets are reduced totype-1 fuzzy sets as follows.

Let U = a1 , a2 , a3,W = b1 , b2 , b3 , b4, R ∈ F(U ×W ), A ∈ F(W ). Let µR (ai, bj ) = R(i, j), i = 1 ∼ 3, j =1 ∼ 4. R and A are given as follows:

R =

0.9 0.3 0.5 0.60.8 0.4 0.4 0.70.8 0.4 0.5 0.6

A = 0.8/b1 + 0.4/b2 + 0.5/b3 + 0.7/b4 .

For every x ∈ U,R∗(A)(x), R∗(A)(x) can be calculatedby using Definition 6. For simplicity, here only x = a1 isconsidered

R∗(A)(a1) = ∧y∈W

[(1 − R(a1 , y)) ∨ µA (y)] = 0.5

R∗(A)(a1) = ∨y∈W

[R(a1 , y) ∧ µA (y)] = 0.8.

For all x ∈ U , the results are

R∗(A) = 0.5/a1 + 0.6/a2 + 0.5/a3

R∗(A) = 0.8/a1 + 0.8/a2 + 0.8/a3 .

IV. IT2 FUZZY-ROUGH ATTRIBUTE REDUCTION

According to Jensen and Shen [4], rough set theory has asignificant advantage in finding a reduct within the originalattributes that are the most informative. However, traditional at-tribute reduction methods can only be operated effectively with



the data sets containing discrete values. It is inadequate to dealwith the case where values of attributes are real-valued. As mostof the data sets contain real-valued attributes, it is necessary toperform a discretization step beforehand. This is typically im-plemented by standard fuzzification techniques. Nevertheless,the membership degrees of attribute values to fuzzy sets are notexploited in the attribute reduction process.

To overcome this shortcoming, Jensen and Shen [4] proposeda fuzzy-rough attribute reduction algorithm. According to theirresearch, the fuzzy-rough reduction is more powerful than theconventional rough-set-based approach. However, the fuzzy-rough attribute reduction is based on the concept of the type-1fuzzy sets, and it cannot deal with the type-2 fuzzy cases. In thissection, the fuzzy-rough attribute reduction will be extended toIT2 fuzzy environments. The results of fuzzy-rough attributereduction reported by Jensen and Shen [4] are recalled at thebeginning. Subsequently, the definitions of IT2 fuzzy-rough at-tribute reduction are introduced. Lastly, an example is given todemonstrate IT2 fuzzy-rough attribute reduction.

A. Fuzzy-Rough Attribute Reduction

Jensen and Shen [4] noted that just as crisp equivalenceclasses were central to rough sets, fuzzy equivalence classeswere central to the fuzzy rough set approach. For typical appli-cations, this means that decision and conditional values may allbe fuzzy. According to Dubois and Prade [3], the family of nor-mal fuzzy sets produced by a fuzzy partitioning of the universeof discourse can play the role of fuzzy equivalence classes.

Let P be a subset of the attribute set A and U/P =F1 , . . . , Fk be the fuzzy partition of U , then the fuzzy P -lower and P -upper approximations are defined by Dubois andPrade [3] as

µP X (Fi) = infx∈U

max1 − µFi(x), µX (x) (25)

µP X (Fi) = supx∈U

minµFi(x), µX (x) (26)

where i = 1, . . . , k,X ∈ F(U) is a fuzzy concept to be ap-proximated, and Fi is a fuzzy equivalence class that belongs toU/P . For the case of attribute reduction, the fuzzy P -lower andP -upper approximations are redefined as

µP X (x) = supF ∈U/P

min

µF (x), infy∈U

max1 − µF (y),

µX (y)

(27)

µP X (x) = supF ∈U/P

min

µF (x), supy∈U

minµF (y), µX (y)(28)

where x ∈ U .Fuzzy-rough attribute reduction builds on the notion of fuzzy

lower approximation to enable reduction of data sets containingreal-valued attributes. Let P and Q be two subsets of the attributeset A, then the membership of an object x ∈ U that belongs tofuzzy positive region is defined by

µPOSP (Q)(x) = supX∈U/Q

µP X (x). (29)

The fuzzy-rough dependency function is defined by

γ′P (Q) =

|µPOSP (Q)(x)||U | =

∑x∈U µPOSP (Q)(x)

|U | . (30)

For the case of multiple attributes, let P contain multiple at-tributes, e.g., P = a, b. In the crisp case, U/P contains setsof objects grouped together that are indiscernible according toboth attributes a and b. In the fuzzy case, objects may belongto many fuzzy equivalence classes. Hence, the Cartesian prod-uct of U/IND(a) and U/IND(b) should be considered indetermining U/P .

In general, U/P =⊗

a ∈ P : U/IND(a) and each setin U/P denotes a fuzzy equivalence class. For example,if P = a, b, U/IND(a) = Na,Za and U/IND(b) =Nb, Zb, then U/P = Na ∩ Nb,Na ∩ Zb, Za ∩ Nb, Za ∩Zb. The extent to which an object belongs to a fuzzy equiva-lence class is calculated by using the conjunction of constituentfuzzy equivalence classes, e.g., Fi, i = 1, . . . , n

µF1 ∩...∩Fn(x) = min(µF1 (x), . . . , µFn

(x)). (31)

B. IT2 Fuzzy-Rough Attribute Reduction

Similar to the fuzzy-rough attribute reduction, IT2 fuzzyequivalence classes will be central to the IT2 fuzzy-rough at-tribute reduction under IT2 fuzzy environments. The family ofIT2 fuzzy sets produced by an IT2 fuzzy partitioning of theuniverse of discourse can play the role of IT2 fuzzy equivalenceclasses. Fig. 1(a) illustrates two IT2 fuzzy equivalence classes,i.e., N (denoting the word “Negative”) and Z (denoting theword “Zero”).

Definition 14: Let P be a subset of attribute set A,U/P =F1 , . . . , Fk be the IT2 fuzzy partition of U , and X ∈ FIT2(U)be an IT2 fuzzy concept to be approximated. The IT2 fuzzyP -lower and P -upper approximations are defined as

µP X (Fi) ≡∫

u∈DP X (Fi )1/u

µP X (Fi) ≡∫

u∈DP X (Fi )1/u, i = 1, . . . , k (32)

where

DPX(Fi) ≡[

infx∈U

max1 − µDFi(x), µDX (x),

infx∈U

max1 − µDFi(x), µDX (x)

](33)

DPX(Fi) ≡[supx∈U

minµDFi(x), µDX (x)

supx∈U

minµDFi(x), µDX (x)

]. (34)

For the case of attribute reduction, the IT2 fuzzy P -lower andP -upper approximations are redefined as

µP X (x) ≡∫

u∈DP X (x)1/u (35)

µP X (x) ≡∫

u∈DP X (x)1/u, i = 1, . . . , k (36)



where

DPX(x) ≡[

supF ∈U/P

minµDF (x), infy∈U

max1 − µDF (y),

µDX (y), supF ∈U/P

minµDF (x),

infy∈U

max1 − µDF (y)µDX (y)], x ∈ U

DPX(x) ≡[

supF ∈U/P

minµDF (x), supy∈U

minµDF (y),

µDX (y), supF ∈U/P

minµDF (x),

supy∈U

minµDF (y), µDX (y)], x ∈ U.

Just as the fuzzy-rough attribute reduction builds on the no-tion of fuzzy lower approximation, the IT2 fuzzy-rough attributereduction also builds on the notion of IT2 fuzzy lower ap-proximation to enable reduction of data sets under IT2 fuzzyenvironments.

Definition 15: Let P and Q be two subsets of the attributeset A. The membership of an object x ∈ U that belongs to IT2fuzzy positive region is defined by

µPOSP (Q)(x) ≡ X∈U/QµP X (x), x ∈ U. (37)

The lower and upper membership functions of µPOSP (Q)(x)are denoted as µ

POSP (Q)(x) and µPOSP (Q)(x). The IT2 fuzzy-

rough dependency functions are defined by

γP

(Q) ≡|µ

POSP (Q)(x)|

|U | γP (Q) ≡|µPOSP (Q)(x)|

|U | . (38)

For the case of multiple attributes, the Cartesian product ofIT2 fuzzy equivalence classes should be considered. In general,U/P =

⊗a ∈ P : U/IND(a), where each set in U/P de-

notes an IT2 fuzzy equivalence class. The extent to which anobject belongs to an IT2 fuzzy equivalence class is calculatedby using the meet of constituent IT2 fuzzy equivalence classes,e.g., Fi , i = 1, . . . , n

µF1 ∩...∩Fn(x) ≡ n

i=1µFi(x). (39)

C. IT2 Fuzzy-Rough Attribute Reduct Computation

In the conventional fuzzy-rough QuickReduct algorithm in[4], the dependency function γ′ is employed to choose the at-tributes that can be added to the current reduct candidate. The al-gorithm terminates when the addition of any remaining attributedoes not increase the dependency. Similar to the fuzzy-rough

TABLE IEXAMPLE OF INFORMATION SYSTEM

QuickReduct algorithm, an IT2 fuzzy-rough QuickReduct al-gorithm is proposed as follows:

1. R ← , γbest

← 0, γbest ← 0, γprev

← 0, γprev ← 0

2. do

3. T ← R

4. γprev

← γbest

, γprev ← γbest

5. for each θ ∈ (C − R)

6. if γR∪θ(Q) > γ

T(Q)

7. T ← R ∪ θ8. γ

best← γ

T(Q)

9. R ← T

10. for each θ ∈ (C − R)

11. if γR∪θ(Q) > γT (Q)

12. T ← R ∪ θ13. γbest ← γT (Q)

14. R ← T

15. until γbest

≤ γprev

or γbest ≤ γprev

Here, C is a set of conditional attributes and Q is a set of decisionattributes. The IT2 fuzzy-rough dependency functions γ and γare employed to choose the attributes that can be added to thecurrent reduct candidate. The algorithm terminates when theaddition of any remaining attribute does not increase the IT2fuzzy-rough dependency degrees.

D. IT2 Fuzzy-Rough Attribute Reduct Example

Following the example by Jensen and Shen [4], an informa-tion system IS = (U,C ∪ Q) defined in Table I is considered,where U = u1 , . . . , u6 is a set of finite objects, A = C ∪ Qis a set of attributes such that for every a ∈ A, a : U → Va, Va

being the value set of attribute a,C = c1 , c2 , c3 is a set of con-ditional attributes, and Q = d is a set of decision attributes.

For each object ui ∈ U (i = 1, . . . , 6) and each conditionalattribute cj ∈ C (j = 1, 2, 3), cj (ui) is a real value, which can beviewed as an objective measured data. For the decision attribute,the corresponding result for each object is a range belonging toI . Here, a range, instead of a crisp value, is used to describethe extent to which one likes or dislikes a certain object. Forexample, range [0, 0.5) stands for fuzzy judgment “No,” range(0.5,1] stands for fuzzy judgment “Yes,” where 0 means totally“No” and 1 means totally “Yes.” Therefore, with respect to



the decision attribute set Q = d, there are two IT2 fuzzyequivalence classes, i.e.

U/Q = Jno , Jyes.

Here

DJno = 0/u1 + 0/u2 + 0.1/u3 + 0/u4 + 0/u5 + 0.3/u6

DJno = 0.1/u1 + 0/u2 + 0.2/u3 + 0/u4 + 0/u5 + 0.4/u6

DJyes = 0/u1 + 0.5/u2 + 0/u3 + 0.7/u4 + 0.6/u5 + 0/u6

DJyes = 0/u1 + 0.6/u2 + 0/u3 + 0.8/u4 + 0.7/u5 + 0/u6 .

Using the IT2 fuzzy sets defined in Fig. 1(a) (forall conditional attributes), the following IT2 fuzzy equiva-lence classes are obtained: U/c1 = Nc1 , Zc1 , U/c2 =Nc2 , Zc2 , U/c3 = Nc3 , Zc3 . For simplicity, only thelower membership functions of Nc1 and Zc1 are listed

DNc1 = 0.6/u1 + 0.6/u2 + 0.4/u3 + 0/u4

+ 0/u5 + 0/u6

DZc1 = 0/u1 + 0/u2 + 0.25/u3 + 0.25/u4

+ 0.5/u5 + 0.5/u6 .

The first step of the IT2 fuzzy-rough attribute reduction isto calculate IT2 fuzzy lower approximations of c1, c2, andc3. For simplicity, only c1 will be considered, i.e., usingc1 to approximate Q.

For the first IT2 fuzzy equivalence class Jno , µc1 Jn o(x) =∫

u∈Dc1 Jn o (x) 1/u, x ∈ U , where

Dc1Jno(x) = [µD c1 Jn o(x), µD c1 Jn o

(x)], x ∈ U

µD c1 Jn o(x) = sup

F ∈U/c1 minµDF (x),

infy∈U

max1 − µDF (y), µDJn o(x)

µD c1 Jn o(x) = sup

F ∈U/c1 minµDF (x),

infy∈U

max1 − µDF (y), µDJn o(x).

Considering the IT2 fuzzy equivalence classes of c1 (i.e.,Nc1 and Zc1 ), for object u1 , the following hold:

minµDNc 1(u1), inf

y∈Umax1 − µDNc 1

(y), µDJn o(y)

= min0.6, inf0.4 ∨ 0, 0.4 ∨ 0, 0.6 ∨ 0.1, 1 ∨ 0,

1 ∨ 0, 1 ∨ 0.3= min0.6, 0.4 = 0.4,

minµDZc 1(u1), inf

y∈Umax1 − µDZc 1

(y), µDJn o(y)

= min0, inf1 ∨ 0, 1 ∨ 0, 0.75 ∨ 0.1, 0.75 ∨ 0,

0.5 ∨ 0, 0.5 ∨ 0.3= min0, 0.5 = 0.

Thus, µD c1 Jn o(u1) = 0.4 ∨ 0 = 0.4.

Hence

minµDNc 1(u1), inf

y∈Umax1 − µDNc 1

(y), µDJn o(y)

= min0.6, inf0.4 ∨ 0.1, 0.4 ∨ 0, 0.6 ∨ 0.2, 1 ∨ 0,

1 ∨ 0, 1 ∨ 0.4= min0.6, 0.4 = 0.4,

minµDZc 1(u1), inf

y∈Umax1 − µDZc 1

(y), µDJn o(y)

= min0, inf1 ∨ 0.1, 1 ∨ 0, 0.75 ∨ 0.2, 0.75 ∨ 0,

0.5 ∨ 0, 0.5 ∨ 0.4= min0, 0.5 = 0.

Thus, µD c1 Jn o(u1) = 0.4 ∨ 0 = 0.4.

Therefore, Dc1Jno(u1) = [µD c1 Jn o(u1), µD c1 Jn o

(u1)]= 0.4.Calculating the IT2 fuzzy c1-lower approximation of Jno

for every object gives

Dc1Jno(u1) = 0.4 Dc1Jno(u2) = 0.4

Dc1Jno(u3) = 0.4 Dc1Jno(u4) = 0.25

Dc1Jno(u5) = 0.5 Dc1Jno(u6) = 0.5.

The corresponding values for Jyes can also be calculated asfollows:

Dc1Jyes(u1) = 0.4 Dc1Jyes(u2) = 0.4

Dc1Jyes(u3) = 0.4 Dc1Jyes(u4) = 0.25

Dc1Jyes(u5) = 0.5 Dc1Jyes(u6) = 0.5.

Using these values, the membership of an object x ∈ Ubelonging to IT2 fuzzy positive region can be calculated byusing

µPOSc 1 (Q)(x) = X∈U/Qµc1 X (x), x ∈ U.

This results in

µPOSc 1 (Q)

(u1) = 0.4 µPOSc 1 (Q)(u1) = 0.4

µPOSc 1 (Q)

(u2) = 0.4 µPOSc 1 (Q)(u2) = 0.4

µPOSc 1 (Q)

(u3) = 0.4 µPOSc 1 (Q)(u3) = 0.4

µPOSc 1 (Q)

(u4) = 0.25 µPOSc 1 (Q)(u4) = 0.25

µPOSc 1 (Q)

(u5) = 0.5 µPOSc 1 (Q)(u5) = 0.5

µPOSc 1 (Q)

(u6) = 0.5 µPOSc 1 (Q)(u6) = 0.5.



Therefore, γc1 (Q) = γc1 (Q) = 2.45/6. Calculations for

c2 and c3 gives

γc2 (Q) = 2.2/6 γc2 (Q) = 2.4/6

γc3 (Q) = 0.4/6 γc3 (Q) = 0.4/6.

It can be observed that attribute c1 will cause the greatestincrease in IT2 fuzzy-rough dependency degrees. Hence, thisattribute is chosen and added to the potential reduct set. Theprocess iterates and produces two IT2 fuzzy-rough dependencydegrees as follows:

γc1 ,c2 (Q) = 2.45/6 γc1 ,c2 (Q) = 2.45/6

γc1 ,c3 (Q) = 2.55/6 γc1 ,c3 (Q) = 2.65/6.

Since a larger increase of IT2 fuzzy-rough dependency de-grees is produced by adding attribute c3 to the reduct candi-date set, the new reduct candidate set becomes c1 , c3. Lastly,adding c2 to the potential reduct set gives

γc1 ,c2 ,c3 (Q) = 2.45/6 γc1 ,c2 ,c3 (Q) = 2.45/6.

As the IT2 fuzzy-rough dependency degrees will not be in-creased by adding c2 to the potential reduct set, the algorithmstops and outputs the IT2 fuzzy-rough reduct c1 , c3.

However, for the case of fuzzy-rough attribute reductionin [4], the reduct is c1 , c2, and γ′

c1 ,c2 = 3.4/6. This ex-ample shows that IT2 fuzzy-rough attribute reduction generatesdifferent results when compared with conventional fuzzy-roughattribute reduction, where:

1) the dependency degrees for the case of the IT2 fuzzy-rough attribute reduction are decreased from γ′ = 3.4/6to γ = 2.55/6, γ = 2.65/6;

2) the reduct set is changed from c1 , c2 to c1 , c3;3) lower IT2 fuzzy-rough dependency degrees may be pro-

duced by using larger conditional attribute sets. Forexample γc1 ,c2 ,c3

(Q) < γc1 ,c3 (Q), γc1 ,c2 ,c3 (Q) <

γc1 ,c3 (Q).

V. CONCLUSION

This paper has presented a detailed investigation on the prop-erties of IT2 fuzzy rough sets, including proposed definitionsfor IT2 fuzzy rough sets as well as a method of attribute re-duction within the IT2 fuzzy rough set framework. This studydiffers from [7] in two aspects: 1) the starting point here is ageneral IT2 fuzzy binary relation, whereas the study by Jensenand Shen [7] was based on an interval-valued fuzzy tolerancerelation and 2) the definition of IT2 fuzzy rough sets in thisstudy is different from that of the interval-valued fuzzy roughsets in the work by Jensen and Shen [7].

According to Klawonnm [25], transitivity may not be a de-sirable property for a fuzzy similarity relation. Because the def-inition of IT2 fuzzy rough sets is based on a general IT2 fuzzybinary relation (see Definitions 7–11) instead of a fuzzy simi-larity relation, Klawonnm’s result has no impact on this paper.

Note that a considerable reference to [4] has been made inthis paper. Recently, Jensen and Shen [6] have reported thatthere are several problems with respect to [4]. For example, the

complexity of calculating the Cartesian product of fuzzy equiv-alence classes becomes unacceptable for large-feature subsets.In some situations, the fuzzy lower approximation might not bea subset of the fuzzy upper approximation. To overcome theseshortcomings, Jensen and Shen [6] proposed three techniquesbased on fuzzy lower approximation, fuzzy boundary region,and fuzzy discernibility matrix. In the future research, it mightbe useful to combine the model of IT2 fuzzy rough sets withthese three techniques as well. In this way, the Cartesian productof the IT2 fuzzy equivalence classes might no longer be used,and the explosive growth in the number of the considered IT2fuzzy equivalence classes might be avoided.

On the other hand, there are at least two approaches for thedevelopment of the fuzzy rough set theory—constructive andaxiomatic [10]. Since this paper focuses on the constructiveapproach, in the future, it might be useful to investigate the IT2fuzzy rough sets using the axiomatic approach, i.e., using thelower and upper approximation operators as primitive notions.

APPENDIX

Proposition 2: For any A, B ∈ FIT2(W ), the following hold:

1) D(A ∪ B) ⊆ DA ∪ DB;

2) D(A ∩ B) ⊇ DA ∩ DB;

3) A A ∪ B;

4) A ∩ B A.

Proof:1) For any y ∈ W,DA(y) = [lA (y), rA (y)],DB(y) =

[lB (y), rB (y)]. Thus, D(A ∪ B)(y) = [lA (y) ∨ lB (y),rA (y) ∨ rB (y)]

∀u ∈ D(A ∪ B)(y)

⇒ u ≥ lA (y) ∨ lB (y), u ≤ rA (y) ∨ rB (y)

⇒ u ≥ lA (y) ∨ lB (y), u ≤ rA (y), or

u ≥ lA (y) ∨ lB (y), u ≤ rB (y)

⇒ u ≥ lA (y), u ≤ rA (y), or u ≥ lB (y), u ≤ rB (y)

⇒ u ∈ DA(y), or u ∈ DB(y)

⇒ u ∈ DA(y) ∪ DB(y).

Thus, D(A ∪ B)(y) ⊆ DA(y) ∪ DB(y) ∀y ∈ W . There-fore, D(A ∪ B) ⊆ DA ∪ DB.

2) The proof procedure is similar to 1).3) For any y ∈ W,DA(y) = [lA (y), rA (y)], D(A ∪ B)

(y) = [lA (y) ∨ lB (y), rA (y) ∨ rB (y)].Thus, µ

DA(y) = lA (y) ≤ lA (y) ∨ lB (y) = µ

D (A∪B )(y).Therefore

DA ⊆ D(A ∪ B). (40)

Similarly we can obtain DA ⊆ D(A ∪ B). According toDefinition 9, there exists A A ∪ B.



4) For any y ∈ W,DA(y)=[lA (y), rA (y)], D(A ∩ B)(y)=[lA (y) ∧ lB (y), rA (y) ∧ rB (y)].

Thus, µDA

(y) = lA (y) ≥ lA (y) ∧ lB (y) = µD (A∩B )(y).

Therefore

DA ⊇ D(A ∩ B). (41)

Similarly we can obtain DA ⊇ D(A ∩ B). According toDefinition 9, there exists A ∩ B A.

Proposition 3: Let R ∈ FIT2(U × W ), and f and f be thelower and upper IT2 fuzzy rough approximation operators, forany A, B ∈ FIT2(W ), the following hold:

1) A ⊆ B ⇒ f(A) ⊆ f(B);

2) A ⊆ B ⇒ f(A) ⊆ f(B).

Proof:1) According to Proposition 1, A ⊆ B ⇒ µ

DA(y) ≥

µDB

(y), µDA

(y) ≤ µDB

(y), y ∈ W . Therefore, for any

x ∈ U

∀u ∈ Df(A)(x)

⇒ u ∈ [µf (DA)(x), µ

f (DA)(x)]

⇒ u ≥ µf (DA)(x), u ≤ µ

f (DA)(x)

⇒ u ≥ ∧y∈W

[(1 − µDf (x)(y)) ∨ µDA

(y)]

u ≤ ∧y∈W

[(1 − µDf (x)(y)) ∨ µDA

(y)]

⇒ u ≥ ∧y∈W

[(1 − µDf (x)(y)) ∨ µDB

(y)]

u ≤ ∧y∈W

[(1 − µDf (x)(y)) ∨ µDB

(y)]

⇒ u ≥ µf (DB )(x) u ≤ µ

f (DB )(x)

⇒ u ∈ Df(B)(x).

Thus,

Df(A)(x) ⊆ Df(B)(x) ∀x ∈ U

⇒Df(A) =⋃x∈U

Df(A)(x) ⊆⋃x∈U

Df(B)(x) = Df(B)

⇒f(A) ⊆ f(B) (by Proposition 1).

2) The proof procedure is similar to 1). Proposition 4: Let R ∈ FIT2(U × W ), f and f be lower and

upper IT2 fuzzy rough approximation operators, for any A, B ∈FIT2(W ), the following hold:

1) f(A ∩ B) f(A) ∩ f(B);

2) f(A ∩ B) f(A) ∩ f(B);

3) f(A) ∪ f(B) f(A ∪ B);

4) f(A) ∪ f(B) f(A ∪ B).

Proof:1) Let Y = f(A) ∩ f(B), then according to formula (9) andDefinition 11, for any x ∈ U

DY (x) = [µf (DA)(x) ∧ µ

f (DB )(x), µf (DA)(x)

∧ µf (DB )(x)]

⇒ DY (x) = µf (DA)(x) ∧ µ

f (DB )(x)

Df(A ∩ B)(x) = [µf (D (A∩B ))(x), µ

f (D (A∩B ))(x)]

⇒ Df(A ∩ B)(x) = µf (D (A∩B ))(x).

According to formula (41), there exists: DA ⊇ D(A ∩ B);hence, µ

DA(y) ≥ µ

D (A∩B )(y), y ∈ W . Therefore

µf (DA)(x) = ∧

y∈W[(1 − µDf (x)(y)) ∨ µ

DA(y)]

≥ ∧y∈W

[(1 − µDf (x)(y)) ∨ µD (A∩B )(y)]

= µf (D (A∩B ))(x).

Similarly, the following holds:

µf (DB )(x) ≥ µ

f (D (A∩B ))(x).

Thus,µ

f (DA)(x) ∧ µf (DB )(x) ≥ µ

f (D (A∩B ))(x)

⇒ DY (x) ≥ Df(A ∩ B)(x) ∀x ∈ U

⇒ D[f(A) ∩ f(B)] ⊇ Df(A ∩ B).

Similarly, we can derive D[f(A) ∩ f(B)] ⊇ Df(A ∩ B).Therefore, according to Definition 9, the following holds:f(A ∩ B) f(A) ∩ f(B).

2) The proof procedure is similar to 1).3) Let Y = f(A) ∪ f(B), then according to formula (8) and

Definition 11, for any x ∈ U ,

DY (x) = [µf (DA)(x) ∨ µ

f (DB )(x), µf (DA)(x)

∨ µf (DB )(x)]

⇒ DY (x) = µf (DA)(x) ∨ µ

f (DB )(x)

Df(A ∪ B)(x) = [µf (D (A∪B ))(x), µ

f (D (A∪B ))(x)]

⇒ Df(A ∪ B)(x) = µf (D (A∪B ))(x).

According to formula (40), there exists DA ⊆ D(A ∪ B).Hence, µ

DA(y) ≤ µ

D (A∪B )(y), y ∈ W .

Therefore

µf (DA)(x) = ∧

y∈W[(1 − µDf (x)(y)) ∨ µ

DA(y)]

≤ ∧y∈W

[(1 − µDf (x)(y)) ∨ µD (A∪B )(y)]

= µf (D (A∪B ))(x).



Similarly, the following holds:

µf (DB )(x) ≤ µ

f (D (A∪B ))(x).

Thus,µ

f (DA)(x) ∨ µf (DB )(x) ≤ µ

f (D (A∪B ))(x)

⇒ DY (x) ≤ Df(A ∪ B)(x) ∀x ∈ U

⇒ D[f(A) ∪ f(B)] ⊆ Df(A ∪ B).

Similarly, we can derive D[f(A) ∪ f(B)] ⊆ Df(A ∪ B).Therefore, according to Definition 9, the following holds:

f(A) ∪ f(B) f(A ∪ B).

4) The proof procedure is similar to 3). Proposition 5: Let R ∈ FIT2(U × U), and f and f be the

lower and upper IT2 fuzzy rough approximation operators. If Ris reflexive, then for any A ∈ FIT2(U), the following hold:

1) f(A) A;

2) A f(A).

Proof: Since R is reflexive, then for any x ∈ U , there alwayshold that µR ((x, x), 1) = 1, and µR ((x, x), u) = 0 for 0 ≤ u <1, i.e., µf (x)(x, 1) = 1, and µf (x)(x, u) = 0, for 0 ≤ u < 1.

Thus, according to formula (18),

µDf (x)(x) = µDf (x)(x) = 1.

1) According to Definition 11, for any x ∈ U

µDf (A)(x) = µ

f (DA)(x)

= ∧y∈U

[(1 − µDf (x)(y)) ∨ µDA

(y)]

= ∧y∈U,y =x

[(1 − µDf (x)(y)) ∨ µDA

(y)]

∧ [(1 − µDf (x)(x)) ∨ µDA

(x)]

= ∧y∈U,y =x

[(1 − µDf (x)(y)) ∨ µDA

(y)]

∧ [0 ∨ µDA

(x)]

= ∧y∈U,y =x

[(1 − µDf (x)(y)) ∨ µDA

(y)] ∧ µDA

(x)

≤ µDA

(x).

Thus, Df(A) ⊆ DA. Similarly we can derive Df(A) ⊆DA. Therefore, according to Definition 9, there exists f(A) A.

2) The proof procedure is similar to 1). Proposition 6: Let R ∈ FIT2(U × U), and f and f be the

lower and upper IT2 fuzzy rough approximation operators. If Ris symmetric, then for any x, z ∈ U , the following hold:

1) µDf (x)(z) = µDf (z )(x), µDf (x)(z) = µDf (z )(x);2) µf (1U \z )

(x) = µf (1U \x )(z);

3) µf (1z )

(x) = µf (1x )

(z)

where

µ1U \x (y, v) =

1, y = x, v = 10, otherwise

µ1x(y, v) =

1, y = x, v = 10, otherwise.

Proof:1) According to the definitions of µ1U \x

(y, v) and µ1x(y, v),

the following hold:

µD 1U \x (y) = µD 1U \x

(y) =

1, y = x0, y = x

µD 1x(y) = µD 1x

(y) =

1, y = x0, y = x.

Since R is symmetric, then for any x, z ∈ U, J(x,z ) = J(z ,x) ,and µR ((x, z), u) = µR ((z, x), u) for all u ∈ J(x,z ) . Thus,µf (x)(z, u) = µR ((x, z), u) = µR ((z, x), u) = µf (z )(x, u), soµDf (x)(z) = µDf (z )(x), µDf (x)(z) = µDf (z )(x).

2) According to Definition 11, for any x ∈ U

Df(1U \z)(x) = [µf (D 1U \z )(x), µf (D 1U \z )

(x)],

where

µf (D 1U \z )(x) = ∧

y∈U[(1 − µDf (x)(y)) ∨ µD 1U \z

(y)]

= ∧y∈U,y =z

[(1 − µDf (x)(y)) ∨ µD 1U \z (y)]

∧ [(1 − µDf (x)(z)) ∨ µD 1U \z (z)]

= ∧y∈U,y =z

[(1 − µDf (x)(y)) ∨ 1]

∧ [(1 − µDf (x)(z)) ∨ 0]

= 1 − µDf (x)(z).

Similarly, µf (D 1U \z )(x) = 1 − µDf (x)(z). Hence, Df

(1U \z)(x) = 1 − µDf (x)(z).Similarly, we can obtain

Df(1U \x)(z) = 1 − µDf (z )(x).

According to 1),

µDf (x)(z) = µDf (z )(x)

and thus

Df(1U \z)(x) = Df(1U \x)(z)∀x, z ∈ U.

Therefore,

µf (1U \z )(x) = µf (1U \x )

(z)∀x, z ∈ U.

3) According to Definition 11, for any x ∈ U,

Df(1z )(x) = [µf (D 1z )

(x), µf (D 1z )

(x)]



where

µf (D 1z )

(x) = ∨y∈U

[µDf (x)(y) ∧ µD 1z(y)]

= ∧y∈U,y =z

[µDf (x)(y) ∧ µD 1z(y)]

∨ [µDf (x)(z) ∧ µD 1z(z)]

= ∧y∈U,y =z

[µDf (x)(y) ∧ 0] ∨ [µDf (x)(z) ∧ 1]

= µDf (x)(z).

Similarly,

µf (D 1z )

(x) = µDf (x)(z).

Hence

Df(1z )(x) = µDf (x)(z). (42)

Similarly, we can obtain

Df(1x)(z) = µDf (z )(x).

According to 1),

µDf (x)(z) = µDf (z )(x)

and thus

Df(1z )(x) = Df(1x)(z)∀x, z ∈ U.

Therefore,

µf (1z )

(x) = µf (1x )

(z)∀x, z ∈ U.

Proposition 7: Let R ∈ FIT2(U × W ). Then, for any x ∈

U, z ∈ W,X ⊆ W , the following hold:

1) Df(1z )(x) = rR (x, z);

2) Df(1W \z)(x) = 1 − lR (x, z);

3) Df(1X )(x) = ∨y∈X

rR (x, y);

4) Df(1X )(x) = ∧y /∈X

[1 − lR (x, y)]

where

µ1X(y, v) =

1, y ∈ X, v = 10, otherwise.

Proof:1) According to formula (42) and (18), for any x ∈ U,

Df(1z )(x) = µDf (x)(z) = rR (x, z).2)

µ1X(y, v) =

1, y ∈ X, v = 10, otherwise

⇒ µD 1X(y) = µD 1X

(y) =

1, y ∈ X0, y /∈ X.

For any x ∈ U

Df(1W \z)(x) = [µf (D 1W \z )(x), µf (D 1W \z )

(x)],

µf (D 1W \z )(x) = ∧

y∈W[(1 − µDf (x)(y)) ∨ µD 1W \z

(y)]

= ∧y∈W,y =z

[(1 − µDf (x)(y)) ∨ µD 1W \z (y)]

∧ [(1 − µDf (x)(z)) ∨ µD 1W \z (z)]

= ∧y∈W,y =z

[(1 − µDf (x)(y)) ∨ 1]

∧ [(1 − µDf (x)(z)) ∨ 0]

= 1 − µDf (x)(z) = 1 − lR (x, z),

Similarly,

µf (D 1W \z )(x) = 1 − µDf (x)(z) = 1 − lR (x, z).

Therefore,

Df(1W \z)(x) = 1 − lR (x, z).3) For any x ∈ U

Df(1X )(x) = [µf (D 1X )

(x), µf (D 1X )

(x)]

µf (D 1X )

(x) = ∨y∈W

[µDf (x)(y) ∧ µD 1X(y)]

= ∨y∈X

µDf (x)(y) = ∨y∈X

rR (x, y).

Similarly,

µf (D 1X )

(x) = ∨y∈X

rR (x, y).

Therefore,

Df(1X )(x) = ∨y∈X

rR (x, y).

4) For any x ∈ U

Df(1X )(x) = [µf (D 1X )(x), µf (D 1X )(x)]

µf (D 1X )(x) = ∧y∈W

[(1 − µDf (x)(y)) ∨ µD 1X(y)]

= ∧y /∈X

[1 − µDf (x)(y)]

= ∧y /∈X

[1 − lR (x, y)].

Similarly,

µf (D 1X )(x) = ∧y /∈X

[1 − lR (x, y)].

Therefore,

Df(1X )(x) = ∧y /∈X

[1 − lR (x, y)].

Proposition 8: Let R ∈ FIT2(U × W ). If R is serial, thefollowing hold:

1) f(1W ) = 1U ;

2) f(0W ) = 0U ;

3) f(A) f(A)



where

µ0W(y, v) =

1, y ∈ W, v = 00, otherwise

µ0U(y, v) =

1, y ∈ U, v = 00, otherwise.

Proof:1) For any x ∈ U, R is serial

⇒ ∃y′ ∈ W, µR ((x, y′), 1) = 1, and

µR ((x, y′), u) = 0, 0 ≤ u < 1

⇒ ∃y′ ∈ W, lR (x, y′) = rR (x, y′) = 1

⇒ ∨y∈W

rR (x, y) = 1.

According to Proposition 7,

Df(1W )(x) =

∨y∈W

rR (x, y)

= 1 ∀x ∈ U.

Therefore, f(1W ) = 1U .2)

µ0W(y, v) =

1, y ∈ W, v = 00, otherwise

⇒ µD 0W(y) = µD 0W

(y) = 0.For any x ∈ U

µf (D 0W )(x) = ∧y∈W

[(1 − µDf (x)(y)) ∨ µD 0W(y)]

= ∧y∈W

[1 − µDf (x)(y)]

= ∧y∈W

[1 − lR (x, y)].

Since R is serial, then ∃y′ ∈ W such that lR (x, y′) = 1;hence, ∧

y∈W[1 − lR (x, y)] = 0, i.e., µf (D 0W )(x) = 0 ∀x ∈ U .

Similarly, we can obtain

µf (D 0W )(x) = 0 ∀x ∈ U.

Thus,

Df(0W )(x) = [µf (D 0W )(x), µf (D 0W )(x)] = 0 ∀x ∈ U.

Therefore,

f(0W ) = 0U .

3) R is serial ⇒ ∃y′ ∈ W such that lR (x, y′) = rR (x, y′) =1 ⇒ µDf (x)(y

′) = µDf (x)(y′) = 1.

For any x ∈ U

µDf (A)(x) = µ

f (DA)(x)

= ∧y∈W

[(1 − µDf (x)(y)) ∨ µDA

(y)]

= ∧y∈W,y =y ′

[(1 − µDf (x)(y)) ∨ µDA

(y)]

∧ [(1 − µDf (x)(y′)) ∨ µ

DA(y′)]

= ∧y∈W,y =y ′

[(1 − µDf (x)(y)) ∨ µDA

(y)]

∧ [0 ∨ µDA

(y′)]

= ∧y∈W,y =y ′

[(1 − µDf (x)(y)) ∨ µDA

(y)] ∧ µDA

(y′)

≤ µDA

(y′)

µDf (A)

(x) = µf (DA)

(x)

= ∨y∈W


(y)]

= ∨y∈W,y =y ′


(y)]

∨ [µDf (x)(y′) ∧ µ

DA(y′)]

= ∨y∈W,y =y ′


(y)] ∨ [1 ∧ µDA

(y′)]

= ∨y∈W,y =y ′


(y)] ∨ µDA

(y′)

≥ µDA

(y′).

Hence,

µDf (A)(x) ≤ µ

DA(y′) ≤ µ

Df (A)(x).

Thus,

Df(A) ⊆ Df(A).

Similarly, we can derive Df(A) ⊆ Df(A). Therefore, ac-

cording to Definition 9, f(A) f(A).

Proposition 9: Let R ∈ FIT2(U × U), and f and f be thelower and upper type-2 fuzzy rough approximation operators. IfR is transitive, then for any A ∈ FIT2(U), the following hold:

1) f(A) f(f(A));

2) f(f(A)) f(A).

Proof: Since R is transitive, then for any x, y, z ∈ U

lR (x, z) ≥ ∨y∈U

[lR (x, y) ∧ lR (y, z)]

rR (x, z) ≥ ∨y∈U

[rR (x, z) ∧ rR (y, z)].

According to formula (18)

µDf (x)(y) = lR (x, y) µDf (x)(y) = rR (x, y).

Thus, for any x, z ∈ U ,

µDf (x)(z) ≥ ∨y∈U

[µDf (x)(y) ∧ µDf (y )(z)] (43)

µDf (x)(z) ≥ ∨y∈U

[µDf (x)(y) ∧ µDf (y )(z)]. (44)



1) For any x ∈ U ,

µDf (f (A))(x)

= µf (Df (A))(x) (by Definition 11)

= ∧y∈U

[(1 − µDf (x)(y)) ∨ µDf (A)(y)]

= ∧y∈U

[(1 − µDf (x)(y)) ∨ µf (DA)(y)] (by Definition 11)

= ∧y∈U

(1 − µDf (x)(y)) ∨ ∧z∈U

[(1 − µDf (y )(z)) ∨ µDA

(z)]

= ∧y∈U

∧z∈U

(1 − µDf (x)(y)) ∨ (1 − µDf (y )(z)) ∨ µDA

(z)

= ∧z∈U

∧y∈U

[(1 − µDf (x)(y)) ∨ (1 − µDf (y )(z))] ∨ µDA

(z).

On the other hand

µDf (A)(x) = µ

f (DA)(x) = ∧z∈U

[(1 − µDf (x)(z)) ∨ µDA

(z)].

According to formula (43), there exists

1 − µDf (x)(z)) ≤ ∧y∈U

[(1 − µDf (x)(y))

∨ (1 − µDf (y )(z))] ∀z ∈ U.

Thus,

µDf (A)(x) ≤ µ

Df (f (A))(x) ∀x ∈ U

i.e.,

Df(A) ⊆ Df(f(A)).

Similarly we can derive

Df(A) ⊆ Df(f(A)).

Therefore, according to Definition 9,

f(A) f(f(A)).

2) For any x ∈ U

µDf (f (A))

(x)

= µf (Df (A))

(x) (by Definition 11)

= ∨y∈U

[µDf (x)(y) ∧ µDf (A)

(y)]

= ∨y∈U

[µDf (x)(y) ∧ µf (DA)

(y)] (by Definition 11)

= ∨y∈U

µDf (x)(y) ∧ ∨z∈U

[µDf (y )(z) ∧ µDA

(z)]

= ∨y∈U

∨z∈U

µDf (x)(y) ∧ µDf (y )(z) ∧ µDA

(z)

= ∨z∈U

∨y∈U

[µDf (x)(y) ∧ µDf (y )(z)] ∧ µDA

(z).

On the other hand

µDf (A)

(x) = µf (DA)

(x) = ∨z∈U

[µDf (x)(z) ∧ µDA

(z)].

According to formula (44),

µDf (x)(z)) ≥ ∨y∈U

[µDf (x)(y) ∧ µDf (y )(z))] ∀z ∈ U.

Thus,

µDf (A)

(x) ≥ µDf (f (A))

(x) ∀x ∈ U,

i.e.,

Df(A) ⊇ Df(f(A)).

Similarly, we can derive

Df(A) ⊇ Df(f(A)).

Therefore, according to Definition 9,

f(f(A)) f(A).

Proposition 10: Let R ∈ FIT2(U × U), and f and f be the

lower and upper IT2 fuzzy rough approximation operators. IfR is reflexive and transitive, then for any A ∈ FIT2(U), thefollowing hold:

1) f(A) = f(f(A));

2) f(f(A)) = f(A).

Proof: 1) Since R is reflexive, then according toProposition 5, f(f(A)) f(A). Since R is transitive, then ac-

cording to Proposition 9, f(A) f(f(A)).Therefore, according to Definition 9, the following holds

f(A) = f(f(A)).2) The proof procedure is similar to 1). It can be seen that except (FR1), properties at the end of

Section II have their IT2 fuzzy counterparts: (FR2)–(FR6) cor-respond to Propositions 4, 5, 6, 8 and 9, respectively.

ACKNOWLEDGMENT

The authors wish to thank the anonymous referees for theirhelpful comments. H. Wu is very grateful to F. Chen (Alice),L. Chen, S. Chen, Z. Li, Wan-wan (Apple), Dou-dou (Lily), andMiao-miao (Cindy) for their great support.

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Haoyang Wu received the B.S. degree in computerscience and the Ph.D. degree in electronic engineer-ing from Xi’an Jiaotong University, Xi’an, China, in1994 and 1999, respectively.

From 2000 to 2002, he was a Postdoctoral Re-searcher at Tsinghua University, Beijing, China. Cur-rently he is a Visiting Researcher at Xi’an JiaotongUniversity. His current research interests include ar-tificial intelligence, data mining, game theory, andquantum computation.

Yuyuan Wu received the B.S. degree in power en-gineering from Xi’an Jiaotong University, Xi’ian,China, in 1966, and the Ph.D. degree in cryogenicsfrom the University of Southampton, Southampton,U.K., in 1984.

Currently he is a Full Professor at Xi’an JiaotongUniversity. His current research interests include airconditioning and cryogenics.

Jinping Luo received the B.S. degree in computerscience from Airforce Engineering College, Xi’an,Shaanxi, China, in 1990, and the Ph.D. degree inComputer Engineering from the National Universityof Defense Technology, Changsha, Hunan, China, in2000.

Currently he is with Kunlun Technology Indus-try Corporation (a software company), Hangzhou,China. His current research interests include dis-tributed computing and E-government.



Theory of Extended Fuzzy Discrete-Event Systemsfor Handling Ranges of Knowledge Uncertainties

and SubjectivityXinyu Du, Student Member, IEEE, Hao Ying, Senior Member, IEEE, and Feng Lin, Senior Member, IEEE

Abstract—In 2001, we originated a theory of fuzzy discrete-event systems (FDESs) that generalized the conventional/crispdiscrete-event systems (DESs). Vagueness and imprecision con-cerning states and event transitions of DESs were represented bymembership grades and computed via fuzzy logic. Our applicationof the FDES theory to computerized human immunodeficiencyvirus/acquired immune deficiency syndrome treatment regimenselection, although preliminarily successful, suggests that a morecomprehensive FDES theory is needed to address two general is-sues critically important not only to biomedical applications, butalso to real-world problems in other industries. First, domain ex-perts should have means other than point estimates and type-1fuzzy sets mandated in the current framework to describe uncer-tainties, subjectivity, and imprecision in their (complex) knowledgeand experience. Second, when a group of experts with distinct opin-ions is involved, they should not be forced to reach consensus forthe sake of system development. This is because collective consen-sus may not be achievable, which is often the case in medicine,where individual experts’ opinions should be equally respectedsince the underlying ground truth is unknown most of the time. Thetheory of extended FDES presented in this paper addresses both theproblems and contains the FDES theory as a special case. Expertsare now allowed to use interval numbers and type-1 and type-2fuzzy sets to intuitively and quantitatively express their diverseknowledge and experience, which will then be processed by the newtheory to form fuzzy state vectors and fuzzy event transition ma-trices. Accordingly, we have established mathematical operationsthat cover the computations of fuzzy states, fuzzy event transitions,and parallel composition. Numerical examples are provided.

Index Terms—Automata, discrete-event systems (DESs), fuzzylogic, type-2 fuzzy systems.

I. INTRODUCTION

THERE exist countless (complex) systems that cannot beeffectively described, at a higher level, by differential

equations, but can be by traces (or sequences) of events thatrecord significant qualitative changes in the state of the sys-tem. These states are logical or symbolic rather than numeri-cal. These systems can be described as discrete-event systems(DESs) whose behavior consists of sequence of occurrences ofdistinct events. These events, for instance, can describe send-ing or receiving messages in computer networks, or processinga part in a manufacturing plant. Comprehensive study and de-

Manuscript received September 7, 2007; revised April 18, 2008 and October2, 2008; accepted December 4, 2008. First published December 22, 2008; currentversion published April 1, 2009.

The authors are with the Department of Electrical and Computer Engineering,Wayne State University, Detroit, MI 48202 USA (e-mail: [email protected];[email protected]; [email protected]).



velopment of DES theory is a recent endeavor. Only after theproliferation of complex systems such as computer systems andnetworks, have DESs been systematically studied. DES theoryaddresses the issues of modeling, control, and optimization ofDES [1]–[3]. It has been applied to practical systems such as au-tomated manufacturing systems, database concurrency control,feature interactions in telecommunication networks, protocolverification and synthesis in communication networks, and pro-tocol conversion and gateway synthesis in computer networks.Although seemingly sufficient for many application domains,crisp DES theory is not adequate for some important fields suchas biomedicine in which the state and state transition of a system(e.g., a person’s health status) are always somewhat uncertainand vague even in a deterministic sense. Subjective human ob-servation, judgment, and interpretation (e.g., by a physician or apatient) invariably play a significant role in describing the statusof state, usually not crisp.

Definitions of fuzzy state, fuzzy event, and fuzzy automatawere proposed and studied as early as in the late 1960s [4], [5].However, only a few studies followed in the next 30 years orso [6]–[8]. A comprehensive theory of fuzzy DESs (FDESs)was yet to be established, so were many fundamentally impor-tant concepts, methods, and theorems in the traditional DES,such as controllability, observability, and optimal control. Toeffectively represent deterministic uncertainties and vaguenessas well as human subjective observation and judgment encoun-tered in many real-world problems especially those in medicine,we recently originated a more comprehensive theory of FDES[9], [10]. We introduced fuzzy states and fuzzy event transitionand generalized conventional/crisp DES to FDES. The largelygraph-based framework of DES was unsuitable for the expan-sion, and we thus reformulated it using state vectors and eventtransition matrices that could be extended to fuzzy vectors andmatrices by allowing their elements to take values in [0, 1].We also extended optimal control and observability of DES toFDES. The new observability allows one to determine whetheror not the system output observed is sufficient for decision mak-ing. The FDES theory is consistent with the DES theory, bothat conceptual and computation levels, in that the former con-tains the latter as a special case when the membership gradesare either 0 or 1. Other researchers have expanded different as-pects of the FDES theory [11]–[20]. Other independent studiesinclude [21] and [22]. More recently, we further developed theFDES theory so that it possessed self-learning capability [23].

As the first application, we have successfully applied theFDES theory to develop a novel system that prescribes drugsfor human immunodeficiency virus (HIV)/acquired immune

1063-6706/$25.00 © 2009 IEEE


DU et al.: THEORY OF EXTENDED FDESs FOR HANDLING RANGES OF KNOWLEDGE UNCERTAINTIES AND SUBJECTIVITY 317

deficiency syndrome (AIDS) patients who are about to receivetheir first combination antiretroviral treatment (the only effec-tive long-term treatment to date) [24], [25]. The United Nationsestimates that 38 million people worldwide are infected withHIV and that more than 22 million have died. HIV/AIDS treat-ment is, unfortunately, among one of the most complex treat-ments for any disease. Although the U.S. Department of HumanHealth and Services HIV/AIDS treatment guidelines cover thefirst-round combination antiretroviral therapy, they do not pro-vide individualized treatment advice. In this regard, computersoftware that can utilize clinical information to recommend reg-imens (a regimen is a combination of different drugs) that willbe sufficiently potent, well tolerated, and taken on scheduleby an HIV-infected person would be a great advance. Accord-ingly, the objective of our project was to build such a systemwhose regimen choice for any given patient will match expertAIDS physician’s selection to produce the (anticipated) optimaltreatment outcome. Preliminary retrospective evaluation of ourprototype system using patients treated in our institution’s AIDSClinical Center demonstrated encouraging results when the sys-tem operated in either self-learning mode or nonlearning mode.

Our first-hand application experience indicates that a morecomprehensive FDES theory is needed to address two generalissues critically important not only to biomedical applications,but also to applications in other industries. First, extractingapplication-specific knowledge and experience from domainexperts is a crucial but challenging task for both the systemdeveloper and the domain expert. It is usually not easy for theexpert to explicitly express his/her ideas qualitatively. Doing thesame quantitatively is obviously even harder. From the systemdeveloper’s standpoint, how to accurately translate the domainexpertise into an FDES-usable form is a major technical hur-dle. The expert should have means other than point estimatesand type-1 fuzzy sets mandated in the current framework todescribe uncertainties, subjectivity, and imprecision in his/herknowledge and experience. Let us use our HIV/AIDS applica-tion as a concrete example to illustrate this issue.

A physician must consider many factors when selecting an an-tiretroviral treatment regimen for a patient. In our FDES scheme,the primary four clinical parameters germane to this decision areas follows: 1) anticipated potency of the regimen; 2) anticipatedadherence for the patient under the regimen; 3) prognosis foradverse events under the regimen; and 4) expected future drugoptions due to the potential for development of resistance tothe regimen. We used the literature-cited percentage of patientswho achieved plasma HIV ribonucleic acid (RNA) less than400 copies/mL after 48 weeks of treatment as our measure ofpotency. Adherence is a very complicated issue involving manyfactors impacting the patient’s ability to comply with the pre-scribed regimen. We defined adherence as expected percentageof drug doses prescribed by the physician that were actuallytaken by the patient weekly for each regimen. The parameteradverse events was defined as undesirable side effects and tox-icities. Finally, the factor of future drug options meant drugsavailable after the current treatment was no longer viable. In theknowledge acquisition phase, one of the tasks of the two AIDSexpert physicians (i.e., domain experts) in our team was to es-

Fig. 1. Type-1 fuzzy sets “medium” and “high” for potency of regimens ofinterest as defined by a single AIDS expert or as a consensus of a group ofexperts. Also shown is how a potency of 85% for a particular regimen leads tothe two memberships of the two fuzzy sets.

timate, based on their experience and knowledge, a percentagenumber to each of the four factors for all the treatment regimensunder consideration (e.g., potency is 85% for regimen X, ad-herence is 80% for regimen Y, and adverse events is 20% forregimen Z). This turned out to be a difficult task for the physi-cians. While the percentages were subjective summarization ofthe clinical possibilities for the parameters and represented theexperts’ best point estimates, there were uncertainties as thetrue values did not exist in the literature, and consequently,the ground truth was (and still is) unknown. It is obvious thatallowing the experts to use interval numbers (e.g., potency is[82%, 88%] for regimen X) or fuzzy numbers (e.g., potencyis around 85%), impossible in the current FDES theory, wouldmake the knowledge acquisition process much easier and moreintuitive for the experts with more realistic knowledge represen-tation. A fuzzy set can be considered as a fuzzy number if a setof conditions described later is met.

The second issue is that when a group of experts with dis-tinct opinions is involved, they should not be forced to reachconsensus for the sake of system development. This is becausecollective consensus may not be achievable, which is often thecase in medicine, where individual experts’ opinions should beequally respected since the underlying ground truth is unknownmost of the time. This can also be true for industrial applications.Continue the aforementioned HIV/AIDS treatment example. Ifphysician A thinks potency of regimen X to be 83%, but physi-cian B judges it to be 87%, under the present FDES framework,they are forced to compromise and/or reach a consensus (e.g.,potency is 85%). The adoption of interval values or fuzzy num-bers (e.g., potency is [82%, 88%] or around 85%) would betterrepresent the diverse estimates.

A similar scenario occurs when the expert physicians definedfuzzy sets to linguistically describe different levels of the fourclinical parameters. For instance, two fuzzy sets “high” and“medium,” as shown in Fig. 1, were chosen to characterize thepossibilities of the system changing from the initial state (i.e.,



untreated mode) to “high” or “medium” state as far as potency isconcerned. The fuzzy sets represented the consensus of the twoexperts. The knowledge acquisition could not have been accom-plished had the experts failed to reach consensuses (differentexperts may insist to use different fuzzy sets to reflect their ownexperiences and knowledge). Clearly, a much better way is thatexperts will not be forced to reach consensus to begin with.An effective approach is to use a type-2 fuzzy set [26], whichutilizes a lower primary membership function and an upperprimary membership function to bound an infinite number oftype-1 membership functions to characterize imprecision anduncertainties. In addition, a secondary membership functioncan be defined over the membership grades at each value ofpotency.

These issues motivated us to expand the FDES theory so thatthey would be effectively dealt with. It turns out that all the ele-ments in fuzzy state vectors and fuzzy event transition matricesmust be fuzzy numbers, as opposed to crisp numbers under thecurrent framework. The operations of the fuzzy numbers arebased on interval number operations. To differentiate the newFDES from the existing ones, we call them the extended FDES(EFDES). We have also extended the theory to cover the parallelcomposition of EFDES. Like the transition from DES to FDESwhere the latter contains the former as a special case, FDES isa special case of EFDES because a crisp number is a specialcase of a fuzzy number. That means the theory developed in thispaper can be applied to FDES as well.

II. THEORY OF EFDESS

We first point out that multiple FDESs or EFDES runningconcurrently are required when the problem of interest is notvery simple. This is the case for our HIV/AIDS application, forexample. To make the presentation better in Section II-A andSection II-B1–II-B3, we will focus only on a single FDES orEFDES system when describing the theories.

Before developing the EFDES theory, we will first provide abrief overview of the FDES theory. More detailed informationon the theory is available in [10].

A. Outline of the FDESs Theory

A general FDES is modeled by a fuzzy automation G

G = (Q, Σ, δ, q0) (1)

where Q is the set of all possible fuzzy state vectors andq0 ∈ Q is the initial state vector. The state vector qk =[k v1 ,

k v2 , . . . ,k vN ] ∈ Q with N being the total number of

states and k = 1, 2, . . . , indicates time instance. Subscripts andsuperscripts are positive or nonnegative integers unless pointedout otherwise. The state variable k vi ∈ [0, 1] represents thepossibility (i.e., membership) of the system being in state i. Σis the set of all possible events. The jth event is represented byan event transition matrix σj = [j amn ]N ×N where the elementj amn ∈ [0, 1] describes the likelihood (i.e., membership) of

the system moving from state m to state n. δ : Q Σ → Q isa transition mapping that describes what event can occur at thecurrent time and what the resulting new state is. Here, the sym-bol represents some fuzzy operation specified by δ. It includes,but not limited to, the max–product or max–min fuzzy infer-ence operation in the fuzzy set theory [27]. Which operation touse is application-dependent and is up to the system developer.Max–product or max–min is an operation in which the opera-tors max( ), product( ), and min( ) are used. Product( ) meansmultiplication of the arguments while max( ) [or min( )] op-eration picks up the largest (or smallest) argument among thearguments involved. If the current fuzzy state vector is qk andthe event σj occurs, then the new fuzzy state vector can becomputed: qk+1 = qk σj .

We also extended the parallel composition of DES to FDES[10]. In the interest of brevity, they will not be presented here.

In the FDES framework, all the memberships in the fuzzystate vectors and fuzzy event transition matrices are crisp num-bers in [0, 1]. While this type of membership representation isadequate for many applications, there are situations where it maynot be powerful enough to handle complex uncertainties (recallthe issues mentioned in Section I and we will demonstrate morespecific cases later). To address this problem, we propose touse type-1 fuzzy sets to characterize the memberships, leadingto the EFDES theory presented next. We will use the notationsestablished in this section and expand them to present the newtheory in a parallel fashion.

B. EFDESs Theory

1) Conceptual and Notational Extensions From the FDESTheory: Like FDES, a general EFDES is also modeled by afuzzy automaton

G = (Q, Σ, δ, q0) (2)

where Q is the set of all fuzzy state vectors. Note that thenotations for (2) and those for (1) are similar with the onlydifference being the symbol∼ not adopted in (2). This will be thecase for the remaining notations for the EFDES in this section—all the notations for the EFDES will not use this symbol todistinguish them from their FDES counterparts. The state vectorqk at time instance k is mathematically represented as

qk = [ kV1kV2 · · · kVN ].

The variable q0 is used to signify the initial state vector. Here,N is the dimension of the state vector. Unlike the FDES, thestate variable kVi is generally a fuzzy set, but can be a numberin [0, 1] or an interval as special cases (in the rest of the paper, aninterval or an interval number will mean a special type-1 fuzzyset whose membership function is always 1 over the interval and0 outside the interval). The universe of discourse of kVi (i.e., thex-axis) is [0, 100%] representing all the possible likelihoods thatthe EFDES can be in state i. The vertical axis of kVi is [0, 1] torepresent the degree of truth. A linguistic label may be assigned



to each state variable for easier representation and interpretation(e.g., the system is in the state “high” with possibility kVi).

The set of all events is Σ and the event σj ∈ Σ is formallyrepresented by a fuzzy event transition matrix

σj =

jA11 · · · jA1N

.... . .

...jAN 1 · · · jAN N

, 1 ≤ j ≤ M

where M is the total number of events. jAmn , 1 ≤ m,n ≤ N ,is a fuzzy set characterizing the behavior of the EFDES movingfrom state m to state n when the jth event takes place. Theuniverse of discourse of jAmn is [0, 100%] to represent all thepossibilities for the system’s transition between the two states.The vertical axis ranges from 0 to 1 to show the degree oftruth. Each of state m and state n has a linguistic label. In thematrix row direction and the column direction, they should beconsistent with the linguistic labels of the corresponding statevectors. Note that in our previous effort [10], all the elementsin the event transition matrices are real numbers in [0, 1] (i.e.,membership grades), not fuzzy sets. We should also mentionthat an event itself is always crisp-–it either takes place or doesnot happen. But unlike the conventional DES where the systemcan be only in one state for any given time, an FDES or EFDESis allowed to be partially in more than one state at the same time.This is realized by fuzzy event transition matrices.

The parameter δ in (2) is a transition mapping that describeshow to generate a new state vector from the current fuzzy statevector and a fuzzy event transition matrix. Like the FDES theory,the mapping is mathematically described by

δ : Q Σ → Q

and the new fuzzy state vector can be computed by qk+1 =qk σj . Unlike the FDES theory though, the arguments of theoperation are no longer restricted to numbers. Instead, theycan be fuzzy sets.

Through the aforementioned extensions, it should be obviousthat the FDES theory is contained in the EFDES theory as aspecial case because a crisp number is a special case of a fuzzyset.

2) Computation of New Fuzzy States: In qk+1 = qk σj ,the arguments of the operation can be fuzzy sets or their spe-cial cases—interval numbers and numbers (an interval should beexpressed as a fuzzy set whose membership is constant 1 over theinterval while a number should be expressed as a singleton fuzzyset). Different kinds of fuzzy sets may be used in a mixture fash-ion. The most common operations are the max–product oper-ation and the max–min operation. To avoid possible confusionregarding the type of the arguments, we use maxS –productS[i.e., maxS ( ) and productS ( )] and maxS– minS when the ar-guments are fuzzy sets, and maxI–productI and maxI – minIwhen the arguments are interval numbers. When the argumentsare numbers, just max–product and max–min will be utilized,which are the conventions in the literature.

When fuzzy set arguments are involved, k+1Vn in qk+1 canbe calculated by

k+1Vn = maxS1≤i≤N

(kVi × jAin

)= maxS(kV1× jA1n , kV2× jA2n , . . . , kVN × jAN n )

(3)

where × means productS( ). Alternatively, if the maxS–minSoperation is preferred, k+1Vn is obtained by

k+1Vn= maxS1≤i≤N

(minS(kVi ,

jAin ))

= maxS(minS(kV1 ,jA1n ),

minS(kV2 ,jA2n ), . . . ,minS(kVN , jAN n )). (4)

To carry out the operations of productS( ), maxS ( ), andminS ( ), the arithmetic based on Zadeh’s extension principlemay be employed [28]. Generally speaking, however, usinga computer program to automatically produce a closed-formmathematical solution can be very challenging, if not impossi-ble, if the fuzzy sets are allowed to be of arbitrary functions. Thisis primarily due to the nature of the method in which the mem-bership functions of the fuzzy sets have to be compared witheach other to determine which function is the largest or smallestand in which parts of the universes of discourse. Alternatively,one may discretize the universes of discourse of the fuzzy setsso that an approximate numerical solution can be generated bya computer program in an automation fashion.

While permitting kVi and jAmn to be arbitrary fuzzy setsmay be desirable from the standpoint of a theory and its com-prehensiveness, it may not be reasonable for practical applica-tions. Take the theory of fuzzy control and fuzzy modeling asan example. Theoretically speaking, arbitrary fuzzy sets are al-lowed, but only certain types of fuzzy sets (i.e., fuzzy numbers)are practically sensible, and hence have been heavily used in al-most all the studies in the literature. Fuzzy numbers are a classof general fuzzy sets that satisfy the following four require-ments [27]: 1) it is a normal (continuous) type-1 fuzzy set (i.e.,its height must be 1); 2) its support set is bounded; 3) its α-cutset is a closed interval; and 4) the support set of its strong 0-cutset is bounded. The restrictions induced by the requirementsare quite mild, but the benefit is substantial—eliminating manyof the general fuzzy sets that do not make practical sense inthe first place. Obviously, there are an infinite number of typesof fuzzy numbers, all of which meet these four requirements.Using fuzzy numbers for kVi and jAmn should be adequatenot only because they are very diverse and flexible to representvarious knowledge and uncertainties, but also because they haveclearer and more intuitive meanings than fuzzy sets of arbitraryshapes do. Widely used fuzzy number types include the triangu-lar and the trapezoidal. Any type of fuzzy numbers is allowed inthe EFDES theory and a specific choice depends on the systemdesigner.

An even more important advantage of using fuzzy numbersfor kVi and jAmn as opposed to general fuzzy sets is that itis much easier to obtain closed-form mathematical results for



the operations (3) and (4) using an arithmetic procedure that isbased on the α-cut set approach [28]. Better yet, the procedurecan be automatically executed by a computer program using asoftware package capable of symbolic mathematical operations(e.g., Mathematica and Maple software on the market). We pointout that even though the complexity of their execution is signifi-cantly different, the extension principle-based arithmetic and theα-cut-set-based approach generate the same result when fuzzynumbers are involved. We now outline the latter method.

According to the first decomposition theorem [27] in fuzzyset theory, a continuous fuzzy number X can be represented asthe union of its α-cut sets

X =⋃

α∈[0, 1]

α ⊗ [x(α), x(α)]

where [x(α), x(α)] is an α-cut set of X, which is an interval (i.e.,an interval number) containing all the elements in the universeof discourse of X whose membership grades are greater than orequal to the specified membership grade α. Here, x(α) and x(α)are the lower and upper terminal points whose values dependon α. And

⋃is the Zadeh fuzzy union operation [i.e., max( )]

and α ⊗ [x(α), x(α)] means that the membership grade for[x(α), x(α)] is α.

Based on the α-cut representation, the operations of fuzzynumbers × [i.e., products( )], maxS( ), and minS( ) in (3) and(4) can be expressed as [27]kVi × jAin

=⋃

α∈[0, 1]

α ⊗ ([kvi(α), k vi(α)] × [jain (α), j ain (α)])

(5)

maxS(kVi ,jAin )

=⋃

α∈[0, 1]

α ⊗ maxI([kvi(α), k vi(α)], [jain (α), j ain (α)])

(6)

minS(kVi ,jAin )

=⋃

α∈[0, 1]

α ⊗ minI([kvi(α), k vi(α)], [jain (α), j ain (α)])

(7)

where kvi(α), k vi(α), jain (α), and j ain (α) are numbers in[0, 1], and × in (5) is the multiplication operation for intervalnumbers. According to the interval number operations [27], theprevious expressions lead to the following:kVi × jAin

=⋃

α∈[0, 1]

α ⊗ [kvi(α) × jain (α), k vi(α) × j ain (α)] (8)

maxS(kVi ,jAin )

=⋃

α∈[0, 1]

α ⊗[max(kvi(α), jain (α)), max(k vi(α), j ain (α))]

(9)

minS(kVi ,jAin )

=⋃

α∈[0, 1]

α ⊗[min(kvi(α), jain (α)), min(k vi(α), j ain (α))]

(10)

where × in (8) is the standard multiplication operation for num-bers. Therefore, (3) becomes

k+1Vn

=⋃

α∈[0, 1]

α⊗[

max1≤i≤N

(kvi(α) × jain(α)), max1≤i≤N

(k vi(α) × j ain(α))]

(11)

and (4) can be written as

k+1Vn

=⋃

α∈[0, 1]

α⊗[

max1≤i≤N

(min(kvi ,jain )), max

1≤i≤N(min(k vi ,

j ain ))].

(12)

We point out that because fuzzy numbers can be representedas the union of their α-cut sets and this representation can berealized by a symbolic mathematical software package automat-ically and quickly with the result stored in the computer priorto the computations in (5)–(12), the computation burden of thisapproach is low.

3) Construction of Fuzzy Event Transition Matrices: Fuzzyevent transition matrices play a central role in the theory ofEFDES. They are the core of EFDES and capture and representmuch of expert knowledge/experience that may be subjective,vague, and imprecise in practical applications. Furthermore,they are used to compute future states from the present one for aparticular system variable of interest, say x (e.g., potency). Theprocess for generating a fuzzy event transition matrix in EFDESis more complex than that in FDES partially because fuzzy sets,jAmn , are now needed to be produced to represent the fuzzyprobabilities in the end of the process instead of crisp numbers inthe case of FDES. To construct a fuzzy event transition matrix isto determine all the fuzzy sets, each of which is an element of thematrix jAmn . The elements should capture expert knowledgeon state transitions and represent the possibility of the systemmoving from one state to another when a system event happens(e.g., patient is treated by a regimen).

For an N × N event transition matrix, there are N differentstates. The element jAmn indicates the fuzzy possibility for thesystem variable to transfer from state m (supposedly “medium”)to state n (supposedly “high”). The system developer may usereal (measurement) data to compute the fuzzy possibility. How-ever, this can be difficult because the states themselves are onlyvaguely defined (e.g., “medium” and “high”); thus, quantifyingthe possibility can be challenging. Additionally, the real datamay not be available in many real-world applications. Alterna-tively, the developer may determine the fuzzy possibilities in amore subjective manner. One possible approach is to directlywrite out the fuzzy sets (or in special cases, membership gradesand interval numbers) in an ad hoc fashion. Nevertheless, it can



be hard to manually generate N × N fuzzy sets systematicallyand consistently without contradictions if N is large and/or morethan one expert’s knowledge is to be captured but experts withconflicting or opposite opinions disagree with one another with-out consensus (a norm in medicine). To resolve the difficulty, wepropose the following systematic method for determining jAmn .

With the help of a group of domain experts, the system devel-oper first defines a fuzzy set, designated as D, over the universeof discourse of x, denoted by X, that represents the experts’knowledge concerning the possibility of the system transitingfrom state m to state n for different values of x (say, potency)after the jth event occurs. (For instance, D can be a fuzzy setdescribing the possibility of general patient population movingfrom “medium” potency state to “high” potency state after theystart regimen ABC (i.e., the jth event) whose potency rangesfrom, say, 70% to 95%). This fuzzy set can be either type-1 ortype-2, depending on the extent of experts’ disagreement. It doesnot need to be defined over the entire universe of discourse; onlypart of the universe of interest would suffice (e.g., 60%–100%in the aforementioned example). Then, the developer needs toassign a value, designated as F, which can be a crisp number,an interval, a type-1 fuzzy set, or an type-2 fuzzy set, to x (i.e.,the potency value of regimen ABC) that is associated with theparticular event.

Without loss of generality, assume that D is a type-2 fuzzyset represented by a 3-D membership function z = ϕD(x, y),where x ∈ X and y is a variable representing the membershipgrade for x. The relationship between x and y is characterizedby two membership functions (instead of one in the case oftype-1 fuzzy sets)—an upper membership function and a lowermembership function. For a given value of x = x′, the upperand lower membership functions will intersect with the verticalline of x = x′, and respectively produce an upper point anda lower terminal point of an interval of membership grades.The interval is denoted by µD

x ′ . Obviously, µDx ′ is a subinterval

of [0, 1] (i.e., µDx ′ ⊆ [0, 1]). A type-1 fuzzy set can be defined

over µDx ′ , which mathematically is z = ϕD(x′, y). Therefore,

z represents the membership of y. The collection of all thesetype-1 fuzzy sets for the entire universe of discourse of x iscalled a secondary membership function [29], which is z =ϕD(x, y). All the intervals µD

x ′ due to different values of x formso-called primary memberships µD

x (obviously, µDx ⊆ [0, 1]).

The collection of all the left and right terminal points of µDx ′

forms the upper and lower membership functions, respectively.Note that the two functions along with the area bounded by themare the primary memberships. Formally, D can be represented by

D = ϕD(x, y)|∀x ∈ X, ∀y ∈ µDx . (13)

In the interest of generality, we suppose that F is a type-2 fuzzyset (crisp number, interval number, and type-1 fuzzy set are allits special cases)

F = ϕF(x, y)|∀x ∈ X, ∀y ∈ µFx . (14)

We then apply a fuzzy intersection operator on D and F.Generally speaking, the result is a type-2 fuzzy set. Denote it

by T that is calculated according to [30]

T = ϕT(x, y)|ϕT(x, y)

= supy=yD ∩yF

(ϕD(x, yD) ∩ ϕF(x, yF)), ∀x ∈ X

y = yD ∩ yF ∀yD ∈ µDx ⊆ [0, 1], ∀ yF∈ µF

x ⊆ [0, 1](15)

where ∩ can be any fuzzy AND operator such as the Zadeh AND

operator [29] and sup( ) is the operation to pick up the largestelement (i.e., the max( ) operator). The primary membershipgrade y (=yD ∩ yF ) and the secondary membership grade(=ϕD(x, yD) ∩ ϕF(x, yF)) are computed. If more than onevalue of ϕD(x, yD) ∩ ϕF(x, yF) is produced for the same valueof y, the largest value will be selected as the result by the sup( )operation.

A fuzzy set type reducer is then needed to convert T to atype-1 fuzzy set, which is jAmn . Note that jAmn represents therelationship between y and z, instead of x and y needed for othertype-2 applications in the literature. There exist several typereducers, and we modify the so-called height type-reductionmethod for our purpose. Consequently, the membership functionfor jAmn , denoted by jAmn (y), defined over the universe ofdiscourse [0, 1] is [29]

jAmn (y) = supx

ϕT(x, y). (16)

Since F can be a crisp number, an interval, a type-1 fuzzy set, ora type-2 fuzzy set, and D can be a type-1 fuzzy set or a type-2fuzzy set, there exist eight combinations. Formula (15) coversthe most general case—both F and D are type-2 fuzzy sets. Theformulas for the rest of the seven situations can be derived from(15) as a special case. The results are summarized in Table I.Note that situation No. 8 is the FDES.

There lacks a method in the current literature to produce a so-lution for (15) when both D and F are general type-2 fuzzy sets.If they are interval type-2 fuzzy sets, numerical solutions canbe obtained via algorithm for situations Nos. 1–5 [29]. The re-maining three situations are readily computable as they involvetype-1 fuzzy sets only. For better illustration, in Section III, wewill use the HIV/AIDS example to demonstrate how to carryout (15) and (16) for situations Nos. 4 and 6.

4) Parallel Composition of EFDESs: Like the DES andFDES theories, to model a (complex) EFDES, a convenientway is often to model each subsystem first, and then synthe-size them by parallel composition. Since the memberships ofall the states and events have been extended from numbers tofuzzy sets, we need to modify the operations of the parallelcomposition accordingly. Suppose two EFDES are given

Gm 1 = (Qm 1 ,Σm 1 , δm 1 ,q0) Gm 2 = (Qm 2 ,Σm 2 , δm 2 ,q′0).

Then, through parallel composition ‖, the combined EFDESGm = (Qm , Σm , δm , q∗

0) is

Gm = Gm 1 ||Gm 2

= (Qm 1 × Qm 2 ,Σm 1 ∪ Σm 2 , δm 1 × δm 2 , (q0 ,q′0))



TABLE IFORMULAS FOR COMPUTING THE FUZZY NUMBERS jAm n IN FUZZY EVENT

TRANSITION MATRICES

where m, m1 , and m2 are positive integers. We now describehow the parallel composition of EFDES is executed.

There are two steps: 1) compute the state vectors for Gm

and 2) compute the fuzzy event transition matrices for Gm . Forthe first step, assume that qh and q′

i are fuzzy state vectors inQm 1 and Qm 2 , respectively, and hVj (1 ≤ j ≤ N ) and iV′

k

(1 ≤ k ≤ N ′) are elements of qh and q′i , respectively. Here, h

and i are nonnegative integers, and j, k, N , and N ′ are positiveintegers, where N and N ′ are the dimensions of fuzzy state vec-tors in Gm 1 and Gm 2 , respectively. The dimension of the statevectors in Gm is one row and N × N ′ column. The composi-tion of the jth element in qh (i.e., hVj ) with the kth element inq′

i (i.e., iV′k ) yields the element in the composite state vector

located in the column corresponding to (j, k). The value of thatnew element is a fuzzy set calculated by

minS(hVj ,iV′

k ). (17)

In the second step, we determine the event transition matricesof Gm . Designate M ∗, M , and M ′ as the total number of eventsfor Gm , Gm 1 , and Gm 2 , respectively. Suppose that β1 Ak1 k2 isan element of the event matrix σβ1 (1 ≤ β1 ≤ M ) in Gm 1 andβ2 A′

j1 j2is an element of the event matrix σ′

β2(1 ≤ β2 ≤ M ′)

in Gm 2 . After the parallel composition, the dimension of σβ1

and σ′β2

will no longer be the same as that in Gm 1 and Gm 2 ,respectively. The dimension of all the events in Gm should be(N × N ′) × (N × N ′). Let σ∗

β1signify the representation of

σβ1 in Gm (discussion concerning σ∗β2

would be similar and isomitted for brevity). Suppose that β1 A∗

kj (1 ≤ k, j ≤ N × N ′)is an element in σ∗

β1that represents a concurrent transition from

state k1 to state k2 (1 ≤ k1 , k2 ≤ N ) in Gm 1 and from state j1to state j2 (1 ≤ j1 , j2 ≤ N ′) in Gm 2 . It can be determined asfollows.

The first scenario is that Gm 2 does not have the event σβ1 .Then

β1 A∗kj =

β1 Ak1 k2 , if j1 = j2

Z, if j1 = j2 .(18)

For those β1 A∗kj corresponding to the unchanged state in Gm 2

(i.e., j1 = j2), the behavior of the event transition from k to jin Gm should be the same as that from k1 to k2 in Gm 1 . Hence,β1 A∗

kj = β1 Ak1 k2 . For those β1 A∗kj representing state change

in Gm 2 (i.e., when j1 = j2), β1 A∗kj = Z, where Z represents a

type-1 singleton fuzzy number 0 whose membership is 1 at 0 inthe universe of discourse and is 0 elsewhere (by fuzzy numbernotation, this fuzzy number can be expressed as 1/0). This isbecause Gm 2 does not have the event σβ1 ; hence, any statechange in Gm 2 should not appear in Gm , and thus Z.

The second scenario is that Gm 2 also has the event σβ1 . Then

β1 A∗kj = minS (β1 Ak1 k2 ,

β2 A′j1 j2 ).

This is because σβ1 now takes place in both Gm 1 and Gm 2 tothe extents of β1 Ak1 k2 and β2 A′

j1 j2, respectively. Therefore,

the fuzzy AND operation should be applied (e.g., the Zadeh AND

operator used here).We point out that state explosion problem due to parallel com-

position is universal to all DESs and EFDES is no exception.



Many methods have been proposed in the DES literature to ad-dress this problem. They include modular control, decentralizedcontrol (e.g., [31]), hierarchical control [32], online control withlimited and variable look-ahead policies [33], and various tech-niques for model abstractions. These methods can also be usedin EFDES.

III. ILLUSTRATIVE EXAMPLES

To make the aforementioned theoretical development easier tounderstand, we now provide three detailed numerical examplesthat are related to medical applications.

Example 1: Construct fuzzy event transition matrices ofEFDES for HIV/AIDS treatment. Continue the HIV/AIDS treat-ment application mentioned in Section I. We will consider twoof the eight scenarios in Table I (more specifically, situationsNos. 4 and 6).

There were three states for regimen’s potency—Initial,Medium, and High. When the FDES theory was applied to atreatment-naive patient, the state vector was

Initial Medium High

q0 = [ 1 0 0 ]

which means that the patient was not in Medium or High potencystate but in the Initial state because he/she had never receivedtreatment before. For clearer illustration, we list the state nameson the top of the vector. Using the EFDES theory presentedbefore, the three crisp membership values become three fuzzynumbers as follows:

Initial Medium High

q0 = [ 0V10V2

0V3 ] (19)

where let 0V1 = 1/1 and 0V2 = 0V3 = Z. Z is the singletonfuzzy number 0 defined in Section II-B4 while 1/1 representsa type-1 singleton fuzzy number 1. One may use other fuzzynumbers 0 and 1 (e.g., the trapezoidal function), which willlikely make the example complicated in computation.

Now, we need to create the fuzzy event transition matrix forpotency, which is 3 × 3. The process for the FDES is illustratedin Fig. 1 where the regimen’s potency is assumed to be 85%.Using the two membership grades in Fig. 1 (i.e., the intersectionsof the vertical line of 85% potency and the two fuzzy sets) canconstruct the fuzzy event transition matrix for the FDES as

Initial Medium High

σ1 =

0 0.4296 10 0 00 0 0

Initial

Medium

High.

(20)

The corresponding fuzzy event transition matrix σ1 for theEFDES is

Initial Medium High

σ1 =

Z 1A121A13

Z Z ZZ Z Z

Initial

Medium

High.

(21)

Fig. 2. Same two fuzzy sets “medium” and “high,” as shown in Fig. 1. Alsoshown is type-1 fuzzy number P defining potency of the particular regimenbeing “around 85%” and fuzzy number PM resulted from fuzzification usingthe Zadeh fuzzy intersection operation.

We now discuss how to produce the fuzzy numbers 1A12 and1A13 under different conditions.

A. First Scenario

The first scenario is that a group of experts all agree to usethe same type-1 fuzzy sets [corresponding to the fuzzy set D in(13)] “Medium” and “High” for potency (the fuzzy sets describethe possibilities of the system changing from the initial state to“Medium” or “High” state). But they fail to reach a consensuson the exact potency value of the regimen. Hence, the regimen’spotency should not be a crisp number but a fuzzy number. Thisis situation No. 6 in Table I. Fig. 2 shows such a hypotheticalexample where a symmetrical triangular fuzzy number P cen-tered on 85% is used to represent diverse opinions of the expertson the potency value [corresponding to the fuzzy set F in (14)].The membership function of P is

µP(x) =

x − 0.820.03

, 0.82 < x < 0.85

0.88 − x

0.03, 0.85 ≤ x < 0.88

0, elsewhere in [0, 1]

where x is the potency value. Using Table I, one can obtain1A12 and 1A13 . By applying an intersection operation (min( )is used here), two fuzzy numbers, PM and PH , are resulted[corresponding to the fuzzy set T computed by (15)]. PM isproduced by intersecting P with “Medium” point by point alongthe potency axis. The membership function of the resulting PM ,shown as a bold curve in Fig. 2, is

µPM (x) =

x − 0.820.03

, 0.82 < x < 0.8354

e−12 ( x −0 . 7 2

0 . 1 )2

, 0.8354 ≤ x < 0.8703

0.88 − x

0.03, 0.8703 ≤ x < 0.88

0, elsewhere in [0, 1].



The minimal and maximal membership values of µPM (x) are0 and 0.5138, respectively (Fig. 2). Therefore, according toTable I, the fuzzy number 1A12 is an interval number [0, 0.5138].In our case, PH happens to be identical to P (thus, it is not markedin Fig. 2). Subsequently, the minimal and maximal membershipvalues after the intersection are 0 and 1, respectively, whichmeans that 1A13 is an interval number [0, 1]. Hence, the fuzzyevent transition matrix σ1 in (21) is

σ1 =

Z [0, 0.5138] [0, 1]

Z Z Z

Z Z Z

. (22)

Compared to the FDES event transition matrix σ1 in (20), thenonzero elements in (22) are interval numbers instead of crispnumbers. The practical implication is that the uncertainty onthe exact potency value has been conveniently expressed bythe domain expert (i.e., the physician), and has been effectivelyprocessed by the EFDES approach.

B. Second Scenario

This scenario is the opposite of the first scenario—the expertsagree to use the exact same (crisp) value to represent the regi-men’s potency, but they have different definitions for the term“Medium” (and/or “High”). If the inclusion of all the differ-ent opinions of the experts is desired, “Medium” and “High”should be defined as type-2 fuzzy sets [29]. This is situationno. 4 in Table I. Fig. 3 shows an example. The primary mem-berships of a type-2 fuzzy set consists of an upper membershipfunction and a lower membership function to bound a range ofopinions technically called footprint of uncertainty [Fig. 3(a)].The primary memberships for “Medium” and “High” are math-ematically defined in Table I, and the two bounded regions (i.e.,the footprints of uncertainty) are shown as two gray areas withdistinct intensities in Fig. 3(a).

According to Table I, we can obtain 1A12 and 1A13 as fol-lows. Draw a vertical line at a potency value [85% in Fig. 3(a)].The line will intersect with the upper and lower membershipfunctions of the two type-2 fuzzy sets, resulting in two intervalsof the membership grades—one for “Medium” and the other for“High.” They are actually two results after applying the Zadehfuzzy intersection operation to “Medium” and the 85% potencyline, and “High” and the same line, respectively. The intervalsthen serve as the universes of discourse for secondary member-ship functions of the respective type-2 fuzzy sets at the potencyvalue where the vertical line is put. A secondary membershipfunction describes the likelihood whether each of the member-ship grades will take place. Like the primary memberships andfootprint of uncertainty, the secondary membership functionsare developed by the experts/developer. For instance, they maybe defined in such a way to reflect the experts’ individual prefer-ence/bias exhibited when they define the primary memberships.In Fig. 3(b), we suppose that the secondary membership function

Fig. 3. Define “medium” and “high” potency as type-2 fuzzy sets to coverdiverse opinions of a group of AIDS experts. (a) Primary memberships withthe footprints of uncertainty of the type-2 fuzzy sets “medium” and “high.” (b)Secondary membership function for the fuzzy set “medium” at potency of 85%is supposedly triangular (solid line) or if the type-2 fuzzy set is assumed to bean interval type-2 fuzzy set (dotted line).

for “Medium” at 85% potency is of the triangular type

ϕM(y) =

2(y − yM)yM − yM

, yM

< y <yM + yM

2

2(yM − y)yM − yM

,yM + yM

2≤ y < yM


where y, the horizontal axis of Fig. 3(b), is the membershipgrade for potency in Fig. 3(a). Similarly, the secondary mem-bership function for the type-2 fuzzy set “High” is supposed to



TABLE IIMATHEMATICAL DEFINITIONS OF THE LOWER AND UPPER PRIMARY

MEMBERSHIP FUNCTIONS OF THE TYPE-2 FUZZY SETS FOR POTENCY

SHOWN IN FIG. 3(A)

be triangular

ϕH(y) =

2(y − yH)yH − yH

, yH

< y <yH + yH

2

2(yH − y)yH − yH

,yH + yH

2≤ y < yH


Note that yM , yM , yH , and yH are marked in Fig. 3(a) and theirvalues can be determined using the functions in Table II. Givenpotency of 85%, yM = 0.3247, yM = 0.5461, yH = 0.9802, andyH = 1.

Assigning ϕM(y) and ϕH(y) to the fuzzy numbers 1A12 and1A13 , respectively, completes the process of producing the ma-trix. The fuzzy number 1A12 is triangular

ϕ1A1 2(y) =

9.0334y − 2.9331, 0.3247 < y < 0.43544.9331 − 9.0334y, 0.4354 ≤ y < 0.54610, elsewhere in [0, 1]

and the fuzzy number 1A13 is also triangular

ϕ1A1 3(y) =

101.0101y − 99.0101, 0.9802 < y < 0.9901101.0101 − 101.0101y, 0.9901 ≤ y < 10, elsewhere in [0, 1].

Consequently, the fuzzy event transition matrix σ1 is

σ1 =

Z ϕ1A1 2 ϕ1A1 3

Z Z ZZ Z Z

. (23)

In contrast to the FDES event transition matrix σ1 in (20)where all the elements are crisp numbers, all the elements in (23)are fuzzy numbers. From practice perspective, the advantage issubstantial—the EFDES method is capable of simultaneouslycapturing and representing a range of opinions contributed bya group of domain experts that do not agree with one another.This is something unachievable under the framework of FDES.

So far, we have considered the case when the type-2 fuzzysets are general in the sense that their primary memberships andsecondary membership functions can be arbitrary but reasonableshapes. If the fuzzy set is an interval type-2 fuzzy set, whichcan be in the case if the experts want to treat the likelihoods of

membership grades for a parameter of interest (e.g., potency) ateach parameter value equally or uniformly, we can simplify theaforementioned results by letting ϕM(y) = 1 or ϕH(y) = 1. Tomake a concrete example, if the secondary membership functionfor “Medium” is as shown in Fig. 3(b), then the fuzzy number1A12 reduces to an interval

ϕ1A1 2 (y) = [0.3247, 0.5461] .

Example 2: Computing new fuzzy states using fuzzy eventtransition matrices. Assume that the fuzzy event transition ma-trix is (23). It can then be used to compute new fuzzy state vectorq1 from the current state, which is the initial state q0 in (19)

q1 = q0 σ1 =[ 1

1Z Z

]

Z 1A121A13

Z Z ZZ Z Z

= [ Z 1A12

1A13 ] .

The maxS–productS operation is used for as all the ele-ments are type-1 fuzzy numbers.

To make this example nontrivial, let us now assume that theHIV/AIDS patient has failed the first-round treatment and mustreceive a second-round treatment. Without loss of generality,suppose the potency event transition matrix for a second-roundregimen that the patient takes is obtained via one of the twoapproaches described earlier

σ2 =

Z Z ZZ 2A22 ZZ 2A32

2A33

where the type-1 fuzzy numbers 2A22 , 2A32 , and 2A33 have thefollowing triangular membership functions:

ϕ2A2 2(y) =

10y − 2, 0.2 < y < 0.34 − 10y, 0.3 ≤ y < 0.40, elsewhere in [0, 1]

ϕ2A3 2(y) =

10y − 6, 0.6 < y < 0.78 − 10y, 0.7 ≤ y < 0.80, elsewhere in [0, 1]

ϕ2A3 3(y) =

10y − 8, 0.8 < y < 0.910 − 10y, 0.9 ≤ y < 10, elsewhere in [0, 1]

.

The new potency state after the patient is treated is

q2 = q1 σ2

= [ Z 1A121A13 ]

Z Z ZZ 2A22 ZZ 2A32

2A33

= [Z maxS(Z, 1A12 × 2A22 ,

1A13 × 2A32)

maxS(Z, Z, 1A13 × 2A33)]

= [Z maxS(1A12 × 2A22 ,1A13 × 2A32) 1A13 × 2A33]

= [ Z 2V22V3 ]



Fig. 4. Graphical representations of the triangular fuzzy numbers 2 V2 and2 V3 .

where 2V2 and 2V3 are type-1 fuzzy numbers whose member-ship functions are (Fig. 4)

ϕ2 V2(y) =

√

2162.7 + 1010.1y − 52.5051, 0.5881 < y < 0.6931

−√2162.7 + 1010.1y + 54.5051, 0.6931≤ y < 0.8


ϕ2 V3(y) =

√

2070.7 + 1010.1y − 53.5051, 0.7842 < y < 0.8911

−√2070.7 + 1010.1y + 55.5051, 0.8911≤ y < 1


Using the key building blocks illustrated in Examples 1 and 2,we are developing an EFDES for HIV/AIDS treatment that willinclude a system validation against the actual patient treatmentrecords. A complete description of the system is beyond thescope and space limit of this paper. It would require a full-lengthpaper as was the case for our FDES HIV/AIDS system [24].

Example 3: Parallel composition of EFDESs. This example isunrelated to the first two examples and HIV/AIDS. However, ina sense it is realistic in that it is related to two important humanorgans—heart and lung. Suppose an EFDES is composed of twosubsystems G1 and G2 . G1 describes patient’s lung condition

and G2 heart condition. In G1 , the lung condition is classifiedas “Excellent” (E, for short), “Fair” (F), and “Poor” (P), whichis represented by a fuzzy state vector

qk = [ kV1kV2

kV3 ]

where kV1 , kV2 , and kV3 are fuzzy numbers charactering thestates “Excellent,” “Fair,” and “Poor,” respectively. There aresix events denoted as σj , j = 1, . . . , 6. Patient’s heart conditionis also classified as “Excellent” (E′, for short), “Fair” (F′), and“Poor” (P′), which is described by a fuzzy state vector in G2

q′i = [ iV′

1iV′

2iV′

3 ].

The three elements are fuzzy numbers concerning “Excellent,”“Fair,” and “Poor.” Suppose that there are six events in total forthe heart as well; they are labeled as αn , n = 1, . . . , 6. Also,suppose that G1 and G2 do not share a common event, i.e., thetreatment will not affect heart and lung simultaneously.

The dimension of the resultant state vector of G1‖G2 is 9(i.e., 3 × 3) that covers the following nine combinations of thestates: (E, E′), (E, F′), (E, P′), (F, E′), (F, F′), (F, P′), (P, E′),(P, F′), and (P, P′). The value of each composite state can becalculated according to (17)

(qk , q′i) = [(E, E′), (E, F′), (E, P′), (F, E′), (F, F′),

(F, P′), (P, E′), (P, F′), (P, P′)]

= [minS(kV1 , iV′1), minS(kV1 , iV′

2), minS(kV1 , iV′3),

minS(kV2 ,iV′1), minS(kV2 , iV′

2), minS(kV2 , iV′3),

minS(kV3 ,iV′1), minS(kV3 , iV′

2), minS(kV3 , iV′3)].

Next, we reformulate all the event transition matrices in G1 andG2 . Without loss of generality, let us focus on event σ1 only,which is related to change of lung condition from “Excellent” to“Fair.” It is represented by the following fuzzy transition matrix:

σ1 =

1B11

1B121B13

1B211B22

1B23

1B311B32

1B33

.

Elements in σ∗1 , which is σ1 after G1 ‖G2 , can be computed

using (18) since G2 does not contain event σ1 . The result is,(∗), as shown at the bottom of this page.

(E, E′), (E, F′), (E, P′), (F, E′), (F, F′), (F, P′), (P, E′), (P, F′), (P, P′)

σ∗1 =

1B11 Z Z 1B23 Z Z 1B13 Z Z

Z 1B11 Z Z 1B23 Z Z 1B13 Z

Z Z 1B11 Z Z 1B12 Z Z 1B131B21 Z Z 1B22 Z Z 1B23 Z Z

Z 1B21 Z Z 1B22 Z Z 1B23 Z

Z Z 1B21 Z Z 1B22 Z Z 1B231B31 Z Z 1B32 Z Z 1B33 Z Z

Z 1B31 Z Z 1B32 Z Z 1B33 Z

Z Z 1B31 Z Z 1B32 Z Z 1B33

(E,E′)

(E,F′)

(E,P′)

(F,E′)

(F,F′)

(F,P′)

(P,E′)

(P,F′)

(P,P′).

(*)



For better illustration, we list all the state combinations onthe top and right side of the matrix. We now explain how theelements are obtained via (18). Take the situation when lungcondition changes from “Excellent” to “Fair” but heart conditionremains the same as an example. It involves three elements inσ∗

1 : (E, E′)→ (F, E′), (E, F′)→ (F, F′), and (E, P′)→ (F, P′).Clearly, the value of all these three elements is 1B21 accordingto (18). On the other hand, consider the situation when bothlung and heart conditions change from “Excellent” to “Fair.”It involves only one element in σ∗

1 corresponding to (E, E′)→(F, F′). Given that j1 = j2 and G2 does not contain event σ1 ,extent for the behavior of event transition (E, E′)→ (F, F′) inGm should be Z.

IV. CONCLUSION

We have extended the theory of FDES to a theory of EFDESthat can effectively deal with the two general and critically im-portant issues concerning the representation and acquisition ofexperts’ knowledge and subjectivity. Under the new framework,elements in the fuzzy state vectors and fuzzy event transitionmatrices can be type-1 fuzzy numbers as opposed to numbers inthe original FDES theory. The extended theory not only enablesdomain experts to more conveniently and realistically expresstheir uncertain and imprecise knowledge and subjective judg-ments but also makes their capture and representation possible.We have established mathematical operations that cover the con-struction of the event transition matrices and the computation ofthe new fuzzy states and new fuzzy event transitions. Further-more, we have developed the corresponding parallel composi-tion. Three detailed numerical examples are provided to showhow to use the key components of the extended theory.

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Xinyu Du (S’05) received the B.S. and M.S.degrees in automation from Tsinghua University,Beijing, China, in 2001 and 2004, respectively. Heis currently working toward the Ph.D. degree in theDepartment of Electrical and Computer Engineering,Wayne State University, Detroit, MI.

His current research interests include the devel-opment of novel biomedical acoustic sensor sys-tems, type-2 fuzzy systems, and fuzzy discrete eventsystems.



Hao Ying (S’88–M’90–SM’97) received the B.S.and M.S. degrees in electrical engineering fromDonghua University (formerly China Textile Univer-sity), Shanghai, China, in 1982 and 1984, respec-tively, and the Ph.D. degree in biomedical engineer-ing from the University of Alabama, Birmingham, in1990.

In 2000, he joined Wayne State University (WSU),Detroit, MI, where he is currently a Professor in theDepartment of Electrical and Computer Engineering.Between 1992 and 2000, he was a faculty member at

the University of Texas Medical Branch, Galveston. Between 1998 and 2000,he was an Adjunct Associate Professor of the Biomedical Engineering Programat the University of Texas, Austin. He has authored the book Fuzzy Controland Modeling: Analytical Foundations and Applications (IEEE Press, 2000).He has authored or coauthored 89 peer-reviewed journal publications and over120 conference papers. He is an Associate Editor, an Area Editor, or a memberof the Editorial Board for 11 international journals.

Prof. Ying is a member of the Fuzzy Systems Technical Committee of theIEEE Computational Intelligence Society and is also the Chair of its Task Forceon Competitions. He is an elected board member of the North American FuzzyInformation Processing Society (NAFIPS). He was the Program Chair for theNAFIPS Conference in 1994 and 2005 and the Publication Chair for the 2000IEEE International Conference on Fuzzy Systems. He was a Program Commit-tee Member for 35 international conferences.

Feng Lin (S’86–M’87–SM’07) received the B.Eng.degree in electrical engineering from Shanghai Jiao-Tong University, Shanghai, China, in 1982, and theM.A.Sc. and Ph.D. degrees in electrical engineeringfrom the University of Toronto, Toronto, ON, Canada,in 1984 and 1988, respectively.

From 1987 to 1988, he was a Postdoctoral Fel-low at Harvard University, Cambridge, MA. Since1988, he has been with the Department of Electricaland Computer Engineering, Wayne State University,Detroit, MI, where he is currently a Professor. His

current research interests include discrete-event systems, hybrid systems, ro-bust control, and image processing. He was a consultant for GM, Ford, Hitachi,and other auto companies. He is the author of a book titled Robust ControlDesign: An Optimal Control Approach.

Prof. Lin received a George Axelby Outstanding Paper Award from theIEEE Control Systems Society. He was also the recipient of a Research Initi-ation Award from the National Science Foundation, an Outstanding TeachingAward from Wayne State University, a Faculty Research Award from ANRPipeline Company, and a Research Award from Ford. He was an Associate Ed-itor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL.



A Hybrid Approach for Design of Stable AdaptiveFuzzy Controllers Employing Lyapunov Theory and

Particle Swarm OptimizationKaushik Das Sharma, Amitava Chatterjee, and Anjan Rakshit

Abstract—This paper proposes a new approach for designingstable adaptive fuzzy controllers, which employs a hybridizationof a conventional Lyapunov-theory-based approach and a parti-cle swarm optimization (PSO) based stochastic optimization ap-proach. The objective is to design a self-adaptive fuzzy controller,optimizing both its structures and free parameters, such that thedesigned controller can guarantee desired stability and can simul-taneously provide satisfactory performance. The design methodol-ogy for the controller simultaneously utilizes the good features ofPSO (capable of providing good global search capability, requiredto provide a high degree of automation) and Lyapunov-based tun-ing (providing fast adaptation utilizing a local search method).Three different variants of the hybrid controller are proposed inthis paper. These variants are implemented for benchmark simu-lation case studies and real-life experimentation, and their resultsdemonstrate the usefulness of the proposed approach.

Index Terms—Adaptive fuzzy logic controllers (AFLCs), hybridapproaches, Lyapunov theory, particle swarm optimization (PSO).

I. INTRODUCTION

FUZZY logic controllers (FLCs) evolved as a popular do-main of control system and control engineering more than

two decades ago [17]. In recent times, more and more attention isbeing paid to systematic design of FLCs from the point of viewof stable behavior and optimization of performance indexes.Several such stable and optimized adaptive FLCs have beenproposed utilizing, e.g., neural network [1], Lyapunov meth-ods [2], [21], genetic algorithm (GA) [3], [4], ant colony opti-mization [14], etc.

Lyapunov-based strategies have been utilized in conventionalcontrol theory for a long time to design controllers that canguarantee closed-loop stability of the system [21]–[24]. In [5]and [6], two of the earliest design methods of Lyapunov-baseddirect adaptive fuzzy controllers were proposed. The main prob-lem in Lyapunov-based method is that it requires a considerableamount of time to track the reference input and also the inherentapproximation error in striving to achieve an unknown theoreti-

Manuscript received November 29, 2007; revised September 15, 2008; ac-cepted December 1, 2008. First published January 6, 2009; current version pub-lished April 1, 2009. This work was supported by the University Grants Commis-sion, India, under Major Research Project Scheme [Grant 32-118/2006(SR)].

K. D. Sharma is with the Department of Electrical Engineering, Future Insti-tute of Engineering and Management, West Bengal University of Technology,Kolkata 700150, India, and also with Jadavpur University, Kolkata 700032,India (e-mail: [email protected]).

A. Chatterjee and A. Rakshit are with the Department of Electrical Engineer-ing, Jadavpur University, Kolkata 700032, India (e-mail: [email protected];[email protected]).


cal control law leads the system toward larger integral absoluteerror (IAE) [2]. A major drawback of many existing designsof adaptive FLCs (AFLCs) is that they are developed for sys-tems with unlimited actuation authority [21]. In [5], one of theearliest methods to propose Lyapunov-based learning for stableAFLCs, the controller had a serious drawback that it neededadaptation of controller parameters for every variation in refer-ence signal, which implied that for temporally varying referencesignals, the controller parameters would never converge. In [6],it was demonstrated that a modified form of stable AFLC canbe proposed where the controller parameters would convergeeven in the presence of a temporally varying reference signal.However, the controllers in [6] were proposed based on controllaws adapting a parameter vector containing the positions of thefuzzy sets of the controller outputs only. The basic structure ofthe FLC was assumed to be a static one. A further improvementof this scheme was proposed in [12], where a GA-based strategywas hybridized with a Lyapunov-based strategy to adapt bothcontroller structures and free parameters, containing center lo-cations of input membership functions (MFs), scaling gains,learning rate, etc., in addition to the positions of the singletonsdescribing output fuzzy sets of the FLC. But the problem asso-ciated with this hybrid model is that this GA-based Lyapunovmethod needs several thousands of fitness evaluations to achieveparameter convergence and guarantee closed-loop stability.

This paper proposes a systematic design of stable AFLCsemploying a hybridization of the Lyapunov-strategy-based ap-proach (LSBA) and particle swarm optimization (PSO) basedapproach, called the PSO-based approach (PSOBA) [8], [9],[11]. It attempts to combine strong points of both methods toevolve a superior method. The objective of this design strategyis to perform simultaneous adaptation of both the FLC struc-ture [15], [22] and its free parameters, so that two competingrequirements can be fulfilled: 1) to guarantee stability of the con-troller design and 2) to achieve very high degree of automationin the process of controller design by employing a global searchmethod. The LSBA helps to guarantee closed-loop stability withimproved speed of convergence. On the other hand, the PSOBAhelps to achieve desired automation for the design methodologyof the AFLC by getting rid of many manually tuned parametersin LSBA design procedure. However, a well-known drawback ofthese global optimization techniques is that they are essentiallyslow in their search procedure to optimize a criterion function.This motivated us to choose PSO among these available stochas-tic methods because of their simple functioning, and generally,they can provide relatively faster, satisfactory solution.

1063-6706/$25.00 © 2009 IEEE



In this paper, three types of hybridization of locally opera-tive LSBA and a global version of PSOBA have been proposedand simulated for benchmark nonlinear plants and also testedfor a real-life experiment. These approaches are called hybrid-adaptation-strategy-based approaches (HASBA). The first twohybrid approaches are the cascade and concurrent combinationsof LSBA and PSOBA. In cascade hybrid approach, the structureof the controller and the values of the free parameters for LSBAare determined by the PSOBA, i.e., the tedious manual tuningpart of the conventional LSBA is replaced. So, the drawbacksof conventional LSBA are not fully taken care of. In concurrenthybrid approach, the LSBA performs within the supervision ofPSOBA, with the objective of keeping the inherent approxima-tion error limited and the tracking of the input reference signal,may be a temporal one, and is expected to be good enough.In some cases, our experimentations show that PSOBA aloneperforms better than the concurrent hybrid approach. Hence, wehave proposed a third hybrid strategy, called preferential hybridapproach. In this case, the movements of the particles in themultidimensional global search space are sometimes governedby the concurrent hybrid method and sometimes by the PSOBAmethod, depending on the performance of the correspondingfitness evolutions. Therefore, the best candidate solution in thispreferential hybrid approach evolves from a second level of hy-bridized process that, according to our knowledge and belief,is a novel idea to design direct AFLCs. The proposed hybridapproaches have all been implemented for different benchmarkcase studies and also experimented on a real-life problem, andthe three variants of HASBA are compared with controllersdesigned by LSBA and PSOBA alone. The obtained results il-lustrate, on the whole, the superiority of the preferential hybridapproach over the other approaches.

The rest of the paper is organized as follows. Section IIpresents a discussion about the design of stable adaptive fuzzycontrollers. Section III presents the conventional adaptationstrategies and the newly proposed hybrid adaptation strategies,while in Section IV, simulation and experimental case studiesfor both conventional and proposed hybrid adaptation strategiesare demonstrated. Section V concludes the paper.

II. DESIGN OF STABLE ADAPTIVE FUZZY CONTROLLERS

Let us consider that our objective is to design an adaptive strat-egy for an nth-order single-input–single-output (SISO) nonlin-ear plant given as

x(n) = f(x) + bu

y = x

(1)

where f(·) is an unknown continuous function, u ∈ R andy ∈ R are the input and output of the plant, and b is an unknownpositive constant. We assume that the state vector is given asx = (x1 , x2 , . . . , xn )T = (x, x, . . . , x(n−1))T ∈ Rn . The sys-tem under control has a reference model given as

x(n)m = f(xm ,w)ym = xm

(2)

where w(t) is the excitation signal input to the referencemodel, and the state vector of the reference model is xm =(xm , xm , . . . , x

(n−1)m )T ∈ Rn.

The control objective is to force the plant output y(t) to followa given bounded reference signal ym (t) under the constraintsthat all closed-loop variables involved must be bounded to guar-antee the closed-loop stability of the system. Thus, the trackingerror e = ym − y.

A. Control Objective

Our objective is to design a stable adaptive fuzzy controllerfor the system described earlier. AFLCs have evolved as popularcontrol solutions for those classes of plants whose input–outputcharacteristics are not precisely known. In case of the system in(1), this is given by the condition when f(·) and b are not pre-cisely known, which is widely prevalent in practical situations.Here, we need to find the structure of the fuzzy controller aswell as a feedback control strategy u = u(x| θ), using a fuzzylogic system and an adaptive law for adjusting the parametervector θ such that the following conditions are satisfied.

The closed-loop system must be globally stable in thesense that all variables, x(t), θ(t), and u(x| θ) must be uni-formly bounded, i.e., |x(t)| ≤ Mx < ∞, |θ(t)| ≤ Mθ < ∞,and |u(x| θ)| ≤ Mu < ∞, where Mx,Mθ , and Mu are set bythe designer [5].

The tracking error e(t) should be as small as possible underthe constraints in (1) and the e − θ space should be stable in thelarge sense for the system [6].

Now, the ideal control law for the system in (1) and (2) isgiven as [2], [12]

u∗ =1b

[−f(x) + y(n)

m + kT e]

(3)

where y(n)m is the nth derivative of the output of the refer-

ence model, e = (e, e, . . . , e(n−1))T is the error vector, andk = (k1 , k2 , . . . , kn )T ∈ Rn is the vector describing the desiredclosed-loop dynamics for the error.

This definition implies that u∗ guarantees perfect tracking,i.e., y(t) ≡ ym (t) if Lt

t→∞e(t) = 0. It indicates that as t ap-

proaches infinity, e(t) tends to zero, and under that condition,output y(t) becomes equal to the reference ym (t) and perfecttracking is achieved.

In practical situations, since f and b are not known precisely,the ideal u∗ of (3) cannot be implemented in real practice. Thus,a suitable solution can be to design a fuzzy logic system toapproximate this optimal control law.

Now, to ensure stability, we assume that the control u(t) isgiven by the summation of a fuzzy control uc(x| θ) and anadditional supervisory control us(x) [2], [5], [25]. Thus, u(t) isgiven as

u(t) = uc(x| θ) + us(x). (4)

Let us assume that the AFLC is constructed using a zero-order Takagi–Sugeno (TS) fuzzy system. Then, uc(x| θ) for the


DAS SHARMA et al.: HYBRID APPROACH FOR DESIGN OF STABLE ADAPTIVE FUZZY CONTROLLERS USING LYAPUNOV THEORY AND PSO 331

Fig. 1. Flowchart representation of LSBA algorithm. k: plant simulation counter; PST: plant simulation time; ∆tc : controller simulation sampling time.

AFLC is given in the form

uc(x| θ) =∑N

l=1 θl ∗ αl(x)(∑Nl=1 αl(x)

) = θT ∗ ξ(x) (5)

where θ = [θ1θ2 · · · θN ]T = the vector of the output singletons,αl(x) =

∏ri=1 µl

i(xi) = the firing degree of rule “l”, N is thetotal number of rules, µl

i(xi) is the membership value of theith input MF in the activated lth rule, ξ(x) = vector contain-ing normalized firing strength of all fuzzy IF–THEN rules =(ξ1(x), ξ2(x), . . . , ξN (x))T , and

ξl(x) =αl(x)(∑Nl=1 αl(x)

) . (6)

Let us define a quadratic form of tracking error as Ve =1/2eT Pe where P is a symmetric positive definite matrix sat-isfying the Lyapunov equation [2]. With the use of us(x), itcan be shown that Ve ≤ −1/2eT Qe ≤ 0, where Q is a positivedefinite matrix [5], [13].

Thus, as P > 0, boundedness of Ve implies the boundednessin x. Hence, the closed-loop stability is guaranteed.

The zero-order T–S-type fuzzy control uc(x|θ) can be soconstructed that it will produce a linear weighted combinationof adapted parameter vector θ. Thus, a simple singleton-basedadaptation law as proposed in [2] and [12] can be given as

θ = νeT pnξ(x) (7)

where ν > 0 is the adaptation gain or learning rate and pn

is thelast column of P .

It should be noted here that when the reference signal ym

is the output of a reference model, excited with a given inputsignal w, and the reference plant has the same dynamics as thatof the controlled plant and can be expressed by k, then the inputstates include w along with x [6], [12].

III. ADAPTATION STRATEGIES FOR THE AFLC

A. Lyapunov-Strategy-Based Approach

Design methodologies for a stable direct adaptive fuzzy con-troller, based on the Lyapunov method, are given in [5], [6],and [21]. The flowchart representation of LSBA is shown inFig. 1. The design methodology can be divided into two parts:1) offline calculations and 2) online adaptation.

The offline calculation deals with the choice of the val-ues of k. A typical method of specifying k is given in[6]. The P matrix is chosen with the constraint P > 0.The adaptation of θ is carried out using (7), if (|θ| < Mθ )or(|θ| = Mθ and eT p

nθT ξ(x) ≤ 0). Otherwise, a projection

operator has to be used for adapting θ [5].

B. PSO-Based Approach (PSOBA)

PSO, originally developed by Kennedy and Eberhart in 1995,is a population-based swarm algorithm [7], [8], [16]. In thePSO computational algorithm, population dynamics simulatesbioinspired behavior, i.e., a “bird flock’s” behavior that involvessharing of information and allows particles to take profit fromtheir own discoveries and previous experience (pBest—personalbest) as well as that of all other particles (gBest—global best)during the search for food.

To design a direct adaptive fuzzy controller with completelyadaptive controller structure and free parameters, we have tochoose the values of the scaling gains, the settings of controllerstructure (i.e., number of MFs in each variable, number of rules,etc.), the supports of input MFs, the positions of output single-tons, etc. The PSOBA design methodology not only tunes thesingletons properly but it can also optimize the values of the scal-ing gains and can determine the optimum controller structure.

In PSOBA design, a particle’s position in solution space is avector containing all required information to construct a fuzzy



controller [12] as

X =

[structural flags for MFs | center locations of the MFs |scaling gains | learning rate | position of the output singletons].

(8)

Each structural flag bears the information about the existenceor nonexistence of an MF in its corresponding variable. It canonly take binary values where 0 indicates the nonexistence and1 indicates the existence of the corresponding MF. However,PSOBA is an algorithm where each entry in the particle vectorcan take continuous values. Hence, for each structural flag inthe vector, the universe of discourse is set as [0, 1] and the flagis set to 0 if the continuous value of the variable is less than 0.5and the flag is set to 1 if the variable is greater than or equal to0.5. For each MF of an input variable, there is a correspondingstructural flag assigned. A specific MF for a given input vari-able will only become active when its corresponding structuralflag is set to 1. When this flag becomes 0, in any given itera-tion, the corresponding MF becomes inactive and all rules, inthe fuzzy rule base, containing that MF in antecedent clauses,become inactive. The total number of 1’s in the structural flagskeeps changing in each iteration, and hence, the total numberof MFs in which an input variable is fuzzified also changeson each iteration. This changes the structure of the AFLC ineach iteration, thus changing the total number of rules of thefuzzy rule base, and hence, changes the total number of ac-tive output singletons. Each input is fuzzified using triangularMFs and the center locations of the MFs in the particle vectorstore the peaks of the MFs. Each input is fuzzified in the range[−1, 1] with two fixed triangular MFs having their peaks fixedat −1 and 1, respectively. All intermediate MFs for that inputvariable are flexible in nature. They can be either active or in-active during an iteration and their peaks are also adapted byPSOBA in each iteration. For each MF, the peaks of the imme-diate adjacent active MFs on either side of its own peak formthe left and right base support of it. Whether the immediate ad-jacent MF is an active MF or not is determined by the contentof its corresponding structural flag. Hence, some of the centerlocations of MFs are ignored while evaluating the AFLC out-put, because their corresponding entries in the structural flagsare zero. A similar logic holds true for the output singletons ina particle vector. Here, those singletons become inactive whoseantecedent parts contain one or more input MFs that becomeinactive in that iteration.

To design an AFLC utilizing PSOBA algorithm first, the pop-ulation of particles in the swarm is chosen, then the algorithmwill decide the structure of the candidate controller according toits structural flags. The candidate controller simulation (CCS)algorithm, as shown in Fig. 3, is carried out on the candidatecontroller to calculate the fitness function, the IAE, which canbe defined as IAE =

∑PSTn=0 e(n)τ , where PST = plant simula-

tion time and τ = step size or sampling time. Then, accordingto the value of the fitness function IAE of each particle in eachiteration, the pBest and gBest candidate controllers are calcu-

lated and the particles’ position and velocity in the search spaceare updated by the global version PSO method [7], [8], [10], asshown in the flowchart representation of the PSOBA algorithmin Fig. 2. The PSOBA algorithm will stop searching the solutionspace when the number of iterations specified by the designeris reached or a prespecified error is attained by the controller.

C. Hybrid-Adaptation-Strategy-Based Approach

In hybrid design methods, the complementary features ofLSBA and the PSOBA are combined in three different waysto get a superior systematic design procedure of direct AFLCsfor a class of SISO systems. The proposed hybrid strategies aredescribed now.

1) HASBA-Concurrent Model: In this design process, theLSBA and the PSOBA run concurrently or in parallel to op-timize the: 1) structure of the controller; 2) scaling gains; 3)learning rate (or adaptation gain); and 4) positions of the outputsingletons.

In this method, a particle X is divided into two subgroups,unlike PSOBA design methodology, and is given as

X = [Ψ| θ] (9)

where

Ψ = [structural flags for MFs | center locations of

the MFs | scaling gains | learning rate]

θ = [position of the output singletons]. (10)

The partition of X vector separates the parameters accord-ingly as Ψ has nonlinear influence and θ has linear influence onuc , respectively [12]. In LSBA, the values of the free parametersin the vector Ψ are defined a priori and the value of the corre-sponding θ minimizing the tracking error can be obtained by theadaptation law as in (7). Here, Ψ is set by a hand-tuned trial-and-error method. In this paper, PSOBA is applied to optimize Ψ andθ in tandem by updating the particle’s position, containing bothΨ and θ, as shown in the flowchart representation of PSOBA inFig. 2. In addition to that, adaptation law, as in (7), is applied toevery updated candidate controller to adjust the value of θ only.So, in this method, θ explores both the local search space andglobal search space simultaneously, in a twofold manner [29].The flowchart representation of HASBA-concurrent model isshown in Fig. 4.

2) HASBA-Cascade Model: In this model, LSBA andPSOBA optimizations are used in cascade. Therefore, in thismodel, the structure [22] and the free parameters of the con-troller are optimized by the PSOBA method as well as the po-sitions of the output singletons are primarily tuned in globalsearch space by the PSOBA method. So, after a primary globaltuning of θ by the PSOBA method, LSBA optimizes θ locallyto further improve the performance of the controller. Thus, inthis hybrid model also, the LSBA method is enhanced by thePSOBA method. However, the basic difference lies in the factthat the LSBA algorithm is applied to the best solution vector de-termined by the PSOBA algorithm. Hence, the LSBA algorithmruns only once at the completion of the PSOBA algorithm. It isin contrast to the HASBA-concurrent model, where the LSBA



Fig. 2. Flowchart representation of PSOBA algorithm. j: particle counter; NOP: number of particles (population size); g: PSO iteration counter; iterm ax :maximum number of PSO iterations.

Fig. 3. Flowchart representation of candidate controller simulation (CCS) algorithm. k: plant simulation counter; PST: plant simulation time; ∆tc : controllersimulation sampling time.

algorithm runs as a submodule of the PSOBA algorithm and at-tempts to locally fine-tune each candidate solution vector, called“particle.” In the cascade model, the learning rate is adjustedin such a fashion that LSBA adaptation searches only the nearneighborhood of the best θ produced by the PSOBA method.

The flowchart representation of HASBA-cascade is shown inFig. 5.

3) HASBA-Preferential Model: In this proposed model, boththe PSOBA method and the HASBA-concurrent method are em-ployed to keep the tracking error as minimum as possible. Here,



Fig. 4. Flowchart representation of HASBA—concurrent model algorithm. j: particle counter; NOP: number of particles (population size); g: PSO iterationcounter; iterm ax : maximum number of PSO iterations.

Fig. 5. Flowchart representation of HASBA—cascade model algorithm.

each particle in the swarm, i.e., each candidate controller, eval-uates the CCS algorithm and the LSBA algorithm separately atthe same time. The fitness values of the candidate controller,evaluated by employing the CCS algorithm and the LSBA al-gorithm separately, are then compared, and then, the algorithmshowing a better performance is taken as the guiding factor forupdating the velocity and the position of that specific particlein that specific iteration, according to the PSO method. If theLSBA algorithm shows a better performance for that particle inthat iteration, then the system will follow the algorithm for theHASBA-concurrent model; otherwise, the system will followthe PSOBA model in that iteration. Thus, in this process, theupdates for each particle in a swarm switch between two algo-rithms, even within the same iteration, to search the solutionspace. This process of evaluation is performed separately for

each particle in each iteration, so that the same particle may fol-low different update rules in different iterations. The flowchartrepresentation of HASBA-preferential is shown in Fig. 6.

IV. SIMULATION AND EXPERIMENTAL STUDIES

To evaluate the effectiveness of the proposed schemes, weconsidered four benchmark case studies, previously utilizedin several research works and a real-life experiment. The pro-cess models are simulated each using a fixed-step fourth-orderRunge–Kutta method with sampling time T = 0.01 s.

A. Case Study I

The controlled plant under consideration is a second-order dcmotor containing nonlinear friction characteristics described bythe following model [6], [12]:

x1 = x2

x2 = − f (x2 )J + CT

J u

y = x1

(11)

where y = x1 is the angular position of the rotor (in radians),x2 is the angular speed (in radians per second), and u is thecurrent fed to the motor (in amperes). The plant parametersare CT = 10 N·m/A, J = 0.1 kg·m2 , and the nonlinear frictiontorque is defined as

f(x2) = 5 tan−1(5x2)N · m. (12)



Fig. 6. Flowchart representation of HASBA—preferential model algorithm. j: particle counter; NOP: number of particles (population size); g: PSO iterationcounter; iterm ax : maximum number of PSO iteration.

The control objective is to make the angular position y followa reference signal given by

ym = 400w(t) − 400ym − 40ym (13)

(i.e., the reference model has a double pole at s1,2 = −20),where w(t) is a square wave of random amplitude, as shown inFig. 7(a), and ym is the output of w(t) filtered by the second-order linear filter [given by (13)] as shown in Fig. 7(b).

The control signal obtained as the output from the fuzzycontroller, in response to input signals, is given as follows:

uc = uc(x,w |X ), for PSOBA and HASBA

uc = uc(x,w | θ ), for LSBA.

(14)

1) Comparison of Design Strategies: In this section, the fol-lowing design strategies are compared.

a) LSBA: In this case, only θ is optimized and the staticΨ is manually set a priori. In our simulation, input MFs aredistributed evenly on the corresponding universe of discourse.The other parameters are chosen as

P =[

p0 p1p2 p2

]=

[5 0.01

0.01 0.001

]k = [ 400 40 ]

such that p1k1 > 0 and p2 > p1/k2 . The value of adaptationgain is chosen as ν = 5, giving a good tradeoff between adap-tation rate and small amplitudes of parameter oscillations thatare due to inherent approximation error. The other manuallyset parameters are b = 100, bL = 95, and fU = 1.5. The in-put and output scaling gains of the AFLC are chosen so as tomap the ranges of the corresponding variables into the interval[−1, 1].

The controller is trained by applying a signal w(t) thatchanges its value randomly in the interval (−1.5, 1.5) in ev-ery 0.5 s. This signal is chosen as it contains infinitely manyfrequencies and leads to good exploration of controller inputspace [6].

The plant is simulated for 210 s. The adaptation starts att = 0 s and continues up to t = 200 s when the adaptation isswitched off. Then, the plant is tested for 10 s with this referencesignal. During this period, adaptation is turned off (i.e., ν = 0)and the controller operates with already adapted parameters.

b) PSOBA: In this case, all the parameters of X = [Ψ | θ]are optimized by the PSOBA. To perform this simulation, a pop-ulation size of ten particles is taken. For each simulation, 200iterations of PSOBA is set. In fixed structure controller design, afixed number of evenly distributed input MFs are used for initialpopulation of PSOBA. The PSOBA algorithm further tunes onlythe free parameters and the positions of the output singletonsto minimize the tracking error. In variable structure controllerdesign, the structural flags, are utilized to automatically varythe structure of the candidate controller in each iteration. Theinput MFs and the corresponding position of the output single-tons are set according to the active structural flags, as stated inSection III-B. IAE between the reference signal ym and themotor output y is taken as the fitness function for optimization.

c) HASBA: It consists of the following models.i) Concurrent hybrid model: In this simulation pro-

cess, the PSOBA-based optimization of parametersfor X = [Ψ | θ] and the LSBA algorithm, which onlyadjusts θ, run concurrently. The position vector Xnow additionally includes the adaptation gain ν in itsΨ portion. The gain ν is varied over the range 14 000–16 000. The structural flags are set and reset in the



Fig. 7. (a) Input signal w(t). (b) Reference signal ym (t).

same manner as is done in PSOBA and utilized toautomatically find out the structure of the candidatecontroller.

ii) Cascade hybrid model: This simulation process startswith the PSOBA simulation. The result obtained fromPSOBA is then passed through the Lyapunov-basedadaptation strategy to further adjust the positions ofthe output singletons θ. The adaptation gain ν duringthe LSBA is set to 5 to explore the near neighborhoodof the best θ as determined by the PSOBA.

iii) Preferential hybrid model: In this hybrid simula-tion process, both concurrent hybrid model and purePSOBA model are implemented for the same particleseparately at the same time, and the candidate con-trollers, i.e., the particles of PSO, are updated accord-ingly. The same particle can take different searchingalgorithms to search the solution space depending onthe value of its fitness function IAE during the entireoptimization process.

The results of simulation studies are tabulated in Table I. Inall these simulations, the plant is evaluated for 10 s after thetraining is completed by different adaptation strategies so as tocompare the results.

TABLE ICOMPARISON OF SIMULATION RESULTS OF CASE STUDY I

(EVALUATION TIME = 10 S)

The simulations are carried out for three fixed structuresof 3 × 3 × 3, 5 × 5 × 5, and 7 × 7 × 7 input MFs and a vari-able structure with a potentially biggest possible structure of7 × 7 × 7 MFs. Table I presents comparison of three possibleHASBA controllers vis-a-vis LSBA and PSOBA controllers forcase study I. Fig. 8 shows temporal variations of system re-sponses for the sample case of fixed structure controllers with5 × 5 × 5 input MFs. Fig. 9(a) shows the temporal variations ofsystem responses of the variable structure HASBA-preferentialmodel for 100 intermediate iterations. In Fig. 9(b)–(d), the sys-tem response for 10 s evaluation phase, the control effort, and theerror plot for this AFLC are shown as a sample case. For fixedstructure controllers, it can be seen that the performances ofthe controllers (demonstrated by the performance index of IAE)improve with increase in the number of rules from a 3 × 3 × 3to 5 × 5 × 5 configuration. However, for a 7 × 7 × 7 configu-ration, results are not better than that for a 5 × 5 × 5 configura-tion. Hence, we can conclude that merely increasing the num-ber of rules in fuzzy controllers does not necessarily improvetheir performance. This is in conformation with the observa-tions presented in [12]. For a given controller MF configuration(e.g., 5 × 5 × 5 or 7 × 7 × 7), among the competing controllers,the HASBA-preferential scheme shows the best performance.Even for the variable structure scheme, HASBA-preferentialcontroller emerges as the best possible solution. Although theIAE value is higher for the variable structure case, it can beseen that the best result, in terms of IAE, is obtained with only100 rules. This is a significant reduction in the total number ofrules and such a simplified structure of an FLC will be veryuseful from the point of view of implementation in real life.Although the HASBA-concurrent model could converge to aneven smaller number of rules (80, in this case), a comparatively



Fig. 8. Responses of the fixed structure (MF: 5 × 5 × 5) adaptive fuzzy controllers of case study I. (a) LSBA. (b) PSOBA. (c) HASBA-concurrent.(d) HASBA-cascade. (e) HASBA-preferential.



Fig. 9. Responses of the variable structure HASBA—preferential adaptive fuzzy controller of case study I. (a) Training phase response after 100 PSO iterations.(b) Evaluation phase response. (c) Evaluation phase control effort. (d) Evaluation phase error plot.

much larger IAE prompted us to conclude that the HASBA-preferential gave the overall superior performance.

B. Case Study II

In this case study, the Duffing’s forced oscillation system,which is itself a chaotic system if unforced, is considered and isgiven as

x1 = x2

x2 = −0.1x2 − x31 + 12cos(t) + u(t)

y = x1

. (15)

The control objective is to track the reference signal ym ,where ym = sin(t). The control signal fed to the fuzzy controlleris

uc = uc(x |X ), for PSOBA and HASBA

uc = uc(x |θ ), for LSBA

. (16)

1) Comparison of Design Strategies: The choice of param-eters for this simulation process is:

P =[

1 0.010.01 0.001

]b = 1

hence, bL = 0.95 and fU = 12 +∣∣x3

1

∣∣ ,K = [2 1], [18]. TheLSBA, PSOBA design strategies, and different hybrid meth-ods are simulated in a similar fashion as in the earlier casestudy.

The adaptation gain ν is set to 2 for LSBA and HASBA-concurrent, 0.5 for HASBA-cascade, and 1 for HASBA-preferential. In a similar fashion of case study I, the simulationsare carried out for three fixed structures of 3 × 3, 5 × 5, and7 × 7 input MFs and a variable structure. In this case study also,the HASBA-preferential design strategy emerges as the bestsolution, for both fixed structure and variable structure config-urations. In fact, for this case study, the best performance resultfor HASBA-preferential controller for variable structure config-uration is obtained with simultaneous achievement of minimumnumber of rules (here, it is 15). Hence, this design strategyevolves as the most superior solution from both the pointsof view of best performance and simplest implementation.Table II shows the results for sample case of variable struc-ture controllers. Fig. 10 shows the response of variable-structureHASBA-preferential controller for case study II as a represen-tative manner.



TABLE IICOMPARISON OF SIMULATION RESULTS OF CASE STUDY II


Fig. 10. Evaluation phase response of the variable structure HASBA—preferential adaptive fuzzy controller of case study II.

C. Case Study III

In this case study, the Duffing’s forced oscillation system withdisturbance is considered and is given as

x1 = x2

x2 = −0.1x2 − x31 + 12cos(t) + u(t) + d(t)

y = x1

(17)

where the external disturbance d(t) is a square wave of randomamplitude within the range [−1, 1] and a period of 0.5 s. Thecontrol objective is to track the reference signal ym , whereym = sin(t). The control signal fed to the fuzzy controller isthe same as in (16).

1) Comparison of Design Strategies: Here also, we choosethe same parameters for this simulation process as was cho-sen in case study II. The design strategies of LSBA, PSOBA,and different hybrid methods are simulated in a similar fash-ion as in case study II. The adaptation gain ν is set to 2 forLSBA and HASBA-concurrent, 0.5 for HASBA-cascade, and1 for HASBA-preferential. In this case study, among the fixedstructure controllers, the HASBA-concurrent shows least av-erage IAE value for 7 × 7 configuration and PSOBA showsbest performance for variable configuration. However, HASBA-preferential configuration emerges as the second best solutionin each configuration. It also achieves the minimum number ofrules for the variable configuration (along with HASBA-cascadeand PSOBA controllers). This is understandable from the fact

TABLE IIICOMPARISON OF SIMULATION RESULTS OF CASE STUDY III


Fig. 11. Evaluation phase response of the variable structure HASBA—preferential adaptive fuzzy controller of case study III.

that, very rarely, a single controller design strategy emerges asthe best solution for many types of systems under several typesof input conditions. Table III shows the results for a sample caseof variable structure controllers. Fig. 11 shows the response ofthe variable-structure HASBA-preferential controller for casestudy III in a representative manner.

D. Case Study IV

All the discussions presented so far are given with reference tothe system in (1), where the control input is given in “bu” form.The discussions hold equally for a more general system, wherecontrol input is given in “g(x)u,” as demonstrated in [6]. Nowwe demonstrate the utility of our proposed controllers for sucha system. We consider the inverted pendulum system [26], [28]as

x1 = x2

x2 = f(x) + g(x)u(t)y = x1

(18)

where

f(x) =mlx2 sinx1 cos x1 − (M + m)g sinx1

ml cos2 x1 − 4/3l(M + m)

and

g(x) =− cos x1

ml cos2 x1 − 4/3l(M + m)

where g = 9.8 m/s2 ,m = 0.1 kg, M = 1 kg, and l = 0.5 m[26], [28]. The control objective is to track the reference signal



TABLE IVCOMPARISON OF SIMULATION RESULTS OF CASE STUDY IV


Fig. 12. Evaluation phase response of the variable structure HASBA—preferential adaptive fuzzy controller of Case Study IV.

ym , where ym = sin(t). The control signal fed to the fuzzycontroller is the same as in (16).

1) Comparison of Design Strategies: The choice of param-eters for this simulation process is

P =[

1 0.010.01 0.001

]b = g(x)

hence, bL = b – 0.1 and fU = 5,K = [2 1]. The design strate-gies of LSBA, PSOBA, and different hybrid methods are simu-lated in a similar fashion as in earlier case studies. The adaptationgain ν is set to 5 for LSBA, 0.001 for HASBA-concurrent andHASBA-preferential, and 0.01 for HASBA-cascade. The maindifference of this case study is that, here, the coefficient of thecontrol input u(t) is not a constant like in earlier cases, but isa function of the states of the inverted pendulum system and isvarying with time. Table IV and Fig. 12 show results in tabularand graphical form for case study IV in a similar manner as incase studies II and III. It is seen from Table IV that the IAE ofthe HASBA-preferential model is close to that of the PSOBAand HASBA-concurrent model but the number of fuzzy rules isminimum (15 in this case). So, the performance of the HASBA-preferential model is good enough to justify it as a superiorcontrol strategy over the other control strategies.

Overall, from our four simulation case studies with differenttypes of temporal reference input variations, it can be concludedthat the HASBA-preferential controller emerges as a superior,feasible control solution.

Fig. 13. (a) Real-life experimental arrangements. (b) and (c) Evaluation phaseresponse of HASBA—preferential adaptive fuzzy controller of Case Study IVwith different reference signals.

E. Case Study V

To demonstrate the usefulness of the proposed algorithms,we have performed a real-life experiment where we attempt toperform the speed control of a real dc motor. Fig. 13(a) showsthe experimental arrangement where the PC-based controllercontrols the dc motor through a parallel port interface in real



TABLE VCOMPARISON OF SIMULATION RESULTS OF CASE STUDY V


time. The scheme employs a power-transistor-based chopper-controlled mechanism to achieve the desired speed in armaturecontrol mode. At first, the motor parameters, i.e., armature re-sistance (Ra ), armature inductance (La ), inertia constant (J),damping constant (B), etc., are identified by exciting the motorin open loop and collecting the input–output data. Here, the dcmotor characteristics are modeled as

x1 = x2

x2 =(

BRa

J La+ Kb KT

J La

)x1 −

(Ra

La+ B

J

)x2 + K KT

J Lau(t)

y = x1

(19)

where [x1x2 ] = [ωω], the output y is the angular speed of themotor (in radians per second), K is the constant of the choppercircuit, and Kb and KT are the constants of the dc motor. Theunknown parameters are learned utilizing PSO, where the PSOalgorithm determines the best vector comprising the unknownparameters for which discrepancy between the model output andthe actual experimental output data, for the same input, is min-imum, considering the entire set of input–output real-life data.When the state model of this real dc motor is identified, our con-troller design is carried out offline for different control strategiesas discussed earlier. Fixed structure controllers with 5 × 5 in-put MFs are implemented for this real-life experimentation forcontrolling the same motor under temporally varying referencespeed conditions for 240 s. Fig. 13(b) and (c) shows the perfor-mance of HASBA-preferential controller, as sample cases, intracking two different temporally varying reference speed sig-nals. Table V shows the sample performances of the controllersin the real implementation phase for the dc motor, for the caseas shown in Fig. 13(b). In each of these experimentations, thespeed measurement is carried out using shaft encoders havingten holes in the wheel mounted on the shaft. Then, the speed sig-nal acquired is filtered in the software developed, employing anexponential averaging technique, to remove noise pickup. It canbe seen that the controllers developed were able to provide sat-isfactory performance. The interrupt-driven software developedfor this sophisticated motor control scheme has a sampling rateof 100 µs within the interrupt mode, whereas each PWM cyclegenerated for control input is of duration 10 ms (i.e., ON-time +OFF-time). The controller output is used to suitably control thisON-time. Table V, reveals that, in this real-life experiment also,the HASBA-preferential controller emerges as the best designstrategy among the LSBA, PSOBA, and other hybrid variants.

V. CONCLUSION

This paper has proposed a new hybrid approach for designingthe AFLCs employing both the conventional Lyapunov theory,to locally optimize the positions of the output singleton, andPSO, which is a stochastic optimization technique. In this pro-posed hybridization process, while LSBA optimizes the posi-tions of the output singleton, PSOBA is utilized to design thestructure of the AFLC and also evaluate the free parameters ofit. Three variants of the hybrid models have been proposed andsimulated for the benchmark case studies. The preferential hy-brid model was evolved as a superior technique as compared tothe LSBA, PSOBA, and other hybrid models. The stability ofthe closed-loop system and the convergence of the plant outputto a desired reference signal are guaranteed precisely. The mainadvantages of these proposed methods are that it requires no apriori knowledge about the controlled plant and the approxima-tion error is reduced greatly. Moreover, the preferential hybridmodel is able to avoid the local minima. In this paper, we havechosen PSO as the representative stochastic global optimizationalgorithm for hybridization with Lyapunov theory. The ratio-nale behind this is that PSO is a relatively simple algorithmamong stochastic optimization algorithms. However, it shouldbe pointed out that other stochastic global algorithms like GA,differential evaluation, etc. [12], [19], [20], [27], can also bepotentially employed to design similar AFLCs. The authors in-tend to design similar AFLCs using other global optimizationalgorithms and compare their performances in the near future.

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Kaushik Das Sharma received the B.Sc. (Hons.)degree in physics, and the B.Tech. and M.Tech. de-grees in electrical engineering from the University ofCalcutta, Kolkata, India, in 1998, 2001, and 2004, re-spectively. He is currently working toward the Ph.D.degree at Jadavpur University, Kolkata.

He is currently an Assistant Professor of the De-partment of Electrical Engineering, Future Institute ofEngineering and Management, West Bengal Univer-sity of Technology, Kolkata. His current research in-terests include fuzzy control system design, stochas-

tic optimization applications, and embedded system design, among others.

Amitava Chatterjee received the B.E., M.E., andPh.D. degrees in electrical engineering from JadavpurUniversity, Kolkata, India, in 1991, 1994, and 2002,respectively.

In 1997, he joined the Department of Electrical En-gineering, Jadavpur University, where he is currentlya Reader. In 2003, he received the Japanese Gov-ernment (Monbukagakusho) Scholarship and wentto Saga University, Saga, Japan. In early 2004, hewas invited as a Teacher–Researcher in the Uni-versity of Paris XII, Val de Marne, France. From

November 2004 to November 2005, he was with the University of Electro-Communications, Tokyo, Japan, on a Japan Society for the Promotion of Science(JSPS) Post-Doctoral Fellowship for Foreign Researchers. His current researchinterests include intelligent instrumentation and control, signal processing, im-age processing, and robotics. He has authored or coauthored more than 50technical papers, including 30 international journal papers.

Anjan Rakshit received the M.E. and Ph.D. degreesin electrical engineering from Jadavpur University,Kolkata, India, in 1978 and 1987, respectively.

He is currently a Professor in the Departmentof Electrical Engineering, Jadavpur University. Hiscurrent research interests include digital signal pro-cessing, Internet-based instrumentation, smart instru-mentation, intelligent control, and design of real-timesystems. He has authored or coauthored almost 70technical papers.



Fuzzy Control for Nonlinear UncertainElectrohydraulic Active Suspensions

With Input ConstraintHaiping Du and Nong Zhang

Abstract—This paper presents a Takagi–Sugeno (T–S) model-based fuzzy control design approach for electrohydraulic activevehicle suspensions considering nonlinear dynamics of the actua-tor, sprung mass variation, and constraints on the control input.The T–S fuzzy model is first applied to represent the nonlinearuncertain electrohydraulic suspension. Then, a fuzzy state feed-back controller is designed for the obtained T–S fuzzy model withoptimized H∞ performance for ride comfort by using the parallel-distributed compensation (PDC) scheme. The sufficient conditionsfor the existence of such a controller are derived in terms of linearmatrix inequalities (LMIs). Numerical simulations on a full-carsuspension model are performed to validate the effectiveness of theproposed approach. The obtained results show that the designedcontroller can achieve good suspension performance despite theexistence of nonlinear actuator dynamics, sprung mass variation,and control input constraints.

Index Terms—Electrohydraulic actuator, input constraint, non-linear dynamic system, Takagi–Sugeno (T–S) fuzzy modeling, un-certainty, vehicle active suspension.

I. INTRODUCTION

ACTIVE suspensions are currently attracting a great deal ofinterest in both academia and industry for improving ve-

hicle ride comfort and road holding performance [1], [2]. Sinceactive suspensions need actuators to provide the required forces,one practical consideration in real-world applications involveschoosing appropriate actuators that can fit into the suspensionpackaging space, and satisfy the practical power and bandwidthrequirements. It has been noted that electrohydraulic actuatorsare regarded as one of the most viable choices for an active sus-pension due to their high power-to-weight ratio and low cost.Therefore, in recent years, many studies have focused on elec-trohydraulic active suspensions, and various control algorithmshave been proposed to deal with the involved highly nonlineardynamics of electrohydraulic actuators [3]–[10].

It is still a challenge to develop an appropriate control strat-egy for dealing with the highly nonlinear dynamics of electro-

Manuscript received December 18, 2007; revised June 19, 2008 andSeptember 11, 2008; accepted December 8, 2008. First published December31, 2008; current version published April 1, 2009. This work was supportedin part by the Early Career Research Grant, University of Technology, Sydney,and by the Australian Research Council’s Discovery Projects funding schemeunder Project DP0773415.

The authors are with the Faculty of Engineering, University of Technology,Sydney, N.S.W. 2007, Australia (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.org.


hydraulic actuators in active suspensions. Generally speaking,the currently proposed control algorithms for electrohydraulicactive suspensions can be divided into two main groups: one isthe two-loop control strategy, in which the outer loop is used toprovide the desired forces and the inner loop is used to makethe electrohydraulic actuators track the desired forces; the otheris the sliding-mode-based control strategy. As proved in [11],pure proportional-integral differential (PID)-like controllers arenot capable of giving satisfactory performance in the actuatorforce tracking problem, and more sophisticated control schemesshould be employed. Hence, some attempts have been made tocompensate for this shortcoming through advanced inner loopforce control algorithms, for example [11]–[14]. Nevertheless,due to their highly nonlinear dynamics, using electrohydraulicactuators to track the desired forces is fundamentally limitedwhen interacting with a dynamic environment [11]. On the otherhand, the chattering phenomenon is inevitable in sliding modecontrol, and it may excite unmodeled high-frequency dynamics,which degrades the performance of the system and may evenlead to instability. Techniques such as adaptive fuzzy slidingcontrol [5] and self-organizing fuzzy sliding control [15] werethen proposed to smooth the chattering phenomenon. However,these approaches need a complicated learning mechanism or aspecific performance decision table, which is designed by a trialand error process, and presents certain difficulties in application.

In practice, the vehicle sprung mass varies with the loadingconditions, such as the payload and number of vehicle occu-pants. The control performance of a vehicle suspension will beaffected if the sprung mass variation is not considered in thecontroller design process. In spite of its importance, this prob-lem has not been explicitly dealt with in any previous studieson electrohydraulic suspensions. Furthermore, the constraint oncontrol input voltage sent to the actuator servo-valve has notyet been considered for controller design, although the inputpower provided to an electrohydraulic actuator is, in practice,limited. Thus, it is surely necessary to develop a new controllerdesign approach that aims at improving the performance of elec-trohydraulic active suspensions while considering the actuatornonlinear dynamics, sprung mass uncertainty, and control inputvoltage limitations.

Following the earlier discussion on electrohydraulic activesuspensions, in this paper, a fuzzy state feedback controllerdesign method is presented to improve the ride comfort per-formance of vehicles with electrohydraulic active suspensionsthrough a Takagi–Sugeno (T–S) fuzzy model approach. In recentdecades, fuzzy logic control has been proposed as an alternative

1063-6706/$25.00 © 2009 IEEE



approach to conventional control techniques for complexnonlinear systems. It was originally introduced and developedas a model-free control design approach, and it has been appliedto active suspensions [5], [8], [16] to deal with the nonlinear-ities associated with the actuator dynamics, shock absorbers,suspension springs, etc. However, the model-free fuzzy logiccontrol suffers from a number of criticisms, such as the lack ofsystematic stability analysis and controller design. It also facesa challenge in the development of fuzzy rules. Recent researchon fuzzy logic control has, therefore, been devoted to model-based fuzzy control systems that guarantee not only stability,but also performance of closed-loop fuzzy control systems [17].The T–S fuzzy system is one of the most popular systems inmodel-based fuzzy control. It is described by fuzzy IF-THENrules that represent local linear input–output relations of a non-linear system. The overall fuzzy model of the nonlinear systemis obtained by fuzzy “blending” of the linear models. The T–Smodel is capable of approximating many real nonlinear systems,e.g., mechanical systems and chaotic systems. Since it employslinear models in the consequent part, linear control theory canbe applied for system analysis and synthesis accordingly, basedon the parallel-distributed compensation (PDC) scheme [18].The T–S fuzzy models are therefore becoming powerful engi-neering tools for the modeling and control of complex dynamicsystems.

To apply the T–S model-based fuzzy control strategy to elec-trohydraulic active suspensions, in this study, the nonlinear un-certain suspension is first represented by a T–S fuzzy model.Then, a fuzzy state feedback controller is designed for the fuzzyT–S model to improve the ride comfort performance by opti-mizing the H∞ performance of the transfer function from theroad disturbance to the sprung mass accelerations. To avoid theproblem of having a large number of inequalities when the in-put saturation constraint is characterized in terms of the convexhull of some linear combination of linear functions and satu-ration functions [19], the norm-bounded approach [20], [21] isused here to handle the saturation nonlinearity. The sufficientconditions for the existence of such a controller are derivedas linear matrix inequalities (LMIs) that can be solved veryefficiently by means of the most powerful tools available todate, e.g., MATLAB LMI Toolbox. The proposed fuzzy statefeedback controller design approach is validated by simula-tions on a full-car electrohydraulic suspension model. A com-parison of the results shows that the designed controller canachieve good suspension performance regardless of the actuatornonlinear dynamics, sprung mass variation, and control inputconstraints.

The rest of this paper is organized as follows. Section IIpresents the model of a full-car electrohydraulic suspension.The T–S fuzzy model of the nonlinear uncertain suspension isgiven in Section III. In Section IV, the computational algorithmfor the fuzzy state feedback controller is provided. Section Vpresents the design results and simulations. Finally, the study’sfindings are summarized in Section VI.

The notation used throughout the paper is reasonably stan-dard. For a real symmetric matrix W, the notation of W > 0(W < 0) is used to denote its positive (negative) definiteness,

‖·‖ refers to either the Euclidean vector norm or the induced ma-trix 2-norm, I is used to denote the identity matrix of appropriatedimensions, and to simplify notation, ∗ is used to represent ablock matrix that is readily inferred by symmetry.

II. ELECTROHYDRAULIC SUSPENSION MODEL

A full-car electrohydraulic suspension model, as shown inFig. 1, is considered in this paper. This is a 7-DOF model wherethe sprung mass is assumed to be a rigid body with freedomsof motion in the vertical, pitch, and roll directions, and eachunsprung mass has freedom of motion in the vertical direction.In Fig. 1, zs is the vertical displacement at the center of gravity, θand φ are the pitch and roll angles of the sprung mass, ms, muf ,and mur denote the sprung and unsprung masses, respectively,and Iθ and Iφ are pitch and roll moments of inertia. The frontand rear displacements of the sprung mass on the left and rightsides are denoted by z1f l , z1rl , z1f r , and z1rr . The front andrear displacements of the unsprung masses on the left and rightsides are denoted by z2f l , z2rl , z2f r , and z2rr . The disturbances,which are caused by road irregularities, are denoted by wf l , wrl ,wf r , and wrr . The front and rear suspension stiffnesses and thefront and rear tyre stiffnesses are denoted by ksf , ksr , andktf , ktr , respectively. The front and rear suspension dampingcoefficients are csf and csr . Four electrohydraulic actuators areplaced between the sprung mass and the unsprung masses togenerate pushing forces, denoted by Ff l , Frl , Ff r , and Frr .

Assuming that the pitch angle θ and the roll angle φ are smallenough, the following linear approximations are applied

z1f l(t) = zs(t) + lf θ(t) + tf φ(t)

z1f r (t) = zs(t) + lf θ(t) − tf φ(t)

z1rl(t) = zs(t) − lr θ(t) + trφ(t)

z1rr (t) = zs(t) − lr θ(t) − trφ(t) (1)

and a kinematic relationship between xs(t) and q(t) can beestablished as

xs(t) = LT q(t) (2)

where q(t) = [ zs(t) θ(t) φ(t) ]T, xs(t) = [z1f l(t) z1f r (t)z1rl(t) z1rr (t)]T , and

L =

1 1 1 1

lf lf −lr −lr

tf −tf tr −tr

.

In terms of mass, damping, and stiffness matrices, the motionequations of the full-car suspension model can be formalized as

Msq(t) = LBs(xu (t) − xs(t))

+ LKs(xu (t) − xs(t)) − LF (t)

Muxu (t) = Bs(xs(t) − xu (t)) + Ks(xs(t) − xu (t))

+ Kt(w(t) − xu (t)) + F (t) (3)

where xu (t) = [ z2f l(t) z2f r (t) z2rl(t) z2rr (t) ]T, w(t) =[wf l(t) wf r (t) wrl(t) wrr (t) ]T, F (t) = [Ff l(t) Ff r (t)


DU AND ZHANG: FUZZY CONTROL FOR NONLINEAR UNCERTAIN ELECTROHYDRAULIC ACTIVE SUSPENSIONS WITH INPUT CONSTRAINT 345

Fig. 1. Full-car suspension model.

Frl(t) Frr (t)]T , and the matrices are given as

Ms =

ms 0 00 Iθ 00 0 Iφ

Mu =

muf 0 0 0

0 muf 0 00 0 mur 00 0 0 mur

Bs =

csf 0 0 00 csf 0 00 0 csr 00 0 0 csr

Ks =

ksf 0 0 00 ksf 0 00 0 ksr 00 0 0 ksr

Kt =

ktf 0 0 00 ktf 0 00 0 ktr 00 0 0 ktr

.

Substituting (2) into (3), we obtain

Mm zm (t) + Bm zm (t) + Km zm (t) = Kmtw(t) + Lm F (t)(4)

where zm (t) = [ qT (t) xTu (t) ]T and

Mm =[

Ms 00 Mu

]Bm =

[LBsL

T −LBs

−BsLT Bs

]Km =

[LKsL

T −LKs

−KsLT Ks + Kt

]Kmt =

[0

Kt

]Lm =

[−L

I

].

The state-space form of (4) can be expressed as

xg (t) = Axg (t) + B1w(t) + B2F (t) (5)

where

xg (t) =[zTm (t) zT

m (t)]T A =[

0 I

−M−1m Km −M−1

m Bm

],

B1 =[

0M−1

m Kmt

]B2 =

[0

M−1m Lm

].

The electrohydraulic actuator dynamics can be expressed as[4], [9], [11]–[13]

Fi(t) = −βFi(t) − αA2s (z1i(t) − z2i(t))

+ γaAs

√Ps −

sgn(xvi(t))Fi(t)As

xvi(t),

xv i(t) =1τ

(−xvi(t) + Kvui(t)) (6)

where xvi(t) is the spool valve displacement, ui(t) is the controlinput voltage to the servo valve, i denotes fl, fr, rl, and rr, re-spectively, As is the actuator ram area, Ps is the hydraulic supplypressure, α = 4βe/Vt, β = αCtm , and γa = αCdωa

√1/ρa ,

where βe is the effective bulk modulus, Vt is the total actuatorvolume, Ctm is the coefficient of total leakage due to pressure,Cd is the discharge coefficient, ωa is the spool valve area gradi-ent, and ρa is the hydraulic fluid density. τ is the time constantof the spool valve dynamics and Kv is the conversion gain.



The dynamics equation (6) of the electrohydraulic actuatorcan be further modeled with the state-space form as

xai(t) = Aai(t)xai(t) + Aaixg (t) + Baiui(t),

Fi(t) = Caixai(t) (7)

where

xai(t) = [ Fi(t) xvi(t) ]T Aai(t) =

−β γaAsfi(t)

0 −1τ

fi(t) =

√Ps −

sgn(xvi(t))Fi(t)As

Bai =

0Kv

τ

Cai = [ 1 0 ]

and Aaixg (t) describes the term of −αA2s (z1i(t) − z2i(t)).

Combining the actuator dynamics equation (7) with the sus-pension model (5), we obtain the electrohydraulic suspensionmodel in state-space form as

x(t) = A(t)x(t) + B1w(t) + B2 u(t) (8)

wherex(t) = [xTg (t) xaf l(t) xaf r (t) xarl(t) xarr (t) ]T

is the state vector and u(t) is the bounded input voltage to theactuator servo valve. In real applications, the input voltageto the servo valve can be bounded as u(t) =sat(u(t)), wheresat(u(t)) is a saturation function of control input u(t), definedas

sat(u(t)) =

−ulim , if u(t) < −ulim

u(t), if − ulim ≤ u(t) ≤ ulim

ulim , if u(t) > ulim

(9)

where ulim is the control input limit. The matrices are

A(t) =[

A B2Ca

Aa Aa(t)

]B1 =

[B1

0

]B2 =

[0

Ba

].

It is noted that the system matrix A(t) is a nonlinear andtime-varying matrix due to the nonlinear time-varying behaviorof the actuator dynamics and the variation of the sprung mass.

III. T–S FUZZY MODELING

The full-car electrohydraulic suspension model (8) incorpo-rates well-characterized and essential actuator nonlinearities,and the controller design is required to consider both the param-eter uncertainty and the control input constraint, which leads toa challenging control problem. In order to design a controller forthe model through the fuzzy approach, the T–S fuzzy modelingtechnique will be applied, and the idea of “sector nonlinear-ity” [18] is employed to construct an exact T–S fuzzy model forthe nonlinear uncertain suspension system (8).

Suppose that the actuator force Fi(t) (where i denotes fl, fr,rl, and rr, respectively) is bounded in practice by its minimumvalue Fimin and its maximum value Fimax ; the nonlinear func-tion fi(t) is then bounded by its minimum value fmin and itsmaximum value fmax . Thus, using the idea of “sector nonlin-

Fig. 2. Membership functions.

earity” [18], fi(t) can be represented by

fi(t) = M1i(ξi(t))fmax + M2i(ξi(t))fmin (10)

where ξi(t) = fi(t) is a premise variable, M1i(ξi(t)) andM2i(ξi(t)) are membership functions, and

M1i(ξi(t)) =fi(t) − fmin

fmax − fminM2i(ξi(t)) =

fmax − fi(t)fmax − fmin

.

(11)Similarly, the uncertain sprung mass ms(t) is bounded by itsminimum value msmin and its maximum value msmax , and canthus be represented by

1ms(t)

= N1(ξm (t))mmax + N2(ξm (t))mmin (12)

where ξm (t) = 1/ms(t) is also a premise variable, mmax =1/msmin , mmin = 1/msmax , and N1(ξm (t)) and N2(ξm (t))are membership functions that are defined as

N1(ξm (t)) =1/ms(t) − mmin

mmax − mmin

N2(ξm (t)) =mmax − 1/ms(t)mmax − mmin

. (13)

For description brevity, we name the aforementioned mem-bership functions M1i(ξi(t)), M2i(ξi(t)), N1(ξm (t)), andN2(ξm (t)), shown in Fig. 2, as ‘big,” “small,” ‘light,” and‘heavy,” respectively. The nonlinear uncertain suspension model(8) can then be represented by a T–S fuzzy model composed of32 (25) fuzzy rules, as listed in Table I, where B, S, L, and Hrepresent ‘big,” ‘small,” ‘light,” and ‘heavy,” respectively. Todescribe the T–S fuzzy model more clearly, several examples ofthe fuzzy IF-THEN rules corresponding to Table I are explainedas follows.

Model Rule 1:

IF ξi(t) (i denotes fl, fr, rl, and rr, respectively)are small and ξm (t) is light,

THEN x(t) = A1x(t) + B1w(t) + B2 u(t)

where matrix A1 is obtained from matrix A(t) in (8) by replacingfi with fmin and 1/ms with mmax .

...



TABLE ILIST OF FUZZY RULES

Model Rule 16:

IF ξi(t) (i denotes fl, fr, rl, and rr, respectively)are big and ξm (t) is light,


where matrix A16 is obtained from matrix A(t) in (8) by replac-ing fi with fmax and 1/ms with mmax .

Model Rule 17:

IF ξi(t) (i denotes fl, fr, rl, and rr, respectively)are small and ξm (t) is heavy,


where matrix A17 is obtained from matrix A(t) in (8) by replac-ing fi with fmin and 1/ms with mmin .

...Model Rule 32:

IF ξi(t) (i denotes fl, fr, rl, and rr, respectively)are big and ξm (t) is heavy,


where matrix A32 is obtained from matrix A(t) in (8) by replac-ing fi with fmax and 1/ms with mmin .

Thus, the T–S fuzzy model that represents exactly the non-linear uncertain suspension model (8) under the assumptionof bounds on actuator forces Fi(t) ∈ [Fmin , Fmax] and sprungmass ms(t) ∈ [msmin ,msmax] is obtained as

x(t) =32∑

i=1

hi(ξ(t))Aix(t) + B1w(t) + B2 u(t) (14)

where

h1(ξ(t)) = M2f l(ξf l(t))M2f r (ξf r (t))M2rl(ξrl(t))

× M2rr (ξrr (t))N1(ξm (t))


× M2rr (ξrr (t))N1(ξm (t))

...


× M1rr (ξrr (t))N2(ξm (t)),

hi(ξ(t)) ≥ 0, i = 1, 2, . . . , 32, and32∑

i=1

hi(ξ(t)) = 1.

In practice, the actuator force Fi(t), the spool valve positionxvi(t), and the sprung mass ms(t) can be measured; thus, theT–S fuzzy model (14) can be realized.

It is noted that the T–S fuzzy model (14) is obtained viathe “sector nonlinearity” approach based on the analysis ofthe nonlinear function fi(t) and the variation of sprung massms, the bounds of which can be estimated in a real operat-ing situation. The construction of a T–S fuzzy model froma given nonlinear dynamic model can also utilize the ideaof “local approximation” or a combination of “sector non-linearity” and “local approximation” [18]. In general, theseare analytic transformation techniques, which can be appliedonly to models described analytically. Since analytic techniquesneed problem-dependent human intuition and cannot be easilysolved in some cases, recently, a higher order-singular-value-decomposition (HOSVD)-based tensor product (TP) modeltransformation approach was proposed to automatically and nu-merically transform a general dynamic system model into a TPmodel form, including polytopic and T–S model forms [22].The TP model representation has shown various advantagesfor LMI-based controller design [23], [24], and relaxed LMIconditions can be further obtained for closed-loop fuzzy sys-tems with TP structure [25]. There is also a MATLAB Tool-box for TP model transformation (available for download to-gether with documentation and examples at http://tptool.sztaki.hu/tpde).

For our problem, in fact, using the convex normalized (CNO)type of TP model transformation can obtain the same member-ship functions as those described before when fi(t) and 1/ms(t)are used as time-varying parameter variables. However, if ms(t)is used as a time-varying parameter variable instead of 1/ms(t),different membership functions can be generated with the TPmodel transformation, as shown in Fig. 3. It can be seen fromFig. 3 that the membership functions are nonlinear, and theyare different from those shown in Fig. 2. Nevertheless, usingthis new type of membership function does not alter the LMI-based controller design process or the design results obtainedwith the membership functions defined in (13). Generally, Fi(t)and xvi(t) can be directly used as the time-varying parametervariables to obtain the T–S model using the TP model transfor-mation. However, the TP model transformation should considerthe tradeoff between approximation accuracy and complexity.For the studied problem, when using the derived membershipfunctions (11) and (13), only 32 fuzzy rules need to be applied.Therefore, in this paper, the derived membership functions (11)and (13) are used. And, despite the authors’ effort, no othertypes of membership functions are found to yield better perfor-mance than the derived membership functions (11) and (13),



Fig. 3. Membership functions for sprung mass obtained from TP modeltransformation.

although there may exist methods to automatically generateaffine decompositions.

IV. FUZZY CONTROLLER DESIGN

In order to avoid the problem associated with having a largenumber of inequalities involved in the controller design, thenorm-bounded approach [20], [21] is used to handle the satu-ration nonlinearity defined in (9). Hence, (14) will be writtenas

x(t) =32∑

i=1

hi(ξ(t))Aix(t) + B1w(t) + B2 u(t)

=32∑

i=1

hi(ξ(t))Aix(t) + B1w(t) + B21 + ε

2u(t)

+ B2

(u(t) − 1 + ε

2u(t)

)= Ahx(t) + B1w(t) + B2

1 + ε

2u(t) + B2v(t) (15)

where Ah =∑32

i=1hi(ξ(t))Ai and v(t) = u(t) − 1+ε2 u(t), 0 <

ε < 1. And for designing the controller, the following lemmawill be used.

Lemma 1 [20]: For the saturation constraint defined by (9),as long as |u(t)| ≤ u l im

ε , we have∥∥∥∥u(t) − 1 + ε

2u(t)

∥∥∥∥ ≤ 1 − ε

2‖u(t)‖ (16)

and hence[u(t)− 1+ ε

2u(t)

]T [u(t)− 1+ ε

2u(t)

]≤

(1− ε

2

)2

uT (t)u(t)

(17)where 0 < ε < 1.

The fuzzy controller design for the T–S fuzzy model (15) iscarried out based on the so-called PDC scheme [18]. For theT–S fuzzy model (15), we construct the fuzzy state feedback

controller via the PDC as

u(t) =32∑

i=1

hi(ξ(t))Kix(t) = Khx(t) (18)

where Kh =∑32

i=1hi(ξ(t))Ki and Ki is the state feedback gainmatrix to be designed.

Since ride comfort is an important performance requirementfor a vehicle suspension and it can usually be quantified bythe sprung mass acceleration, the sprung mass acceleration ischosen as the control output, i.e.

z(t) = zs(t) =32∑

i=1

hi(ξ(t))Cix(t) = Chx(t) (19)

where Ch =∑32

i=1hi(ξ(t))Ci and Ci is extracted from theeighth to the tenth row of matrix Ai, i = 1, 2, . . . , 32.

In order to design an active suspension to perform adequatelyin a wide range of shock and vibration environments, the L2gain of the system (15) with (19) is chosen as the performancemeasure, which is defined as

‖Tzw‖∞ = sup‖w‖2 =0

‖z‖2

‖w‖2(20)

where ‖z‖22 =

∫ ∞0 zT (t)z(t)dt and ‖w‖2

2 =∫ ∞

0 wT (t)w(t)dt,and the supermum is taken over all nonzero trajectories of thesystem (15) with x(0) = 0. Our goal is to design a fuzzy con-troller (18) such that the fuzzy system (15) with controller (18)is quadratically stable and the L2 gain (20) is minimized.

To design the controller, the following lemma will be used.Lemma 2: For any matrices (or vectors) X and Y with appro-

priate dimensions, we have

XT Y + Y T X ≤ εXT X + ε−1Y T Y

where ε > 0 is any scalar.Theorem 3: For a given number γ > 0, 0 < ε < 1, the T–S

fuzzy system (15) with controller (18) is quadratically stable andthe L2 gain defined by (20) is less than γ if there exist matricesQ > 0, Yi, i = 1, 2, . . . , 32, and scalar ε > 0, such that, (21)and (22), as shown at the bottom of the next page.

Moreover, the fuzzy state feedback gains can be obtained asKi = YiQ

−1 , i = 1, 2, . . . , 32.Proof: Let us define a Lyapunov function for the system (15)

as

V (x(t)) = xT (t)Px(t) (23)

where P is a positive definite matrix. By differentiating (23),we obtain

V (x(t)) = xT (t)Px(t) + xT (t)P x(t)

=[Ahx(t)+ B1w(t)+ B2

1+ε

2u(t)+B2v(t)

]T

Px(t)

+ xT (t)P[Ahx(t) + B1w(t) + B2

1 + ε

2u(t)

+ B2v(t)]. (24)



By Lemma 1, Lemma 2, and definition (18), we have

V (x(t)) ≤ xT (t)[AT

h P + PAh +(

B21 + ε

2Kh

)T

P

+ PB21 + ε

2Kh

]x(t)

+ wT (t)BT1 Px(t) + xT (t)PB1w(t) + εvT (t)v(t)

+ ε−1xT (t)PB2BT2 Px(t)

≤ xT (t)Θx(t) + wT (t)BT1 Px(t) + xT (t)PB1w(t)

(25)

where

Θ =[AT

h P + PAh +(

B21 + ε

2Kh

)T

P + PB21 + ε

2Kh

+ ε

(1 − ε

2

)2

KTh Kh + ε−1PB2B

T2 P

](26)

and ε is any positive scalar.Adding zT (t)z(t) − γ2wT (t)w(t) on both sides of (25)

yields

V (x(t)) + zT (t)z(t) − γ2wT (t)w(t)

≤ [xT (t) wT (t) ][Θ + CT

h Ch PB1

BT1 P −γ2I

][x(t)

w(t)

]. (27)

Let us consider

Π =[

Θ + CTh Ch PB1

BT1 P −γ2I

]< 0 (28)

then, V (x(t)) + zT (t)z(t) − γ2wT (t)w(t) ≤ 0 and the L2 gaindefined in (20) is less than γ > 0 with the initial conditionx(0) = 0 [26]. When the disturbance is zero, i.e., w(t) = 0, it

can be inferred from (27) that if Π < 0, then V (x(t)) < 0, andthe fuzzy system (15) with the controller (18) is quadraticallystable.

Pre- and postmultiplying (28) by diag( P−1 I ) and its trans-pose, respectively, and defining Q = P−1 and Yh = KhQ, thecondition Π < 0 is equivalent to

Σ =

QAT

h + AhQ +1 + ε

2Y T

h BT2 +

1 + ε

2B2Yh

+ε(1 − ε

2

)2Y T

h Yh+ ε−1B2BT2 + QCT

h ChQ

B1

BT1 −γ2I

< 0.

(29)By the Schur complement, Σ < 0 is equivalent to (Ψ), as shownat the bottom of this page.

By the definitions Ah =∑32

i=1hi(ξ(t))Ai, Ch =∑32i=1hi(ξ(t))Ci, Yh =

∑32i=1hi(ξ(t))Yi, and the fact

that hi(ξ(t)) ≥ 0 and∑32

i=1hi(ξ(t)) = 1, Ψ < 0 is equivalentto (21).

On the other hand, from (18), the constraint |u(t)| ≤ u l imε can

be expressed as

∣∣∣∣∣32∑

i=1

hi(ξ(t))Kix(t)

∣∣∣∣∣ ≤ ulim

ε. (31)

It is obvious that if |Kix(t)| ≤ u l imε , then (31) holds. Let

Ω(K) = x(t)| |xT (t)KTi Kix(t)| ≤ (u l im

ε )2; then the equiv-alent condition for an ellipsoid Ω(P, ρ) = x(t)| xT (t)Px(t) ≤ρ being a subset of Ω(K), i.e., Ω(P, ρ) ⊂ Ω(K), is [19]

Ki

(P

ρ

)−1

KTi ≤

(ulim

ε

)2. (32)

QATi + AiQ +

1 + ε

2[Y T

i BT2 + B2Yi

]+ ε−1B2B

T2 Y T

i QCTi B1

∗ −ε−1(

21 − ε

)2

I 0 0

∗ ∗ −I 0

∗ ∗ ∗ −γ2I

< 0 (21)

(ulim

ε

)2I Yi

Y Ti ρ−1Q

≥ 0. (22)

Ψ =

QATh + AhQ +

1 + ε

2[Y T

h BT2 + B2Yh

]+ ε−1B2B

T2 Y T

h QCTh B1

∗ −ε−1(

21 − ε

)2

I 0 0

∗ ∗ −I 0

∗ ∗ ∗ −γ2I

< 0. (30)



By the Schur complement, inequality (32) can be written as(ulim

ε

)2I Ki

(P

ρ

)−1

(P

ρ

)−1

KTi

(P

ρ

)−1

≥ 0. (33)

Using the definitions Q = P−1 and Yi = KiQ, inequality (33)is equivalent to (22). This completes the proof.

The minimization of γ can be realized as

min γ subject to LMIs (21) and (22). (34)

This problem can be solved very efficiently by means of theMATLAB LMI Toolbox software.

Remark 1: It is noted that once the solution of problem (34)is feasible, then the value of γ can be obtained using the LMIToolbox software and the designed controller can guarantee theL2 gain (20) to be less than γ in terms of the LMI condition (21).When the designed controller is applied to the system (14), thereal value of the L2 gain (20) for the system (14) under a givendisturbance can be evaluated by measuring the control outputresponse (19) and calculating the value using equation (20).

Remark 2: In the paper, the common quadratic Lyapunovfunction approach, where the Lyapunov function candidate isdefined as in (23), is utilized to derive the controller synthesisconditions. It has been noted that common quadratic Lyapunovfunctions tend to be conservative and, even worse, might not ex-ist for some complex highly nonlinear systems [17] . This is oneof the main limitations of this kind of approach. With regardto overcoming the drawback of common quadratic Lyapunovfunctions, piecewise quadratic Lyapunov functions and fuzzyLyapunov functions have received increasing attention recently[27], [28]. However, controller synthesis conditions based onpiecewise quadratic Lyapunov functions and fuzzy Lyapunovfunctions are generally given by bilinear matrix inequalities(BMIs), which have to be solved by way of, e.g., a cone com-plementarity linearisation approach [28] or a descriptor systemapproach [27]. Therefore, the computation cost and complex-ity of using piecewise quadratic Lyapunov function and fuzzyLyapunov function approaches would be much higher in general.In terms of the possible requirement on real-time computationfor a practical system and the feasible solutions checking onthe given system, this paper keeps using the common quadraticLyapunov function approach regardless of its conservatism.

V. APPLICATION EXAMPLE

In this section, we will apply the proposed approach to designa fuzzy state feedback controller for a full-car electrohydraulicsuspension model, as described in Section II. The full-car sus-pension model parameter values are listed in Table II, and theparameter values for each hydraulic actuator used in the simu-lation are given in Table III.

In this study, we suppose that the input voltage of each spoolvalve is limited to ulim = 2.5 V, and each actuator output forceis limited to 2000 N. The bounds of the nonlinear function fi(t)are estimated as fmin = 2800 and fmax = 4000. The sprung

TABLE IIPARAMETER VALUES OF THE FULL-CAR SUSPENSION MODEL

TABLE IIIPARAMETER VALUES OF THE HYDRAULIC ACTUATOR

mass is assumed to be varied between msmin = 1120 kg andmsmax = 1680 kg, which is ±20% variation of the nominalsprung mass. Using the controller design approach presented inSection IV, and choosing ε = 0.97 by trial and error, we obtainthe controller gain matrices Ki that consist of 32 matrices, eachwith dimensions of 4 × 22. It is noted that the controller gainmatrices are constant matrices that do not need to be recalcu-lated in a real-time implementation and can be easily stored ina microprocessor memory (RAM or ROM). The calculation ofthe control outputs, i.e., input voltages sent to the actuators, interms of the measured state variables and the calculated mem-bership functions is also quite straightforward. Therefore, therequired computational power will not be very high, which en-ables the implementation of the controller on a DSP, e.g., theTexas Instruments TMS320C30.

To compare the suspension performance, an optimal H∞controller is designed for the linear full-car suspension model(5) without considering the electrohydraulic actuator dynamics(7). By defining the control output as the sprung mass acceler-ation and using the bounded real lemma (BRL), this controllergain matrix is obtained as (35), as shown at the bottom of thenext page.

Since the optimal H∞ controller design theory can be foundin many references, the controller design process is omitted herefor brevity.

In the simulation, a test road disturbance, which is given as

zr (t) = 0.0254 sin 2πt + 0.005 sin 10.5πt

+ 0.001 sin 21.5πt (m) (36)

is used first. This road disturbance is close to the car bodyresonance frequency (1 Hz), with high-frequency disturbanceadded to simulate the rough road surface. To observe the rollmotion, this road disturbance is assumed to pass the wheels onthe left side of the car only. The simulation program is realizedby MATLAB/Simulink.

For the nominal sprung mass under the specified road distur-bance (36), the time-domain responses for three suspensions,i.e., passive suspension, active suspension with H∞ controller(35), and active suspension with the electrohydraulic actuators



Fig. 4. Time-domain responses under a test road profile. Solid line is for active suspension with fuzzy controller. Dotted line is for active suspension with H∞controller. Dot-dashed line is for passive suspension.

and the designed fuzzy controller are compared. The responses,consisting of the sprung mass heave acceleration, heave dis-placement, pitch angle, and roll angle, are plotted in Fig. 4. Itis observed from Fig. 4 that the proposed fuzzy control strategyreduces the sprung mass acceleration and displacement mag-nitudes significantly compared to the passive suspension underthe same road disturbance. The active suspension with the de-signed fuzzy controller also achieves a very similar suspensionperformance to the active suspension with an optimal H∞ con-troller. This confirms that the proposed fuzzy control strategycan realize good suspension performance with highly nonlinearelectrohydraulic actuators.

Fig. 5 shows the actuator output forces for the active sus-pension with the designed fuzzy controller. It is observed thatthe four actuators provide different control forces in accordancewith the fuzzy rules and the measurements of the state variables.The proposed fuzzy control strategy provides effective actuatorforces that aim to optimize the sprung mass heave acceleration,pitch acceleration, and roll acceleration to improve ride com-

fort performance. Fig. 6 shows the control input voltages, andFig. 7 shows the nonlinear function outputs. It can be seen fromFigs. 6 and 7 that the control input voltages are within the de-fined input voltage range and the nonlinear function outputs arelocated within the estimated bounds.

Now, consider the case of an isolated bump in an otherwisesmooth road surface. The corresponding ground displacementfor the wheel is given by

zr (t) =

a

2

(1 − cos

(2πv0

lt

)), 0 ≤ t ≤ l

v0

0, t >l

v0

(37)

where a and l are the height and the length of the bump. Wechoose a = 0.1 m, l = 10 m, and the vehicle forward velocityas v0 = 45 km/h.

For the nominal sprung mass under the road disturbance (37),Fig. 9 shows the bump responses of the sprung mass heave

104 ×

−1.0324 −0.2478 −1.2670 0.8961 0.2431 2.2499−1.0324 −0.2478 1.2670 0.2431 0.8961 −2.0930−1.3299 1.3735 −1.5342 0.4602 −0.4434 −0.6042−1.3299 1.3735 1.5342 −0.4434 0.4602 1.6712

−2.0930 0.2879 0.3718 0.0665 0.0044 −0.0023 0.0078 −0.00722.2499 0.2879 0.3718 −0.0665 −0.0023 0.0044 −0.0072 0.00781.6712 0.2097 −0.3700 0.0504 0.0128 −0.0124 0.0110 −0.0081−0.6042 0.2097 −0.3700 −0.0504 −0.0124 0.0128 −0.0081 0.0110

.

(35)



Fig. 5. Actuator output forces for active suspension with fuzzy controller.

Fig. 6. Actuator control voltages for active suspension with fuzzy controller.

acceleration, pitch acceleration, and roll acceleration for thepassive suspension, the active suspension with H∞ controller(35), and the active suspension with the electrohydraulic actua-tors and the designed fuzzy controller. It is again observed thatthe proposed fuzzy control strategy achieves suspension per-

formance very similar to the active suspension with an optimalH∞ controller. The active suspension performance is signifi-cantly improved compared to the passive suspension.

To illustrate the effect of sprung mass variation, Fig. 9 showsthe bump responses of the sprung mass heave acceleration for



Fig. 7. Nonlinear function values for active suspension with fuzzy controller.

Fig. 8. Acceleration responses under a bump road profile. Solid line is for active suspension with fuzzy controller. Dotted line is for active suspension with H∞controller. Dot-dashed line is for passive suspension.

the passive suspension (passive) and the active suspension withelectrohydraulic actuator and fuzzy controller (active) when thesprung mass is 1120 and 1680 kg. It is observed that, despite thechange in sprung mass, the designed fuzzy controller achievessignificantly better performance on heave acceleration, where a

lower peak and shorter settling time are obtained. Figs. 10 and 11show the bump responses of the sprung mass pitch accelerationand roll acceleration, respectively. It can be seen from Fig. 10that the sprung mass affects the pitch acceleration significantly.However, the active suspension can keep the pitch acceleration



Fig. 9. Heave acceleration responses for different sprung mass.

Fig. 10. Pitch acceleration responses for different sprung mass.

low, regardless of the sprung mass variation. In Fig. 11, theactive suspension achieves lower roll acceleration compared tothe passive suspension, although the sprung mass variation doesnot affect roll acceleration due to the symmetric distribution ofthe sprung mass about the vehicle’s roll axis. Figs. 9–11 indicatethat the improvement in ride comfort can be maintained by thedesigned active suspension for large changes in load conditions.

When the road disturbance is considered as random vibration,it is typically specified as a stationary random process that can

be represented by

zr (t) = 2πq0

√G0V ω(t) (38)

where G0 stands for the road roughness coefficient, q0 is thereference spatial frequency, V is the vehicle forward velocity,and ω(t) is zero-mean white noise with identity power spectraldensity. For a given road roughness G0 = 512 × 10−6 m3 anda given vehicle forward velocity V = 20 m/s, the rms valuesfor sprung mass heave acceleration, pitch acceleration, and roll



Fig. 11. Roll acceleration responses for different sprung mass.

Fig. 12. RMS ratios for sprung mass heave acceleration, pitch acceleration, and roll acceleration versus sprung mass.

acceleration are calculated as the sprung mass changes from1120 to 1680 kg. The rms ratios between the active suspensionwith fuzzy controller and the passive suspension are plottedagainst sprung mass in Fig. 12. It can be seen that the designedfuzzy controller maintains a ratio below 1, regardless of thelarge variations in the sprung mass. When the road roughnessand vehicle forward velocity are given different values, verysimilar results are obtained. For brevity, these results are notshown. Fig. 12 further validates the claim that the proposed

fuzzy control strategy can realize good ride comfort perfor-mance for electrohydraulic suspension even when the sprungmass is varied significantly.

VI. CONCLUSION

In this paper, we have presented a fuzzy state feedback controlstrategy for electrohydraulic active suspensions to deal withthe nonlinear actuator dynamics, sprung mass variation, and



control input constraint problems. First, using the idea of “sectornonlinearity,” the nonlinear uncertain electrohydraulic actuatorwas represented by a T–S fuzzy model in defined regions. Thus,by means of the PDC scheme, a fuzzy state feedback controllerwas designed for the obtained T–S fuzzy model to optimize theH∞ performance of ride comfort. At the same time, the actuatorinput voltage constraint was incorporated into the controllerdesign process. The sufficient conditions for designing such acontroller were expressed by LMIs. Simulations were used tovalidate the effectiveness of the designed controller.

ACKNOWLEDGMENT

The authors would like to thank the editors and anonymousreviewers for their invaluable suggestions and comments on theimprovement of the paper. They also would like to thanks. Mr.W. Smith for his careful reading and English modification of themanuscript.

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[9] P. C. Chen and A. C. Huang, “Adaptive sliding control of active suspensionsystems with uncertain hydraulic actuator dynamics,” Veh. Syst. Dyn.,vol. 44, no. 5, pp. 357–368, 2006.

[10] C. Kaddissi, J.-P. Kenne, and M. Saad, “Identification and real-time controlof an electrohydraulic servo system based on nonlinear backstepping,”IEEE/ASME Trans. Mechatronics, vol. 12, no. 1, pp. 12–22, Feb. 2007.

[11] A. Alleyne and R. Liu, “On the limitations of force tracking control forhydraulic servosystems,” J. Dyn. Syst., Meas. Control, vol. 121, no. 2,pp. 184–190, 1999.

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[13] A. Alleyne and R. Liu, “Systematic control of a class of nonlinear systemswith application to electrohydraulic cylinder pressure control,” IEEETrans. Control Syst. Technol., vol. 8, no. 4, pp. 623–634, Jul. 2000.

[14] A. G. Thompson and B. R. Davis, “Force control in electrohydraulic activesuspensions revisited,” Vehicle Syst. Dyn., vol. 35, no. 3, pp. 217–222,2001.

[15] Y.-S. Lu and J.-S. Chen, “A self-organizing fuzzy sliding-mode controllerdesign for a class of nonlinear servo systems,” IEEE Trans. Ind. Electron.,vol. 41, no. 5, pp. 492–502, Oct. 1994.

[16] N. Al Holou, T. Lahdhiri, D. S. Joo, J. Weaver, and F. Al Abbas, “Slidingmode neural network inference fuzzy logic control for active suspensionsystems,” IEEE Trans. Fuzzy Syst., vol. 10, no. 2, pp. 234–246, Apr. 2002.


[18] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analy-sis:Linear Matrix Inequality Approach. New York: Wiley, 2001.

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[21] C.-S. Tseng and B.-S. Chen, “H∞ fuzzy control design for nonlinearsystems subject to actuator saturation,” in Proc. 2006 IEEE Int. Conf.Fuzzy Syst., Vancouver, BC, Canada, pp. 783–788.

[22] P. Baranyi, “TP model transformation as a way to LMI-based controllerdesign,” IEEE Trans. Ind. Electron., vol. 51, no. 2, pp. 387–400, Apr.2004.

[23] P. Baranyi and Y. Yam, “Case study of the TP-model transformation in thecontrol of a complex dynamic model with structural nonlinearity,” IEEETrans. Ind. Electron., vol. 53, no. 3, pp. 895–904, Jun. 2006.

[24] Z. Petres, P. Baranyi, P. Korondi, and H. Hashimoto, “Trajectory trackingby TP model transformation: Case study of a benchmark problem,” IEEETrans. Ind. Electron., vol. 54, no. 3, pp. 1654–1663, Jun. 2007.

[25] C. Arino and A. Sala, “Relaxed LMI conditions for closed-loop fuzzysystems with tensor-product structure,” Eng. Appl. Artif. Intell., vol. 20,pp. 1036–1046, 2007.

[26] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear MatrixInequalities in System and Control Theory. Philadelphia, PA: SIAM,Jun. 1994.


[28] H. Gao, Z. Wang, and C. Wang, “Improved H∞ control of discrete-timefuzzy systems: A cone complementarity linearization approach,” Inf. Sci.,vol. 175, pp. 57–77, 2005.

Haiping Du received the Ph.D. degree in mechanicaldesign and theory from Shanghai Jiao Tong Univer-sity, Shanghai, China, in 2002.

He is currently a Research Fellow in the Fac-ulty of Engineering, University of Technology, Syd-ney, N.S.W., Australia. He was a Postdoctoral Re-search Associate at the University of Hong Kong,Hong Kong, from 2002 to 2003, and at the ImperialCollege London, U.K., from 2004 to 2005. His cur-rent research interests include robust control theoryand engineering applications, soft computing, dy-

namic system modeling, model and controller reduction, and smart materialsand structures.

Dr. Du was the recipient of the Excellent Ph.D. Thesis Prize from ShanghaiProvincial Government in 2004.

Nong Zhang received the Doctorate degree from theUniversity of Tokyo, Tokyo, Japan, in 1989.

In 1989, he joined the Faculty of Engineering, Uni-versity of Tokyo, as a Research Assistant Professor. In1992, he joined the Engineering Faculty, Universityof Melbourne, as a Research Fellow. Since 1995, hehas been with the Faculty of Engineering, Universityof Technology, Sydney, N.S.W., Australia. His cur-rent research interests include experimental modalanalysis, rotor dynamics, vehicle power train dynam-ics, and hydraulically interconnected suspension and

vehicle dynamics.Dr. Zhang is a Member of the American Society of Mechanical Engineers

(ASME) and a Fellow of the Society of Automotive Engineers, Australasia.



Rapid Load Following of an SOFC Power System viaStable Fuzzy Predictive Tracking Controller

Tiejun Zhang, Member, IEEE, and Gang Feng, Fellow, IEEE

Abstract—The solid oxide fuel cell (SOFC) is widely acceptedfor clean and distributed power generation use, but critical opera-tion problems often occur when the stand-alone fuel cell is directlyconnected to the electricity grid or the dc electric user. In order toaddress these problems, in this paper, a data-driven fuzzy modelingmethod is employed to identify the dynamic model of an integratedSOFC/capacitor system. A novel offset-free input-to-state stablefuzzy predictive controller is developed based on the obtained fuzzymodel. Both the rapid power load following and safe SOFC oper-ation requirements are taken into account in the design of theclosed-loop control system. Simulations are also given to demon-strate the load following control performance of the proposedfuzzy predictive control strategy for the SOFC/capacitor powersystem.

Index Terms—Fuel cell, fuzzy systems, identification, input-to-state stability, load following, output tracking, predictive control.

I. INTRODUCTION

A S ONE of the second generations of fuel cells, the solidoxide fuel cell (SOFC) has been demonstrated to be a

promising power generation technology, especially in stationaryapplications [1]–[6]. They are characterized by high operatingtemperature range (600 C–1000 C) with the use of solid-stateceramic electrolyte, which is easier to maintain due to the lackof cell corrosion. The SOFC/gas turbine (GT) based distributiongeneration can provide ancillary services such as load follow-ing and regulation with respect to the current deregulation andunbundling of the energy market [4], [5]. However, load follow-ing problems occur when the response of the fuel cell systemcannot safely meet both the external power load demand andthe balance of parasitic plant power demand [6]. For example,the phenomenon of oxygen starvation will appear in a fuel cellwhen the sudden power load changes greatly [7]. In that case,the partial pressure of oxygen falls dramatically, accompaniedby rapid decrease in cell voltage, which would shorten the lifeof the fuel cell stack. On the other hand, the fuel cell may also bepermanently damaged when the fuel starvation occurs in case

Manuscript received December 22, 2007; revised June 19, 2008 andSeptember 9, 2008. First published December 22, 2008; current version pub-lished April 1, 2009. This work was supported in part by the Hong KongResearch Grant Council under CityU 112806 and by the National Natural Sci-ence Foundation of China under Grant 50640460116.

T. J. Zhang was with the Department of Manufacturing Engineering and Engi-neering Management, City University of Hong Kong, Kowloon, Hong Kong. Heis now with the Center for Automation Technologies and Systems, RensselaerPolytechnic Institute, Troy, NY 12180-3590 USA (e-mail: [email protected]).

G. Feng is with the Department of Manufacturing Engineering and Engi-neering Management, City University of Hong Kong, Kowloon, Hong Kong(e-mail: [email protected]).



of deficient fuel supply [8], [9]. Therefore, an effective controlsystem is in great demand to ensure that the fuel cell systemmeets the time-varying power load demand with high processoperation efficiency [2]–[10].

Model predictive control (MPC) is one of the most powerfuloptimized control approaches to wide-range-constrained mul-tivariable industrial processes with slow dynamics [11]–[15],particularly for power industry [16]. In MPC, the control ac-tions are computed by solving a receding-horizon optimizationproblem at each sampling time instant. This is in contrast to con-ventional optimal control, where only a predetermined controllaw is employed. However, as for large-scale nonlinear pro-cesses, MPC suffers from a couple of important drawbacks inpractical application. First, it is difficult and time-consuming toobtain a reliable and concise first-principles prediction modelfor highly nonlinear and complex plants. Second, for nonlinearplants, especially with fast dynamics, nonlinear MPC suffersgreatly from its intensive computational burden of solving anonlinear nonconvex optimization problem. Hence, it is desir-able to exploit alternative approximate-model-based predictivecontrol approaches [17]–[19] so as to alleviate heavy optimiza-tion burden and restrictive requirement on the precise mathe-matical plant model in nonlinear predictive control [13]–[15].On the other hand, several offset-free MPC design methodshave been developed to meet the output tracking requirementsof constrained linear systems [20]–[22]. In these tracking con-trol systems, an integrating disturbance signal is estimated toaccount for the plant–model mismatch and/or unmodeled plantdisturbances. Then, a steady-state target generator is constructedto remove the effects of estimated disturbances, thus makingsure the zero steady-state offset output tracking through a linearMPC. Obviously, an offset-free linear MPC may not be capableof controlling strongly nonlinear processes operating in a widerange to achieve the steady-state zero offset.

Takagi–Sugeno (T–S) fuzzy dynamic models have beenwidely accepted in control community for its capability to repre-sent nonlinear dynamics [24]–[30]. Fuzzy systems are demon-strated to have very good approximation and interpretation ca-pability to general nonlinear systems [27]–[29]. And there havebeen many offline state feedback and output feedback controllerdesign methods for T–S fuzzy systems [25], [26], [30], whereasthe transient response of the closed-loop control system hasrarely been considered. As a matter of fact, transient perfor-mance is of much more concern in industrial process controland economical plant operation. Fortunately, several T–S-fuzzy-model-based predictive control approaches have been developedmost recently, such as [31]–[35], where the improvement of tran-sient closed-loop control performance in real time is one of themain design objectives.

1063-6706/$25.00 © 2009 IEEE



Fig. 1. System diagram of an SOFC/GT combined cycle plant.

Although the control of nonlinear SOFC system is challeng-ing due to the slow response under tight safe operation con-straints, several predictive control strategies have been proposedfor rapid load following. A novel structure of data-driven lin-ear MPC was used in [8] to deal with systems without fullonline measurement of all output variables, i.e., in the absenceof output measurements for fuel utilization, H2 /O2 ratio, andfuel cell pressure difference between the anode and cathode.As reported in [36], a fuzzy Hammerstein model was identifiedfrom the input–output operation data of an SOFC stack, thenthe associated standard predictive controller was applied to thefuel control of the stand-alone SOFC stack to meet stepwisepower load demands. Unfortunately, the stability issues of theseclosed-loop MPC systems have been largely ignored. However,the control system stability plays a fundamental and vital rolein plant operation and safety, especially for high-temperaturefuel cells. Moreover, the model-reality discrepancy also needsto be considered in the model-based controller design, like thepeer research on input-to-state stable (ISS) MPC [23]. In thispaper, we will focus on the rapid load tracking problem of theSOFC/capacitor power system via offset-free fuzzy MPC strat-egy, where the fuzzy dynamic model is identified from the plantoperation data and the closed-loop control system stability ismaintained.

Notation: x(k + i|k), i > 0, are the predicted system states attime k + i based on the current state estimation x(k|k) or x(k).And for any two vectors x, y ∈ n , x y, means xj − yj ≥0 ∀ j = 1, . . . , n, where xj , and yj denote the jth componentof x, and y, respectively.

II. SOFC/CAPACITOR SYSTEM

The typical diagram of an SOFC/GT system is shown inFig. 1. SOFC has the outstanding fuel flexibility. With the in-

Fig. 2. Schematic view of an SOFC/capacitor power system.

ternal steam reformer, methane can be converted into hydrogen(H2) even in the presence of some extent of carbon monoxide(CO). And syngas from biomass and coal gasification is alsoable to be transformed to hydrogen based on a water–gas shiftreaction (WGSR). In fact, syngas itself may also serve as thefuel gas for SOFC more than pure hydrogen [37]. The followingreactions will occur at the anode:

2H2 + 2O2− −→ 2H2O + 4e (1)

2CO + 2O2− −→ 2CO2 + 4e (2)

while the reaction at the cathode is

O2 + 4e −→ 2O2−. (3)

It should also be noted that a desulfurizer is included in Fig. 1for the compressed fuel gas supplied to fuel cell since sulfur isa poison for SOFCs.

Although the SOFC is widely accepted for stationary powergeneration use, critical operation problems often occur in caseswhere the fuel cell is directly connected to the electricity gridor the dc electric user. The reason is that the voltage of thestand-alone fuel cell would generally fall rapidly with risingcurrent density. In fact, fuel cells are particularly badly regu-lated, whereas most electronic and electrical equipments requirea fairly constant voltage. Moreover, due to the slow response,the fuel cell cannot be directly connected to the load in general.

A parallel capacitor can be integrated so as to compensatethe negative effect of varying external load on the stand-aloneSOFC stack operation, as shown in Fig. 2. The following pointscan be noted.

1) The capacitor will increase the response time of the SOFCstack/capacitor power system to its fuel/oxygen inputchanges. Larger capacitance leads to slower dynamic re-sponse of the integrated power system.

2) The capacitor will guarantee the load tracking of the stackcurrent. In steady state, the stack voltage VS becomes con-stant, VS = 0, and no current flows through the capacitor,IC = 0; thus, IS = ID + IC = ID .

As the first step of our advanced control research, we con-sider here the ideal fuel processor to produce pure hydrogenfor the feed of SOFCs as in [2]–[4], and [8]. Here, hydrogenand oxygen pass through the anode and cathode of fuel cells,respectively. The hydrogen is consumed by an electrochemicalreaction to generate the water and electricity, and the resultingpartial pressures are expressed as pH2 , pO2 , pH2O . Thus, one hasthe following schematic SOFC/capacitor dynamics with respect


ZHANG AND FENG: RAPID LOAD FOLLOWING OF AN SOFC POWER SYSTEM VIA STABLE FUZZY PREDICTIVE TRACKING CONTROLLER 359

TABLE IPARAMETERS IN THE SOFC SYSTEM

to inlet fuel flow rate qF , inlet oxygen flowrate qO , and externalcurrent load ID , as in [2]–[4], and [8]–[10]

VS =1

rC(V0 − VS − rID )

qH =1

TF(qF − qH)

pH2 =1

τH2 KH2

(qH − 2KrIS − pH2 KH2 ) (4)

pH2O =1

τH2OKH2O(2KrIS − pH2OKH2O)

pO2 =1

τO2 KO2

(qO − KrIS − pO2 KO2 )

where VS denotes the stack voltage, qH the hydrogen flow rate,IS = (V0 − VS )/r

V0 = N0

[E0 +

R0T0

2F0ln

pH2 (pO2 /101.325)1/2

pH2O

](5)

and Kr = N0/4F0 . Note that (5) is the well-known Nernstequation [1], and E0 is the electromotive force (EMF) at stan-dard pressure and it is dependent on the change in molarGibbs free energy of formation at different stack temperatureT [1], which can be approximated by an empirical formulaE0 = 1.2586 − 0.000252T [10]. Table I contains the SOFCplant parameters. The nominal stationary voltage/power–currentcharacteristics of a stand-alone SOFC stack are depicted inFig. 3, which shows that SOFC exhibits nonlinear behaviorin a wide operating regime. Obviously, the stack voltage is usu-ally under significant changes, especially in the low and highcurrent load, and the overloaded current would lead to the rapiddeterioration of operating stack voltage. Moreover, the stackcan maintain its maximal power output at some operating point,e.g., when the current load is around 300 A from Fig. 3. As theload current becomes larger, the power output would decreaseabruptly.

Furthermore, the open-loop dynamic responses of theSOFC/capacitor system to stepwise changes of inputs qF , qO ,and ID are given in Figs. 4–6, respectively.

The following variables are also defined for performancemonitoring during SOFC operation.

Fig. 3. Voltage/power–current characteristics of open-loop SOFC.

Fig. 4. SOFC system response to stepwise fuel flow rate change (+0.1 mol/s).

Fig. 5. SOFC system response to stepwise oxygen flow rate change(+0.1 mol/s).



Fig. 6. SOFC system response to stepwise load current change (+8 A).

1) Fuel utilization Uf stands for the ratio between the con-sumed hydrogen flow in reaction and the inlet hydrogenflow, i.e., Uf = qr

H/qinH .

2) Hydrogen/oxygen ratio rH/O stands for the ratio betweeninlet hydrogen flow and oxygen flow, i.e., rH/O = qin

H /qinO .

3) Pressure difference ∆p denotes the difference of hydro-gen/oxygen partial pressures in the fuel cell anode andcathode compartments, i.e., ∆p = pH2 − pO2 .

In order to prevent damage to the electrolyte, the fuel cellpressure difference ∆p between the anode and cathode com-partments should be kept below 4 kPa under normal operationand 8 kPa under transient conditions [4]. Although the stoichio-metric ratio of hydrogen to oxygen in overall fuel cell reactionis 2:1, oxygen excess is always expected in practice to makesure hydrogen reacting with oxygen more completely. For thisSOFC system, it is desirable to maintain the ratio of hydrogento oxygen rH/O around 1.145, as in [4] and [8]. In addition, thenormal operating point of fuel utilization Uf is about 85%, andtoo low or high utilization degree Uf would lead to potentialproblems: if Uf < 70%, a large portion of fuel is underused;thus, the economical efficiency of SOFC would decrease; other-wise, when Uf > 90%, the cells may suffer from fuel starvationand might be at the risk of permanent damage.

III. T–S FUZZY MODELING AND INPUT-TO-STATE STABILITY

In this section, T–S fuzzy system model and the input-tostate stability concept that will facilitate our discussions in thesubsequent sections are introduced.

A. T–S Fuzzy Dynamic Model

Consider a general multiple-input–multiple-output (MIMO)nonlinear system with m inputs, u ∈ m , and p outputs, y ∈ p .This system can be approximated by a set of coupled multiple-input–single-output (MISO) discrete-time T–S fuzzy dynamicmodels with linear consequents, which is described by L fuzzy

rules as in [24]–[30]

Rl : IF ξ1(k) is Ml1 and · · · and ξr (k) is Ml

r , THEN

y(k + 1) = Al(q−1)y(k) +m∑

h=1

Bhl (q−1)uh(k),

h = 1, . . . , m, l = 1, . . . , L (6)

where Rl denotes the lth fuzzy inference rule, L the numberof inference rules, Ml

j are fuzzy sets, y(k) any system outputvariable, u(k) ∈ m the system input variables, ξ(k) ∈ r thepremise measurable variables, (Al , B

hl ) is the lth local model

with the shift operator q−1 defined by q−1y(k) = y(k − 1), and

Al(q−1) = al1 + al2q−1 + · · · + alny

q−ny +1 (7)

Bhl (q−1) = bh

l1 + bhl2q

−1 + · · · + bhlnu

q−nu +1 . (8)

Let µl(ξ(k)) be the normalized membership functionof the inferred fuzzy set Ml , where Ml =

∏rj=1 Ml

j and∑Ll=1 µl(ξ) = 1. By using a center-average defuzzifier, product

inference, and a singleton fuzzifier [24]–[26], the fuzzy model(6) can be expressed as follows:

y(k + 1) = Aµ(q−1)y(k) +m∑

h=1

Bhµ (q−1)uh(k) (9)

where µ := µ(ξ) := (µ1 , . . . , µL ) and

Aµ =L∑

l=1

µlAl(q−1) Bhµ =

L∑l=1

µlBhl (q−1).

Thus, (9) also reads

y(k + 1) =L∑

l=1

µlψT (k)θl (10)

where

ψ(k) = [y(k) · · · y(k − ny + 1) u1(k) · · ·u1(k − nu + 1) · · · um (k) · · · tum (k − nu + 1)]T

(11)

θl = [al1 · · · alnyb1l1 · · · b1

lnu· · · bm

l1 · · · bmlnu

]T .

(12)

Note that the fuzzy model (6) or (9) is in the so-called con-trolled autoregressive moving average (CARMA) form.

B. Input-to-State Stability

Let ,≥0 ,N , and N≥0 denote the real, nonnegative real,integer and nonnegative integer numbers, respectively.

Definition 1 (Input-to-state stability [39]): A discrete-timenonlinear system x(k + 1) = F (x(k), v(k)) is ISS if there exista KL-function β and a K-function γ such that

‖x(k)‖ ≤ β(x(0), k) + γ(‖v‖) (13)

for each k ∈ N≥0 and the initial state x(0).



Definition 2 (ISS Lyapunov Function [39]): A continuousfunction V : n → ≥0 is called an ISS Lyapunov function fora system x(k + 1) = F (x(k), v(k)) if there exist some K∞-functions α1 , α2 , and α3 and a K-function σ such that

α1(‖x‖) ≤ V (x) ≤ α2(‖x‖) (14)

V (F (x, v)) − V (x) ≤ −α3(‖x‖) + σ(‖v‖). (15)

For the definitions of K,K∞, and KL-functions, refer to [38].Lemma 1 (ISS Lyapunov Characterization [39]): If a system

x(k + 1) = F (x(k), v(k)) admits an ISS Lyapunov function,then the system is ISS.

IV. FUZZY-MODEL-BASED PREDICTIVE TRACKING

CONTROLLER SYNTHESIS

A. Identification of Fuzzy Model and its State-Space Form

Defining

φ(k) =

µ1

...

µL

⊗ ψ(k) =

µ1(k)ψ(k)

...

µL (k)ψ(k)

Θ =

θ1

...

θL

where ⊗ is the Kronecker product, it follows that the system ofthe SOFC can be approximated by a fuzzy model

y(k + 1) = φT (k)Θ. (16)

When the MISO dataset (u1(k), . . . , um (k), y(k)), k =1, . . . , N , associated with plant operation is frequency-richenough in the sense that the inverse defined in (17) exists, itis straightforward to apply any standard parameter identifica-tion method such as the least squares method to identify itslocal dynamic model parameters, θl , l = 1, . . . , L, i.e.

Θ = (ΦT Φ)−1ΦT Y (17)

where

Φ =

φT (1)

...

φT (N − 1)

Y =

y(2)

...

y(N)

.

Note that the resulting modeling performance can be measuredby the variance accounted for (VAF) index [31] given by

VAF =

[1 − var(Y − Y )

var(Y )

]× 100% (18)

where Y is the sequence of real plant output measurements andY the fuzzy model output predictions, with Y = ΦΘ.

Then, from the estimated parameters θl and (12), one canidentify the fuzzy CARMA model with a signal of modelingerror d(k)


r , THEN

y(k + 1)=Al(q−1)y(k)+ Bl(q−1)u(k)+ d(k), l = 1, . . . , L

(19)

where u(k) = [u1(k) · · · um (k)]T and

Bl(q−1) = bl1 + bl2q−1 + · · · + blnu

q−nu +1

blj = [ b1lj · · · bm

lj ] , j = 1, . . . , nu .

Note that the system inputs in (19) can include both the manip-ulated variables and the reference variables, which are for thepurposes of feedback and feedforward controls, respectively.For the subsequent controller design, they will be separatedexplicitly as


r , THEN

y(k + 1) = Al(q−1)y(k) + Bl(q−1)u(k)

+ Hl(q−1)v(k) + d(k), l = 1, . . . , L (20)

where u(k) denote the manipulated inputs for feedback control,v(k) the reference inputs for feedforward control, and

Hl(q−1) = hl1 + hl2q−1 + · · · + hlnv

q−nv +1 .

In fact, the previous model with feedforward (20) may alsobe transformed into the equivalent state-space form as follows:


r , THENx(k + 1) = Alx(k) + Blu(k) + Hlw(k) + Eld(k)y(k) = Clx(k) + Dld(k), l = 1, . . . , L

(21)

where x(k) ∈ n denotes the system state variables, d(k) ∈ q

the modeling error signals or model–plant discrepancies, and(Al,Bl, Cl ,Dl, El ,Hl) is the associated local linear model. Inparticular, the MISO state-space model in observable compan-ion form becomes

Al =

0 · · · 0 aln

1. . . 0 aln−1

.... . .

......

0 · · · 1 al1

Bl =

bln

bln−1

...

bl1

Hl =

0 0 00 0 0...

......

hl1 · · · hlnv

El =

00...

1

Cl = C = [ 0 · · · 0 1 ] , Dl = D = [0], l = 1, . . . , L

when ny = nu = n and the reference input sequence

w(k) = [v(k)T v(k − 1)T · · · v(k − nv + 1)T ]T .

Note that Cl in this case is constant and taken as C, and Dl iszero.

With the same fuzzy inference principle as for the fuzzyCARMA model (9), the earlier fuzzy state-space model (21)



can be expressed as follows:x(k + 1) = Aµx(k) + Bµu(k) + Hµw(k) + Eµd(k)

y(k) = Cx(k) + Dd(k)(22)

where

Aµ =L∑

l=1

µlAl Bµ =L∑

l=1

µlBl

Eµ =L∑

l=1

µlEl Hµ =L∑

l=1

µlHl

µ := µ(ξ) := (µ1 , . . . , µL ). (23)

B. Assumptions and Lipschitz Conditions

In this paper, we use the nominal T–S fuzzy dynamic modelF (x, u) =

∑Ll=1 µl(ξ) (Alx + Blu) with an additive uncer-

tainty d to approximate the original nonlinear plant G(x, u)as

x(k + 1) = G(x(k), u(k)) = F (x(k), u(k)) + Ed(k) (24)

where d(k) ∈ D is supposed to be bounded.The following assumptions are made here as in [20]–[22].Assumption 1 (Reference): The trajectory reference yr (k)

is known currently and unknown in the future, and it wouldreach an unknown steady-state value asymptotically, i.e.,limk→∞ yr (k) = yr .

Assumption 2 (Disturbance): The disturbance d(k) is al-ways unknown for each time constant k, but it would reachan unknown steady-state value asymptotically, i.e., limk→∞d(k) = d.

Assumption 3 (Detectability): The augmented pair[( C D ) ,

(Al El

O Iq

)], l = 1, . . . , L (25)

is detectable, and the following condition holds:

rank

[I − Al −El

C D

]= n + q. (26)

Before verifying that the T–S fuzzy dynamic model (21) sat-isfies the Lipschitz condition, the following lemma is introducedfirst.

Lemma 2 [35]: If functions fi(x), i = 1, . . . , r, satisfythe Lipschitz condition ‖fi(x1) − fi(x2)‖ ≤ Li‖x1 − x2‖, i =1, . . . , r, then the following hold.

1) Function F(x) =∑r

i=1 fi(x) satisfies the Lipschitz con-dition

‖F(x1) −F(x2)‖ ≤(

r∑i=1

Li

)‖x1 − x2‖. (27)

2) Function F(x) =∏r

i=1 fi(x) for each bounded functionfi(x) satisfies the Lipschitz condition

‖F(x1) −F(x2)‖ ≤(

r∑i=1

Mi−Li

)‖x1 − x2‖ (28)

where Mi− =∏r

j=1,j =i Mj and Mj = sup ‖fj (x)‖.3) Function F(x) = f1 [f2(x)] satisfies the Lipschitz

condition

‖F(x1) −F(x2)‖ ≤ (L1L2)‖x1 − x2‖. (29)

Assumption 4: For the T–S fuzzy system F (x, u) =∑Ll=1 µl(ξ)[Alx + Blu], each fuzzy membership function

Mli (x) is assumed to be bounded and satisfies the Lipschitz

condition.In particular, the bounded triangle and trapezoid membership

functions are easily proved to be Lipschitz. Based on Lemma 2and Assumption 4, we can obtain the following result.

Theorem 1 [35]: For the T–S fuzzy system x(k + 1) =F (x(k), u(k)) with state and input constraints x ∈ X , u ∈ U , ifthe premise variable functions ξi = Hi(x), i = 1, . . . , r, satisfythe Lipschitz condition, there exists a positive finite Lipschitzconstant LF such that

‖F (x1 , u) − F (x2 , u)‖ ≤ LF ‖x1 − x2‖ (30)

for all x1 , x2 ∈ X , and u ∈ U .

C. Augmented Fuzzy Observer

For the fuzzy system with disturbance (21), consider the cor-responding simple augmented fuzzy observer as[

x(k + 1)

d(k + 1)

]=

L∑l=1

µl

[(Al El

O I

) [x(k)

d(k)

]

+

(Bl

O

)u(k) +

(Hl

O

)w(k)

+

(F 1

F 2

)[y(k) − y(k)]

](31)

y(k) = Cx(k) + Dd(k) (32)

where F = [F 1TF 2T ]T is the overall augmented observer

gain. Defining the estimation error x(k) = x(k) − x(k) andd(k) = d(k) − d(k), one can easily obtain the following ob-server error dynamics:[

x(k + 1)

d(k + 1)

]=

L∑l=1

µl

[(Al El

O I

)

+

(F 1

F 2

)( C D )

][x(k)

d(k)

]. (33)

Then, we have the following stability conditions of the fuzzyerror dynamic system (33).

Theorem 2: The error dynamic system (33) is exponentiallystable with a decay rate α ∈ (0, 1] if there exists a positive



definite matrix Z and a matrix G satisfying the following set oflinear matrix inequalities (LMIs):[

−α2Z + Qe ATl ZT + CT GT

ZAl + GC −Z

]< 0, l = 1, . . . , L

(34)

where

Al =(

Al El

0 I

), C = (C D )

and Qe ≥ 0 is a given matrix parameter. Moreover, the overallobserver gain is given by

F =(

F 1

F 2

)= Z−1G.

Proof: It is easy to prove this theorem by resorting to invokingthe Lyapunov stability theorem [40], which is also a special caseof [35, Th. 2] since both the matrix C and the observer gain Fare constant here. Thus, the proof is omitted.

Remark 1: As shown in Theorem 2 and its proof, the sta-ble overall fuzzy observer error system implies that each localobserver error system (Al + FC), l = 1, . . . , L, is also stable.

In fact, we also have the following result for the augmentedfuzzy observer.

Lemma 3: Under Assumption 3, if the number of disturbancestates is equal to the number of outputs, q = p, then the partialobserver gain F 2 in (31) is of full rank.

Proof: The lemma can be proved by the argument of contra-diction similar to that for [35, Lemma 4], and is thus omitted.

D. Fuzzy Predictive Tracking Control

In order to achieve the offset-free output tracking perfor-mance, a constrained target generator has been developed in[20]–[22], where the desired targets of system states and controlinputs in steady state are obtained with respect to the lumpeddisturbance estimates. Here, we extend this idea to the targetgeneration for fuzzy systems.

Meanwhile, from the viewpoint of the SOFC operation, thefuel cell pressure difference between the anode and cathodecompartments should be below 4 kPa under normal operationand 8 kPa under transient conditions [4]. It is desirable tomaintain the ratio of hydrogen to oxygen rH/O around 1.145to keep the fuel cell pressure difference close to zero, i.e.,qF − rH/OqO = u1 − rH/Ou2 = [1 − rH/O]u ≈ 0. This con-trol objective can also be embedded in the corresponding con-strained target generator.

For the overall fuzzy system (22), its steady-state and inputtarget vectors xt and ut can be determined from the solution ofthe following quadratic programming (QP) problem:

minxt ,ut

[(yt − yr )T Qt(yt − yr )] (35)

subject toI − Aµ −Bµ

C O

O [1− rH/O]

[xt

ut

]=

Eµd(k)+ Hµw(k)

yr − Dd(k)

O

(36)

umin ≤ ut ≤ umax (37)

where yr is the desired output trajectory reference, w(k) arethe feedforward sequence of the trajectory reference inputs v,d(k) is the current estimate of the disturbance signal d(k), and(umin , umax) are the input constraints.

In particular, as for the stepwise reference trajectory, the ref-erence inputs v(k) are supposed to remain constant at the currentpoint within a finite prediction horizon.

So, when this target generation problem is feasible, one has

xt = Aµxt + Bµut + Hµwt + Eµd(k). (38)

By subtracting (38) from the first equation of (22), the resultingtransformed system becomes

χ(k + 1) = Aµχ(k) + Bµω(k) + Eµd(k) (39)

where χ := x − xt, ω := u − ut , and d := d − d. Then, onecan use the nominal model of the transformed system for pre-diction as follows:

χ(k + i + 1|k) = Aµχ(k + i|k) + Bµω(k + i|k), i ≥ 0(40)

where χ(k|k) = χ(k). Its infinite horizon predictive control per-formance cost can be defined as

J∞0 (k) =

∞∑i=0

W (χ(k + i|k), ω(k + i|k))

=N −1∑i=0

W (χ(k + i|k), ω(k + i|k))

+∞∑

i=N

W (χ(k + i|k), ω(k + i|k)) (41)

where W (χ(k+ i|k), ω(k+ i|k)) = χ(k+ i|k)T Qχ(k+ i|k)+ω(k + i|k)T Rω(k + i|k).

Define a quadratic function V (χ(k + i|k)) = χ(k + i|k)T

P (k + i|k)χ(k + i|k), and suppose it satisfies

V (χ(k + i + 1|k)) − V (χ(k + i|k))

≤ −W (χ(k + i|k), ω(k + i|k)) ≤ 0. (42)

Summing (42) from i = N to ∞ and requiring χ(∞|k) = 0 orV (χ(∞|k)) = 0, it follows that

J∞N (k) =

∞∑i=N

W (χ(k + i|k), ω(k + i|k))

≤ V (χ(k + N |k))

= χ(k + N |k)T P (k + N |k)χ(k + N |k). (43)



By substituting (43) into (41), one can get

J∞0 (k) ≤

N −1∑i=0

W (χ(k + i|k), ω(k + i|k)) + V (χ(k + N |k))

=: J(χ(k), π(k)). (44)

Here, J(χ(k), π(k)) gives an upper bound for J∞0 (k), so we

can formulate MPC as an equivalent minimization problem onJ(χ, π) with respect to the optimal control sequence

π∗(k) = ω∗(k|k), ω∗(k + 1|k), . . . , ω∗(k + N − 1|k).(45)

With (42) and (43), there always exists a terminal constraint setE such that χ(k + N |k) ∈ E ; thus, one has that the terminal con-trol law ω(k + i|k) = Kχ(k + i|k), i ≥ N , and the associatedterminal cost V (χ(k + N |k)) = χ(k + N |k)T Pχ(k + N |k).For the closed-loop predictive control system, let

Ψ(k) := J(χ(k), π∗(k)) = infπ (k)

J(χ(k), π(k)). (46)

Obviously, Ψ(k) is the optimal performance cost at time k. Also,from the definition of J(χ(k), π(k)), at time instant k + 1, onehas

J(χ(k + 1), π(k + 1)) =N∑

i=1

W (χ(k + i|k), ω(k + i|k))

+ V (χ(k + N + 1|k)) (47)

for the shifted control sequence π(k + 1) = ω∗(k + 1|k), . . . ,ω∗(k + N − 1|k),Kχ(k + N |k).

With the overall predictive fuzzy dynamic model χ(k + i +1|k) = Aµχ(k + i|k) + Bµω(k + i|k) and an open-loop con-trol sequence π(k), we can easily derive the multistep-aheadstate and output prediction

χ(k + N |k) = ANµ χ(k) + AB π(k) (48)

Y (k) = TAχ(k) + TB π(k) (49)

Y (k) =

χ(k|k)

χ(k + 1|k)...

χ(k + N − 1|k)

π(k) =

ω(k|k)

ω(k + 1|k)...

ω(k + N − 1|k)

AB =[AN −1

µ Bµ · · · AµBµ Bµ ]

TA =

In

Aµ

...

AN −1µ

TB =

O · · · O O

Bµ · · · O O

.... . . O O

AN −2µ Bµ · · · Bµ O

.

Then, the MPC performance cost can be reformulated as

J(χ(k), π(k)) = [TAχ(k) + TB π(k)]T Q[TAχ(k) + TB π(k)]

+ π(k)T Rπ(k) + [ANµ χ(k) + AB π(k)]T

× P [ANµ χ(k) + AB π(k)] (50)

where Q = IN ⊗ Q and R = IN ⊗ R. Moreover, with the feed-back control law ω(k + i|k) = Kχ(k + i|k), the closed-loopfuzzy control system within the terminal region becomes

χ(k + i + 1|k) =L∑

l=1

µl(Al + BlK)χ(k + i|k). (51)

Before showing the input-to-state stability of the proposedMPC system, we present two related lemmas.

Lemma 4 (Lipschitz Quadratic Functions): For the quadraticfunction

W (x, u) = xT Qx + uT Ru, Q,R > 0

there exists a finite Lipschitz constant LW > 0 such that

‖W (x1 , u) − W (x2 , u)‖ ≤ LW ‖x1 − x2‖ (52)

for all x1 , x2 ∈ X , and u ∈ U . Similarly, for the quadratic func-tion V (x) = xT Px, P > 0, there exists a finite Lipschitz con-stant LV > 0 such that

‖V (x1) − V (x2)‖ ≤ LV ‖x1 − x2‖ (53)

for all x1 , x2 ∈ X , and u ∈ U .Lemma 5 (Model Prediction Difference Bound): Consider

the predictive fuzzy model χ(k + i + 1|k) := F (k + i|k) =F (χ(k + i|k), ω(k + i|k)), where χ(k|k) = χ(k) such thatTheorem 1 holds. For a sequence of control inputs, the mul-tistep nominal model prediction difference is bounded by

‖χ(k + i|k + 1) − χ(k + i|k)‖ ≤ Li−1F ‖Eµ‖‖d(k)‖ (54)

where LF is the Lipschitz constant of the T–S fuzzy systemand d(k) denotes the disturbance estimation error from the aug-mented observer.

Proof: Due to the Lipschitz condition of the T–S fuzzy model,the result in (54) is readily proved from (39)–(40) and Theorem 2by the induction as in [35].

Now we present the main result in this paper.Theorem 3 (Offset-Free Fuzzy MPC): For system (21) subject

to the input constraints umin u umax , under Assumptions1–3, the closed-loop output feedback fuzzy predictive controlsystem, with objective function J(χ(k), π(k)) in (44), aug-mented fuzzy observer (31)–(32), and target generation pro-cedure (35)–(37) is ISS and achieves the offset-free referencetracking performance if all the following conditions are satis-fied.

1) The LMIs (34) for observer design in Theorem 2 admit apositive definite matrix Z and a matrix G.



2) There exist a positive definite matrix X > 0, a matrix Y ,and γ > 0 satisfying the LMIs

X

AlX + BlY X

X O γQ−1

Y O O γR−1

> 0, l = 1, . . . , L

(55)[u2

j Uj Y

Y T UTj X

]≥ 0, j = 1, . . . ,m

(56)

such that there exists a constraint set

E = z ∈ n |zT Pz ≤ γ (57)

where the tight input constraint uj = min|umax − ut |j ,|umin − ut |j, and Uj is the jth unit row vector with thenonzero element in the jth position Uj = [0 · · · 0 1 0· · · 0].

3) There exist feasible solutions (xt, ut) to the target trackingquadratic programming problem (35) subject to (36)–(37).

4) There exists a feasible solution π to the followingquadratic programming problem at time instant k

minπ

(12πT Θπ + ΞT π

)subject to Aπ B (58)

Θ = 2(T TB QTB + R + AT

B PAB )

Ξ = 2[T TB Q(TAχ(k)) + AT

B Ps(ANµ χ(k))]

A =[

IN ×m

−IN ×m

]B =

[Υ ⊗ [umax − ut ]

−Υ ⊗ [umin − ut ]

]where χ(k) = x(k) − xt and Υ = [1 · · · 1]T .

Meanwhile, the real-time control action is chosen as u(k) =π∗(1 : m) + ut , and the feedback controller and fuzzy observergains are given by

P = γX−1 K = Y X−1 (59)

F =(

F 1

F 2

)= Z−1G. (60)

Proof: As shown in Theorem 2, the sufficient condition 1)can guarantee the observer error system to be asymptoticallystable. Moreover, when both the sufficient conditions 2)–4) aresatisfied for the nominal system (40) with χ(k) = x(k) − xt

and ω(k) = u(k) − ut , by resorting to the Schur complements,

the sufficient conditions (55) can be rewritten as

(AlX + BlY )T X−1(AlX + BlY ) − X−1 + γ−1XT QX

+ γ−1 Y T RY < 0, l = 1, . . . , L. (61)

Multiplying (61) on both sides by γ1/2X−1 with X = γP−1

and Y = KX yields

(Al + BlK)T P (Al + BlK) − P + Q + KT RK < 0.(62)

Similar to the proof of [33, Th. 3.2], for the normalized member-ship function µl(ξ) and the predictive system states χ(k + i|k),the earlier inequalities imply that (63), as shown at the bottom ofthis page, holds where 2XT RY ≤ XT RX + Y T RY ,R > 0,is used. By defining a global quadratic Lyapunov function

V (χ(k + i|k)) = χ(k + i|k)T Pχ(k + i|k) (64)

with the closed-loop fuzzy system (51), (63) implies

V (χ(k + i + 1|k)) − V (χ(k + i|k))

< −W (χ(k + i|k), ω(k + i|k)) ≤ 0. (65)

Summing (65) from i = N to ∞, with χ(∞|k) = 0 orV (χ(∞|k)) = 0, it follows that∞∑

i=N

W [χ(k + i|k), ω(k + i|k)]

< χ(k +N |k)T Pχ(k +N |k)= V (χ(k + N |k))≤ γ (66)

which thus constitutes the constraint set (57) or χ ∈ n |V (χ) ≤ γ.

Now, we check if each element of the control inputs satisfiesthe constraint |uj (k + i|k)| ≤ min|umax |j , |umin |j, i ≥ 0,j = 1, . . . ,m. For any i within the finite horizon N , the in-put constraints are satisfied since Υ ⊗ [umin − ut ] π Υ ⊗[umax − ut ],Υ = [1 · · · 1]T . Otherwise, beyond the finitehorizon i ≥ N , χ(k + i|k) belongs to the constraint set E =z ∈ n |zT X−1z ≤ 1, X > 0, which is guaranteed by the in-equality (57). And in this case, there exists a feedback controllaw ω(k + i|k) = Kχ(k + i|k) = Y X−1χ(k + i|k) such that

maxi≥N

|ωj (k + i|k)|2

= maxi≥N

|(Y X−1χ(k + i|k))j |2

≤ maxz∈E

|(Y X−1/2X−1/2z)j |2

≤ ‖(Y X−1/2)j‖22‖(X−1/2z)j‖2

2 ≤ Uj Y X−1 Y T UTj

≤ u2j , j = 1, . . . ,m (67)

where Uj = [0 · · · 0 1 0 · · · 0] and uj = min|umax −ut |j , |umin − ut |j. Therefore, the input constraints are satis-fied if the LMIs (56) are feasible.

[L∑

l=1

µl(Al + BlK)χ(k + i|k)

]T

P

[L∑

j=1

µj (Aj + BjK)χ(k + i|k)

]− χ(k + i|k)T Pχ(k + i|k)

< −χ(k + i|k)T Qχ(k + i|k) − [Kχ(k + i|k)]T R [Kχ(k + i|k)] < 0 (63)



As mentioned before, it can be found that the difference ofthe optimal cost Ψ(k) also satisfies

∆Ψ(k) = J(χ(k + 1), π∗(k + 1)) − J(χ(k), π∗(k))

≤ V (χ(k + N + 1|k + 1)) − V (χ(k + N |k + 1))

+ V (χ(k + N |k + 1)) − V (χ(k + N |k))

+ W (χ(k + N |k + 1), ω(k + N |k + 1))

− W (χ(k|k), ω∗(k|k))

+N −1∑i=1

[W (χ(k + i|k + 1), ω(k + i|k + 1))

− W (χ(k + i|k), ω∗(k + i|k))].

Taking (54) of Lemma 5 into account, and with Lemma 4, onecan conclude that

‖V (χ(k + N |k + 1)) − V (χ(k + N |k))‖

≤ LV LN −1F ‖Eµ‖‖d(k)‖

where d(k) = d(k) − d(k). From the definition of π(k + 1), itfollows that ω(k + i|k + 1) = ω∗(k + i|k), i = 1, . . . , N − 1;consequently, we have

‖W (χ(k + i|k + 1), ω(k + i|k + 1))

− W (χ(k + i|k), ω∗(k + i|k))‖≤ LW ‖χ(k + i|k + 1) − χ(k + i|k)‖≤ LW Li−1

F ‖χ(k + 1|k + 1) − χ(k + 1|k)‖

≤ LW Li−1F ‖Eµ‖‖d(k)‖.

Then, one can get

∆Ψ(k) ≤ [V (χ(k + N + 1|k + 1)) − V (χ(k + N |k + 1))

+ W (χ(k + N |k + 1), ω(k + N |k + 1))]

+

(LV LN −1

F +N −1∑i=1

LW Li−1F

)‖Eµ‖‖d(k)‖

− W (χ(k|k), ω∗(k|k)). (68)

Since the stability constraint (65) also holds for i = N , the termsin the square brackets of (68) are thus less than zero. Then, onecan conclude

∆Ψ(k) ≤(LV LN −1

F + LWLN −1

F − 1LF − 1

)‖Eµ‖‖d(k)‖

− [χ(k|k)T Qχ(k|k) + ω∗(k|k)T Rω∗(k|k)]

≤ LJ ‖d(k)‖ − a‖χ(k)‖2

where d(k) is the disturbance estimation error

LJ = [LV LN −1F + LW (LN −1

F − 1)/(LF − 1)]‖Eµ‖

and a = λmin(Q). And obviously

Ψ(k) ≥ λmin(Q)‖χ(k)‖2 .

Thus, the optimal cost Ψ(k) serves as an ISS Lyapunov functionfor the closed-loop system by invoking Lemma 1. Hence, theresulting fuzzy MPC control system is ISS.

Then we turn to the proof of offset-free output tracking prop-erty. Suppose that x(∞), and d(∞) denote as the steady-stateestimates of the plant states and lumped disturbances, u(∞)is the steady-state control input by the regulator, and y(∞) isthe measured steady-state process output. Hence, from the aug-mented observer (31)–(32), we have

x(∞) = Aµx(∞) + Bµu(∞) + Eµd(∞)

+ F 1 [Cx(∞) + Dd(∞) − y(∞)] (69)

d(∞) = d(∞) + F 2 [Cx(∞) + Dd(∞) − y(∞)]. (70)

Since F 2 is shown to be of full rank from Lemma 3, (69) implies

y(∞) = Cx(∞) + Dd(∞) (71)

and accordingly

x(∞) = Aµx(∞) + Bµu(∞) + Eµd(∞). (72)

If the target generation procedure in (35) and (37) has a feasiblesolution at steady state, the following equality holds:

xt = Aµxt + Bµut + Eµd(∞). (73)

Subtracting (73) from (72) yields

x(∞) − xt = Aµ [x(∞) − xt ] + Bµ [u(∞) − ut ]. (74)

From the property of stabilizing MPC, the feasibility of theregulator conditions implies that there is a feedback control law

u(∞) − ut = K[x(∞) − xt ]

and substituting this feedback law into (74) results in

L∑l=1

µl(Al + BlK − I)[x(∞) − xt ] = 0. (75)

As shown in the previous controller design, the closed-loopsystem is guaranteed to be stable, and thus, the eigenvalues ofeach local system Al + BlK stand within the unit circle. Hence,the only solution of (75) is

x(∞) − xt = 0.

Also, with the feasible target generation procedure at steadystate, we have

yr = Cxt + Dd(∞). (76)

So, the steady-state output tracking error between (71) and (76)becomes

e = y(∞) − yr = C[x(∞) − xt ] = 0. (77)

Therefore, the offset-free property of the closed-loop system isguaranteed, and thus, the theorem is proved.

Remark 2: In Theorem 3, the tight input constraints areconsidered in the computation of conservative terminal costsince the state/disturbance estimation errors x(k)/d(k) wouldaffect the scope of the associated terminal constraint set. So,



Fig. 7. Excitation input signals to SOFC system.

the smaller feasible terminal region from the tight input con-straints u u is employed to prevent the transient predictivestates out of the hard limit umin , umax . Also, due to the exis-tence of estimation errors x(k)/d(k), the transient performanceof model-based control system is only suboptimal.

Remark 3: Note that the comprehensive explicit reachabilityanalysis is a challenging issue in general for a T–S fuzzy sys-tem due to its nonlinearity nature. It is well known that localreachability and observability do not always imply the globalones of the overall fuzzy system. In this paper, if the sufficientconditions of Theorems 2 and 3 are satisfied, the reachability ofthe system is implicitly ensured.

V. SIMULATION RESULTS

A. Fuzzy Model Identification of SOFC System

To identify the fuzzy model of the SOFC/capacitor sys-tem, some amplitude-modulated pseudorandom binary signals(APRBSs) are employed as excitation signals for the manipu-lated variables, such as fuel input flow qF and oxygen inputflow qO , which are shown in Fig. 7. Here, both qF and qO aresupposed to be of the uniform distribution between limits of 0.5and 0.8. Meanwhile, according to the stationary characteristicsof SOFC, another input variable ID may be chosen as

ID =qF Ui

f

2Kr

where Uif is subject to the uniform distribution between

the lower and upper limits of fuel utilization, 0.7 and 0.9,respectively.

Here, the constraints on qF and qO are used to reflect possiblephysical restriction in practice. Since the fuel cell stack capac-ity is limited, it is impossible to inject an arbitrary amount offuel and oxygen into the anode and cathode with limited spaces.When the fuel cell is shut down, the fuel/oxygen flow rate issupposed to be zero in theory. When the excitation signal is im-posed on SOFC for identification purposes, it is still desirableto keep the pressure difference between the anode and cathode

Fig. 8. Output response signals of SOFC system.

Fig. 9. Monitored variables of SOFC system.

under the safe range; otherwise, it would destroy the cell itself.The detailed safety region is normally provided by the manu-facturer. The maximal value of the pressure difference in thisstudy is assumed to be 8 kPa, as given in [4] and [8].

With the earlier excitation signals for inputs, the correspond-ing output response signals for stack voltage VS , stack currentIS , and power output PO are presented in Fig. 8, while theresponses of the monitored variables, ∆p, Uf , and rH/O areshown in Fig. 9. Obviously, the transient pressure difference∆p is always less than 8 kPa, so SOFC still operates within thesafety region during the identification process.

Choosing the membership functions for the premise variableID as

“Low” : M 1 =

1, if ID ≤ 200

1 − ID − 200250 − 200

, if 200 < ID < 250

0, if ID ≥ 250.



Fig. 10. Membership function of current load ID in fuzzy SOFC model.

“Middle” : M 2 =

1 − M 1 , if ID ≤ 250

1 − M 3 , if ID ≥ 300.

“High” : M 3 =

0 if ID ≤ 250

ID − 250300 − 250

, if 250 < ID < 300

1, if ID ≥ 300.

which are shown in Fig. 10, and using the method described inSection IV-A, it is easy to identify the T–S fuzzy dynamic model(78) of the SOFC/capacitor power system as follows.

Rule 1: IF current load ID is Low, THEN

VS (k + 1) = 2.1219VS (k) − 1.3934VS (k − 1)

+ 0.27077VS (k − 2) + 0.34582qF (k)

+ 0.14722qF (k − 1) + 0.33459qF (k − 2)

+ 2.4031qO(k) − 1.051qO(k − 1)

− 1.6142qO(k − 2) − 0.069504ID (k)

+ 0.1123ID (k − 1) − 0.043102ID (k − 2).

Rule 2: IF current load ID is Middle, THEN

VS (k + 1) = 2.0877VS (k) − 1.3433VS (k − 1)

+ 0.25436VS (k − 2) + 0.032884qF (k)

+ 0.48617qF (k − 1) + 0.0030969qF (k − 2)

+ 2.6161qO(k) − 0.92155qO(k − 1)

− 1.6953qO(k − 2) − 0.069017ID (k)

+ 0.10947ID (k − 1) − 0.040451ID (k − 2).

Rule 3: IF current load ID is High, THEN

VS (k + 1) = 2.1041VS (k) − 1.3698VS (k − 1)

+ 0.26532VS (k − 2) + 0.22085qF (k)

+ 0.3106qF (k − 1) + 0.16448qF (k − 2)

+ 2.6499qO(k) − 0.95128qO(k − 1)

− 1.6032qO(k − 2) − 0.070544ID (k)

+ 0.11077ID (k − 1) − 0.041867ID (k − 2).

(78)

Fig. 11. Dynamic fuzzy modeling performance of SOFC system (solid: orig-inal SOFC voltage; dashed: fuzzy model prediction).

Fig. 12. Validation of fuzzy model for SOFC system under stepwise inputchanges (solid: original SOFC voltage; dashed: fuzzy model prediction).

Then the VAF index in (18) is employed to evaluate the fuzzymodeling performance. Here, with the series–parallel identifi-cation structure [42] between fuzzy model and plant, the corre-sponding dynamic fuzzy modeling results are shown in Fig. 11with the associated VAF index value as 99%. Moreover, step-wise input changes are imposed on both the SOFC plant and itsfuzzy model for further validation, with results shown in Fig. 12.It can be observed that the approximation performance of theidentified fuzzy SOFC model (78) is excellent.

B. Fuzzy Predictive Control of SOFC System

As mentioned in Section II, the external load will directlyaffect the operation of the SOFC power plant. In fact, to bean independent power source candidate, the output voltage ofthe SOFC/capacitor system is expected to be at a desired con-stant value for stable operation of external electrical equipment.



Fig. 13. Diagram of fuzzy MPC/SOFC/capacitor system structure.

Therefore, it is desirable to integrate a controller into the stand-alone SOFC plant to achieve the aforementioned objective.

To develop flexible advanced control strategies, the specificclosed-loop control structure diagram, shown in Fig. 13, hasbeen proposed in this paper. It should be noted in Fig. 13 thatthree system inputs are classified as follows according to theanalysis of SOFC/capacitor system:

1) manipulated inputs: fuel flow qF , oxygen flow qO ;2) feedforward input: load current demand ID .The proposed fuzzy MPC strategy is based on the fuzzy

dynamic model (78), and the detailed predictive controllerdesign method has been presented in Theorem 3. The cor-responding controller parameters are chosen in this case asN = 10, Qt = 0.1I1 , Q = 0.1I2 , and R = 15I2 . Due to thelimited reaction capacity and space in the fuel cell stack, thefuel and oxygen flow rates would be under some practical lim-itations, and it is assumed here that the amplitude constraintson manipulated inputs qF /qO are uj

max = 1.2, ujmin = 0, with

tight constraints uj = 0.5, j = 1, 2, for controller design. Then,by solving the LMI conditions (55)–(56) in Theorem 3, we canobtain the corresponding characteristic matrices P and K toconstruct the terminal cost for MPC

P =

8.0441 7.9584 7.87867.9584 8.9429 9.49897.8786 9.4989 11.0709

K =

[−0.2956 −0.3146 −0.3287−0.0742 −0.1887 −0.2877

].

When α = 0.95 and Qe = 20I3 , the augmented observer gaincan be obtained from Theorem 2 as

F = [−0.2635 1.3689 −2.5093 −0.4062 ]T .

To emulate the practical needs of dc electric users, the powerload is supposed under stepwise changes while the voltage refer-ence is assumed to be 245 V. Accordingly, the current referencefor the SOFC/capacitor power system will also change step-wisely, as shown in the dashed lines of Figs. 14 and 15.

By applying the proposed fuzzy predictive controller designin Theorem 3, the satisfactory control performances are achievedfor the closed-loop SOFC system with the suboptimal transientreference tracking ensured, as shown in Fig. 14. It can be ob-served from the third subfigure of Fig. 14 that the power outputof the integrated SOFC/ccapacitor system (solid line) can followthe power load change more quickly than stack power output(dotted-dash line) due to the existence of an additional capacitor,and the latter is with larger transient overshoots and significantoscillations. On the other hand, with the proposed fuel/oxygen

Fig. 14. Output responses in offset-free fuzzy predictive control systemof SOFC (solid: system output; dotted-dash: stack power output; dashed:reference).

Fig. 15. Manipulated input changes in offset-free fuzzy predictive controlsystem of SOFC.

flow controller, the closed-loop power response to load currentchanges is also much quicker than the open-loop case. In fact, thesettling time is about 10 and 100 s for the closed-loop and theopen-loop systems, respectively. Meanwhile, the closed-loopcontrol input responses are given in Fig. 15, and the monitoredvariables changes are kept within the safety range, as depictedin Fig. 16.

As for the tunning of controller parameters, larger Q/R ratiousually improves the tracking performance, but it sometimesleads to oscillatory control actions and wastes more reactantsupply energy. And with larger prediction horizon length N ,better control performance may be achieved, but at greater op-timization cost.

As shown in Table II (given at the top of the next page), theintegral rms performance index does not change much, whichindicates that the proposed control system is not very sensitive



TABLE IIEVALUATION OF CONTROLLER PARAMETER CHANGES

Fig. 16. Monitoring performances in offset-free fuzzy predictive control sys-tem of SOFC.

TABLE IIICOMPARISON OF OF CONTROL METHODS

to controller parameters. In fact, even for the most conservativecase, the stack power load following performance in Fig. 14 isstill quite satisfactory.

Regarding the comparison in Table III, the fuel/hydrogen flowrate qH and oxygen flow qO in [4] are fixed to be rH/O = 1.145,so their control system has only one active manipulated vari-able and slow load following response. While in this paper andin [8], there are two independent control inputs qF and qO indata-driven control system, in general, more control freedomoffers more opportunities for performance improvement. Forthe predictive controller in [8], evident steady-state load follow-ing errors was observed in [8, Figs. 9 and 11] when the powerdemand takes stepwise changes. Since, in this paper, we havetaken into account the offset-free control performance in con-troller design and synthesis, so our proposed predictive controlsystem can maintain the rapid load following ability with zerotracking error.

VI. CONCLUSION

To improve the load tracking performance of a stand-aloneSOFC stack, a constrained offset-free fuzzy predictive controlapproach has been developed for the integrated SOFC/capacitorpower system based on its fuzzy model from data-driven iden-

tification. Both the rapid power load following and safe SOFCoperation requirements are taken into account in the proposedpredictive tracking control system design. The simulation re-sults have validated the satisfactory load tracking performanceof the proposed SOFC control strategy even in the presenceof input constraints. One future research topic is to develop arobust stable fuzzy model predictive control system for morecomplex SOFC power systems under grid connection, such asthe control of plant-wide systems including SOFC stack withtemperature dynamics, buck–boost or boost dc/dc converters,and forced-commutated dc-ac voltage source inverter. Anotherpossible research is to enhance the fuzzy modeling ability, e.g.,adaptive model identification for time-varying dynamics. Mean-while, it is desirable to study adaptive predictive control whenthe plants are slowly time-varying.

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Tiejun Zhang (S’04–M’08) was born in Wuxi,China, in 1979. He received the B.Eng. and M.Sc.degrees in thermal power engineering from theSoutheast University, Nanjing, China, in 2001 and2004, respectively, and the Ph.D. degree in systemsand control from the City University of Hong Kong,Kowloon, Hong Kong, in 2008.

He is currently a Postdoctoral Research Associateat Rensselaer Polytechnic Institute, Troy, NY. Hiscurrent research interests include output tracking andregulation, fuzzy systems and control, particularly

dynamic modeling and predictive control of clean energy and power systems.Dr. Zhang was the recipient of the Outstanding M.Sc. Dissertation Award of

Jiangsu Province of China.

Gang Feng (S’90–M’92–SM’95–F’09) received theB.Eng. and M.Eng. degrees in automatic control (ofelectrical engineering) from Nanjing Aeronautical In-stitute, Nanjing, China, in 1982 and 1984, respec-tively, and the Ph.D. degree in electrical engineeringfrom the University of Melbourne, Melbourne, Vic.,Australia, in 1992.

Since 2000, he has been with the the City Univer-sity of Hong Kong, Kowloon, Hong Kong, where heis currently a Professor. From 1992 to 1999, he was aLecturer/Senior Lecturer in the School of Electrical

Engineering, University of New South Wales, Kensington, N.S.W., Australia.His current research interests include piecewise-linear systems, robot networks,and intelligent systems and control.

Prof. Feng is an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC

CONTROL and the IEEE TRANSACTIONS ON FUZZY SYSTEMS, and was an Asso-ciate Editor of the IEEE TRANSACTIONS ON SYSTEMS, MAN AND CYBERNETICS,PART C and the Journal of Control Theory and Applications, and on the Confer-ence Editorial Board of the IEEE Control System Society. He was the recipientof the Alexander von Humboldt Fellowship Award in 1997–1998, and the 2005IEEE TRANSACTIONS ON FUZZY SYSTEMS Outstanding Paper Award in 2007.



Diagnosability of Fuzzy Discrete-Event Systems:A Fuzzy Approach

Fuchun Liu and Daowen Qiu

Abstract—In order to more effectively cope with the real-worldproblems of vagueness, fuzzy discrete-event systems (FDESs) wereproposed by Lin and Ying recently. Then we and Cao and Ying in-vestigated the supervisory control of FDESs independently. In thispaper, we are concerned with another important issue of FDESs,the failure diagnosis. More specifically: 1) we propose a “fuzzydiagnosability” approach by introducing a fuzzy diagnosabilityfunction to characterize the diagnosability degree, which takesvalues in the interval [0, 1] rather than 0, 1; 2) based on theobservability of events, we formalize the construction of the diag-nosers that are used to perform fuzzy diagnosis; 3) a number ofbasic properties of the diagnosers are investigated. In particular, wepresent a necessary and sufficient condition for failure diagnosis ofFDESs. Our results generalize the important consequences of thediagnosability for crisp discrete-event systems (DESs) introducedby Sampath et al. The newly proposed approach allows us to dealwith the problem of diagnosability for both crisp DESs and FDESs;4) in addition, a method for checking the fuzzy diagnosability forFDESs is proposed. Also, some examples are provided to illustratethe application of the diagnosability of FDESs.

Index Terms—Discrete-event systems (DESs), failure detection,fuzzy diagnosability, fuzzy discrete-event systems (FDESs), fuzzyfinite automata.

I. INTRODUCTION

D ISCRETE-EVENT systems (DESs) are dynamical sys-tems whose state space is discrete and states can only

change when certain events occur, which have been successfullyapplied to provide a formal treatment of many technological andengineering systems [4]. In order to guarantee performance toa reliable system, the control engineers should design a systemthat runs safely within its normal boundaries. Therefore, failurediagnosis in DESs, which is to detect and isolate the unobserv-able fault events occurring in a system within a finite delay, isof practical and theoretical importance, and has received con-siderable attention in recent years [5], [10], [15], [17]–[19],[22]–[28].

In most engineering applications, the states and events ofDESs are crisp. However, this is not the case in many other

Manuscript received March 29, 2008; revised October 6, 2008; acceptedJanuary 12, 2009. First published January 27, 2009; current version publishedApril 1, 2009. This work was supported in part by the National Natural ScienceFoundation under Grant 60573006 and Grant 60873055, in part by the ResearchFoundation for the Doctoral Program of Higher School of Ministry of Educationunder Grant 20050558015, and in part by the New Century Excellent Talents(NCET) of China.

F. C. Liu is with the Faculty of Computer, Guangdong University of Tech-nology, Guangzhou 510006, China, and also with the Department of Com-puter Science, Zhongshan University, Guangzhou 510275, China (e-mail:[email protected]).

D. W. Qiu is with the Department of Computer Science, Zhongshan Univer-sity, Guangzhou 510275, China (e-mail: [email protected]).


applications such as biomedicine, in which vagueness, impre-ciseness, and subjectivity are typical features. For example, it isvague when a man’s condition of the body is said to be “good.”Moreover, it is imprecise to say at what point exactly a manhas changed from “good” state to “poor” state. In order to moreeffectively cope with those problems, Lin and Ying [11], [12]recently initiated significantly the study of fuzzy discrete-eventsystems (FDESs) by combining fuzzy set theory [31] with crispDESs [4]. Notably, FDESs have been applied to biomedicalcontrol for HIV/AIDS treatment [13], [29], [30] and decisionmaking [14]. More recently, R. Huq et al. have proposed an in-telligent sensory information processing technique using FDESsfor robotic control [8], [9].

Just as Lin and Ying [12] pointed out, a comprehensive the-ory of FDESs still needs to be set up, including many importantconcepts, methods, and theorems, such as controllability, ob-servability, and optimal control. These issues have been partiallyinvestigated. The supervisory control of FDESs was proposedby Qiu [20], the authors [16], [21], and Cao and Ying [1]–[3],respectively. It is worth noting that there are great differencesbetween the above two frameworks. In the works of [1]–[3], theobservability of events is crisp, and the controllable and uncon-trollable event sets are also crisp, although the controllability ofevents is not completely crisp. However, in [16] and [21], wehave established the centralized and decentralized supervisorycontrol of FDESs under partial observations, in which both theobservability and controllability of events are fuzzy instead. Asa continuation of our works, this paper is to investigate anotherimportant issue of FDESs, the failure diagnosis.

In this paper, we propose a “fuzzy diagnosability” approachfor failure diagnosis of FDESs. Although we have respectivelydealt with the uncertain problems in decentralized diagnosis [15]and safe diagnosis of stochastic DESs [17] by means of theframework in [27], and Kilic has also considered the uncer-tain problems in [10], fuzzy diagnosability has not receivedenough attention since most of the research of diagnosabilityin the literature has thoroughly focused on the centralized di-agnosis [24], [25], decentralized diagnosis [5], [22], distributeddiagnosis [26], and its applications in telecommunication net-works [18], [23] and digital circuits [28], and so on.

The framework in this paper is different from those in theliterature. As a continuation of our works [16], [21], in this paperwe still consider the same model as that in [20], while the FDESin [10] is modeled as a fuzzy expert system that contains fuzzyIF–THEN rules. Furthermore, it is well known that the diagnoserapproach introduced by Sampath et al. [24] has been extensivelyadopted in the literature for its efficiency and adaptability [23].The fuzzy diagnosability proposed in this paper is exactly based

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LIU AND QIU: DIAGNOSABILITY OF FUZZY DISCRETE-EVENT SYSTEMS: A FUZZY APPROACH 373

on the diagnoser of FDESs, which also differs from that in [10].This paper is also different from [15], [17], and [27], since [15],[17], and [27] deal with the diagnosability of stochastic DESs,which are modeled by stochastic automata with a probabilisticstructure appending to each transition of states, while this paperis to formalize the diagnosis of FDESs modeled by max–minsystems with fuzzy states and fuzzy events [20], [21].

The present paper attempts to address the diagnosability ofFDESs. We formalize a notion of the fuzzy diagnosability func-tion to characterize the diagnosability degree, which takes valuesin the interval [0, 1] rather than 0, 1, and a “fuzzy diagnos-ability” approach is proposed. The fuzzy diagnosability functionindicates that the system is diagnosable with a certain degree.In particular, if the fuzzy diagnosability function equals one,then the fuzzy system is completely diagnosable; if the fuzzydiagnosability function equals zero, then the fuzzy system iscompletely nondiagnosable. Based on the observability degreeof events, a method for constructing the diagnosers used to per-form fuzzy diagnosis is developed. After investigating a numberof basic properties of the diagnosers, we present a necessary andsufficient condition for diagnosability of FDESs, which gener-alizes the important consequences of failure diagnosis for crispDESs introduced by Sampath et al. [24]. The newly proposeddiagnosability approach allows us to deal with the problem offailure diagnosis for both crisp DESs and FDESs. Furthermore,a computing method to check the fuzzy diagnosability of FDESsis given. Also, some examples are provided to illustrate the ap-plication of the diagnosability of FDESs.

The rest of the paper is organized as follows. Section II recallssome preliminaries concerning diagnosability for crisp DESsand the concept of fuzzy finite automata. In Section III, we for-malize the notion of fuzzy diagnosability for FDESs. Then, inSection IV, a method for constructing the diagnosers used to per-form fuzzy diagnosis is given. After that, some main propertiesof the diagnosers are investigated in Section V. In particular, wepresent a necessary and sufficient condition for diagnosabilityof FDESs. Moreover, a computing process to check the diagnos-ability condition is further provided. In Section VI, a number ofexamples are provided to illustrate the potential applications todiagnosability for FDESs. To conclude, in Section VII, we sum-marize the main results of the paper and address some relatedissues.

II. PRELIMINARIES

In this section, we briefly recall some preliminaries concern-ing diagnosability for crisp DESs proposed in [24] and the no-tions of fuzzy finite automata [20], [21].

A. Diagnosability for Crisp DESs

A crisp DES is usually described by a deterministic automatonG = (Q,E, δ, q0), where Q is the finite set of states with theinitial state q0 , E is the finite set of events, δ : Q × E → Q isthe transition function. Each sequence over E is called a string.E∗ denotes the set of all finite strings over E. For s ∈ E∗, |s| isthe length of s. Especially, s is an empty string in case |s| = 0,which is denoted by ε. The behavior of G is described by the

language generated by G, denoted as L(G) or L. L/s standsfor the set of all possible continuations of string s in L, i.e.,L/s = t ∈ E∗ : st ∈ L.

The event set E is partitioned into two disjoint subsets E =Eo ∪ Euo , where Eo is the set of observable events and Euo isthe set of unobservable events. When a string of events occursin a system, the sequence of observable events is filtered by theprojection P : E∗ → E∗

o , which is defined as: P (ε) = ε, andP (sσ) = P (s)P (σ) for σ ∈ E, s ∈ E∗, where

P (σ) =

σ, if σ ∈ Eo,

ε, if σ ∈ Euo .(1)

The inverse projection of string t ∈ E∗o is given by

P−1(t) = s ∈ L : P (s) = t . (2)

Let Ef denote the set of failure events that are to be di-agnosed. Usually, Ef is partitioned into a set of failure typesf1 , f2 , . . . , fm , i.e.,

Ef = Ef1 ∪ Ef2 ∪ · · · ∪ Efm. (3)

Denote Ψ(Ef ) as the set of all traces that end with a failurebelonging to Ef , i.e.,

Ψ(Ef ) = s ∈ L : sl ∈ Ef (4)

where sl denotes the last event of s.Definition 1 [24]: Let L be a language generated by G =

(Q,E, δ, q0) and let P : E∗ → E∗o be a projection. L is said to

be f -diagnosable if the following holds:

(∃n0 ∈ N)(∀s ∈ Ψ(Ef ))(∀t ∈ L/s)(|t| ≥ n0 ⇒ D(st) = 1)(5)

where the diagnosability function D : E∗ → 0, 1 is definedas follows:

D(st) =

1, if ω ∈ P−1(P (st)) ⇒ Ef ∈ ω

0, otherwise(6)

in which Ef ∈ ω represents that string ω contains a failure eventfrom Ef .

Intuitively, L being f -diagnosable means that, for any traces that ends in a failure event and for its any sufficiently longcontinuation t, every trace with the same observable record asst should contain a failure event too (i.e., D(st) = 1).

B. Fuzzy Discrete-Event Systems

If the crisp state set Q = q0 , q1 , . . . , qn−1, then in the set-ting of FDESs, each fuzzy state q is a vector over the crisp stateset, denoted as [a0 a1 · · · an−1 ], where ai ∈ [0, 1] representsthe possibility of the current state being qi . Similarly, a fuzzyevent σ is a matrix [aij ]n×n , in which aij ∈ [0, 1] indicates thepossibility of the system transforming from state qi to state qj

when σ occurs.Definition 2 [20]: A fuzzy finite automaton is a max–min

system

G = (Q, E, δ, q0) (7)

where Q is a set of fuzzy states with the initial fuzzy stateq0 ; E is the set of fuzzy events; the state transition relation



δ : Q × E → Q is defined as δ(q, σ) = q σ. Note that ismax–min operation: for matrix A = [aij ]n×m and matrix B =[bij ]m×k , define A B = [cij ]n×k , where

cij =m

maxl=1

minail , blj. (8)

Remark 1: As a continuation of our works [16], [20], [21],we still would like to consider max–min operation instead ofmax-product operation [12] in fuzzy finite automata. Moreover,δ can be regarded as a partial function in practice.

Example 1: Let G = (Q, E, δ, q0) be an FDES. If fuzzy stateq = [0.7, 0.3] ∈ Q and fuzzy event

σ =[

0.4 0.50.8 0.1

]∈ E

occurs in q, then

δ(q, σ) = q σ = [0.7, 0.3] [

0.4 0.50.8 0.1

]= [0.4, 0.5].

III. DEFINITION OF DIAGNOSABILITY FOR FDESS

In this section, we formalize the notion of diagnosability forFDESs, in which a fuzzy diagnosability function is introducedto characterize the degree of diagnosability.

Let G = (Q, E, δ, q0) be an FDES. In the framework of ourworks [16], [20], [21], each fuzzy event simultaneously belongsto the observable event set, the unobservable event set, and thefailure event set; only the degrees of belonging to those sets maybe different. So we introduce three fuzzy subsets: the unobserv-able event fuzzy subset Σuo : E → [0, 1], the observable eventfuzzy subset Σo : E → [0, 1], and the failure event fuzzy subsetΣf : E → [0, 1]. Intuitively, Σf (σ) describes the possibility offailure occurring on σ ∈ E, and Σo(σ) and Σuo(σ) show theobservability degree of σ, which satisfies

Σuo(σ) + Σo(σ) = 1. (9)

Furthermore, we define

Σo(σ1 σ2 · · · σm ) = minΣo(σi) : i = 1, 2, . . . ,m (10)

Σf (σ1 σ2 · · · σm ) = maxΣf (σi) : i = 1, 2, . . . ,m (11)

where each σi ∈ E.In order to avoid the case that the event set of the diagnoser

constructed later is null, we introduce the maximal observableevent set Emax−o , which is composed of the events with thelargest observability degree, i.e.

Emax−o = σ ∈ E : (∀σ′ ∈ E)Σo(σ) ≥ Σo(σ′). (12)

The language generated by G, denoted as LG

or L, is definedas follows:

L = s ∈ E∗ : (∃q ∈ Q)δ(q0 , s) = q. (13)

The postlanguage of L after s is the set of continuations of s inG, i.e.,

L/s = t ∈ E∗ : (∃q ∈ Q)δ(q0 , st) = q. (14)

Let L(q) be the set of all traces that originate from state q.Denote

L1(q, σ) = (Emax−o ∪ a ∈ E : Σo(a) > Σo(σ)) ∩ L(q)

(15)

L2(q, σ) = ua ∈ L(q) : a ∈ L1(q, σ) ∧ Σo(σ) ≥ Mo(u)(16)

where Mo(u) = maxΣo(σ) : σ ∈ u; and

L(q, σ) = L1(q, σ) ∪ L2(q, σ).

Intuitively, L1(q, σ) collects all of the single fuzzy events inL(q) whose observability degree is the largest or larger thanΣo(σ); and L2(q, σ) consists of the strings whose observabilityof each event is not larger than Σo(σ) except for the last event.

Definition 3 [21]: For σ ∈ E, the σ-projectionPσ

: E∗ → E∗

is defined as: Pσ(ε) = ε, and P

σ(sa) = P

σ(s)P

σ(a) for a ∈ E

and s ∈ E∗, where

Pσ(a) =

a, if a ∈ Emax−o or Σo(a) > Σo(σ)ε, otherwise.

(17)

The inverse projection operator is given by

P−1σ

(t) = s ∈ E∗ : (∃q ∈ Q)δ(q0 , s) = q ∧ Pσ(s) = t.

(18)Remark 2: σ-projection is due to erasing the fuzzy events of

strings whose observability is not larger than Σo(σ). Especially,if each event is either completely observable or completely un-observable, then all of the σ-projections are just the projectionP : E∗ → E∗

o in crisp DESs, which simply erases the unobserv-able events.

For the sake of making a correct decision of detection, itwould be instructive to specify an upper bound Σf (σ) for eachσ ∈ E, if the possibility of the failure occurring on a string sexceeds the upper bound (i.e., Σf (s) ≥ Σf (σ)), then we thinkthat s is a failure string since the system is in a failed state. In thefollowing, we formalize an approach of “fuzzy diagnosability”to detect the occurrence of those failure strings that exceed thespecified upper bound.

Denote Efail = σ ∈ E : Σf (σ) > 0. For σ ∈ Efail , letΨ

σ(Σf ) be the set of all traces that end with an event whose

possibility of failure occurring is not less than Σf (σ), i.e.

Ψσ(Σf ) = s ∈ E∗ : (∃q ∈ Q)δ(q0 , s)

= q ∧ Σf (sl) ≥ Σf (σ). (19)

Definition 4: For σ ∈ Efail , the fuzzy diagnosability functionis defined as a partial function

FDσ

: E∗ → [0, 1] (20)

where for s ∈ Ψσ(Σf ) and t ∈ L/s,

FDσ(st) =

minΣf (σ), Σf (ω) : ω ∈ P−1σ

(Pσ(st))

Σf (σ). (21)



Remark 3: We compare the fuzzy diagnosability function forFDESs with the diagnosability function for crisp DESs [24] asfollows. If the observable event set and the failure event setare crisp instead of fuzzy, i.e., Σo(a), Σf (a) ∈ 0, 1 for eacha ∈ E, then for σ ∈ Efail (i.e., Σf (σ) = 1), by Remark 2, wehave

FDσ(st) =


(Pσ(st))

Σf (σ)

= min1, Σf (ω) : ω ∈ P−1(P (st))

=

1, if ω ∈ P−1(P (st)) ⇒ Σf (ω) = 10, otherwise.

Thus, in this case, the fuzzy diagnosability function FDσ

:E∗ → [0, 1] degenerates into the diagnosability function D :E∗ → 0, 1 for crisp DESs described in Definition 1.

Definition 5: Let L be a language generated by FDES G.1) G is said to be completely f -diagnosable with respect to σ

(or the diagnosability degree of G equals one with respectto σ), if

(∃n0 ∈ N)(∀s ∈ Ψσ(Σf ))(∀t ∈ L/s)

(|t| ≥ n0 ⇒ FDσ(st) = 1).

(22)

2) G is said to be f -diagnosable with degree λ with respect toσ (or the diagnosability degree of G equals λ with respectto σ) where 0 ≤ λ < 1, if for any n0 ∈ N, we have

minFDσ(st) : s ∈ Ψ

σ(Σf ), t ∈ L/s, |t| ≥ n0 = λ.

(23)

3) In particular, if the above λ = 0, i.e., there are s ∈ Ψσ(Σf )

and t ∈ L/s such that

FDσ(st) = 0 (24)

then G is said to be completely f -undiagnosable withrespect to σ (or the diagnosability degree of G equals zerowith respect to σ).

Remark 4: Definition 5 shows that the fuzzy diagnosabilityfunction FD

σ: E∗ → [0, 1] indicates the diagnosability degree

of G, which can be interpreted as follows:Case 1: If the diagnosability degree of G is equal to one with

respect to σ, then for any s that Σf (sl) ≥ Σf (σ) and its anycontinuation t, we have FD

σ(st) = 1, which makes clear that

the following inequality holds:

Σf (σ) ≤ minΣf (ω) : ω ∈ P−1σ

(Pσ(s t )). (25)

That is to say, if Σf (sl) ≥ Σf (σ), then for every trace ω thatproduces the same observed record as st, the possibility of thefailure occurring on ω is not less than Σf (σ), too. Obviously,if Σo(a), Σf (a) ∈ 0, 1 for each a ∈ E, then (25) degeneratesinto

Σf (σ) = 1 ⇒ Σf (ω) = 1

for all ω ∈ P−1(P (st)), which is exactly consistent with theconcept of the diagnosability for crisp DESs (see Definition 1).

Therefore, the fuzzy diagnosability for FDESs is a generaliza-tion of the diagnosability for crisp DESs introduced by Sampath,et al. [24].

Case 2: If the diagnosability degree of G equals to λ withrespect to σ, then there are s and continuation t such thatFD

σ(st) = λ is minimal, that is, there is ω ∈ P−1

σ(P

σ(st))

satisfying Σf (ω) = λΣf (σ), and Σf (ω′) ≥ λΣf (σ) for otherω′ ∈ P−1

σ(P

σ(st)).

Case 3: In particular, if λ = 0, then there are s and t such thatFD

σ(st) = 0, which means that P−1

σ(P

σ(st)) contains a trace

ω that Σf (ω) = 0, that is, the failure does not possibly occur onω although ω has the same observed record as st and the failureoccurs on st.

Definition 6: If G is completely f -diagnosable with respect toeach σ ∈ Efail , then G is said to be completely f -diagnosable.If there exists σ ∈ Efail such that G is f -diagnosable with re-spect to σ with degree λ, and the diagnosability degree of Gwith respect to other σ′ ∈ Efail is not less than λ, then G issaid to be f -diagnosable with degree λ. If G is completely f -undiagnosable with respect to a certain σ ∈ Efail , then G is saidto be completely f -undiagnosable.

IV. CONSTRUCTION OF DIAGNOSERS FROM AN FDES

In this section, we present a method of constructing the di-agnosers from a given FDES with a single failure type. In Sec-tion V-D, we will extend it to the case of multiple failure types.

Let G = (Q, E, δ, q0) be an FDES. We use the possible fail-ure labels to label the states of diagnosers. For each σ ∈ Efail ,denote the set of possible failure labels as

= F ∪ Nµ : µ = Σf (a), a ∈ E (26)

where F stands for the possibility of which the failure occurringexceeds the specified degree Σf (σ) and the system is in “failed”state, whileNµ means that the possibility of the failure occurringis µ but it does not exceed the specified degree, so the system isstill in “normal” state.

For σ ∈ E, we define a subset of Q as

Qσ

= q0 ∪ q ∈ Q : (∃q′ ∈ Q)(∃a ∈ Eσ)δ(q′, a) = q

(27)where

Eσ

= Emax−o ∪ a ∈ E : Σo(a) > Σo(σ). (28)

Intuitively, Qσ

consists of the initial state q0 and the states reach-able along an event whose observability degree is the largest orlarger than Σo(σ).

Definition 7: Let G be an FDES and σ ∈ Efail . The diagnoserwith respect to σ is constructed as a finite automaton

Gd = (Qd , Ed , δd , 0) (29)

where Qd is the state set of the diagnoser, Ed is the set ofevents, δd is the transition function, and 0 is the initial state.More specifically



1) The state space Qd ⊆ 2Qσ×

is composed of the statesreachable from 0 under δd . A state of Qd is of the form

= (q1 , 1), (q2 , 2), . . . , (qn , n ) (30)

in which each qi ∈ Qσ

and i ∈ .

2) The initial state 0 = (q0 , N0), which means that G is

normal to start with.3) The event set of the diagnoser is E

σ, i.e.,

Ed = Eσ

= Emax−o ∪ a ∈ E : Σo(a) > Σo(σ).(31)

4) δd : Qd × Ed → Qd is the transition function of the diag-noser, which will be defined in Definition 9.

Definition 8: The label propagation function LP : Qσ××

E∗ → is defined as follows. Let q ∈ Qσ, ∈ , and s ∈ E∗.

1) If = F , then LP(q, , s) = F .2) If = Nµ and Σf (s) ≥ Σf (σ), then LP(q, , s) = F .3) If = Nµ , Σf (s) < Σf (σ) and Σf (s) > µ, then

LP(q, , s) = N Σf (s) .4) If = Nµ , Σf (s) < Σf (σ) and Σf (s) ≤ µ, then

LP(q, , s) = Nµ .The above label propagation function is used to describe the

changes of label from one state of diagnoser to another, whichis interpreted as follows. Denote δ(q0 , s0) = q and δ(q, s) = q′.

1) If label F is added to state q (i.e., q is a “failed” statefor Σf (s0) ≥ Σf (σ)), then the successor state q′ is also a“failed” state since Σf (s0 s) ≥ Σf (σ).

2) If state q is a “normal” state, the possibility of the failureoccurring on s0 is µ that does not exceed the specifieddegree Σf (σ), but if the possibility of the failure occurringon s exceeds Σf (σ), then q′ should be labeled F as a“failed” state since Σf (s0 s) ≥ Σf (σ).

3) If state q is a “normal” state labeled by Nµ , where µ ≤Σf (s) < Σf (σ), then q′ is also to be a “normal” state forΣf (s0 s) = µ, which does not exceed the specified degree,

but the label of q′ should be changed to N Σf (s) .4) If state q is a “normal” state labeled by Nµ , and Σf (s) is

smaller than both µ and Σf (σ), then q′ inherits the labelof q for Σf (s0 s) = Σf (s0) < Σf (σ).

Definition 9: The transition function of the diagnoser Gd isa partial function δd : Qd × Ed → Qd , which is defined as: for ∈ Qd and a ∈ Ed ,

δd(, a) =⋃

(q i ,i )∈

⋃s∈L

a(q i ,σ )

(δ(qi , s), LP(qi , i , s))

(32)

where La(qi , σ) collects all of the strings in L(qi , σ) that end

with a, i.e.,

La(qi , σ) = s ∈ L(qi , σ) : sl = a . (33)

V. NECESSARY AND SUFFICIENT CONDITION OF

DIAGNOSABILITY FOR FDESS

In this section, we give some properties of the diagnosers.In particular, a necessary and sufficient condition of the fuzzydiagnosability for FDESs is presented. Also, we propose anapproach to check the fuzzy diagnosability for FDESs.

A. Properties of Diagnosers

Property 1: Let Gd = (Qd , Ed , δd , 0) be the diagnoserwith respect to σ ∈ Efail and ∈ Qd . (q1 , 1), (q2 , 2) ∈ ifand only if there exist s1 , s2 ∈ E∗ such that (s1)l = (s2)l ∈Ed , P

σ(s1) = P

σ(s2), δd(0 , Pσ

(s1)) = , and δ(q0 , sk ) =qk , LP(q0 , N

0 , sk ) = k for k = 1, 2.Proof: Necessity: From ∈ Qd , there are a1 , a2 , . . . , aj ∈

Ed and 1 , 2 , . . . , j−1 ∈ Qd , such that δd(i , ai+1) = i+1 ,where 0 ≤ i ≤ j − 1 and j = . Due to (q1 , 1), (q2 , 2) ∈ ,by Definition 9, there exist (qk

1 , k1 ) ∈ j−1 , and tk1 ∈ L

aj(qk

1 , σ)(k = 1, 2) such that for k = 1, 2

qk = δ(qk1 , tk1 ) k = LP(qk

1 , k1 , tk1 ).

Similarly, note that δd(j−2 , aj−1) = j−1 , and hence, there are(qk

2 , k2 ) ∈ j−2 , and tk2 ∈ L

aj −1(qk

2 , σ) (k = 1, 2) satisfying fork = 1, 2,

qk1 = δ(qk

2 , tk2 ) k1 = LP(qk

2 , k2 , tk2 ).

. . . . . .

With the analogous process, there are (qkj−1 ,

kj−1) ∈ 1 and

tkj ∈ La1

(q0 , σ) (k = 1, 2) such that for k = 1, 2,

qkj−1 = δ(q0 , t

kj ) k

j−1 = LP(q0 , N0 , tkj ).

We take

s1 = t1j t1j−1 · · · t12 t11 s2 = t2j t

2j−1 · · · t22 t21 . (34)

Obviously, δd(0 , Pσ(s1)) = , (s1)l = (s2)l = aj ∈ Ed , and

for k = 1, 2, we have δ(q0 , sk ) = qk , LP(q0 , N0 , sk ) = k .

Moreover Pσ(s1) = a1 a2 · · · aj = P

σ(s2).

Sufficiency: Assume that there exist s1 , s2 ∈ E∗ suchthat (s1)l = (s2)l ∈ Ed , P

σ(s1) = P

σ(s2), δd(0 , Pσ

(s1)) =, and δ(q0 , sk ) = qk , LP(q0 , N

0 , sk ) = k for k = 1, 2. Fromδd(0 , Pσ

(s1)) = , we denote Pσ(s1) = a1 a2 · · · aj , and then

we can obtain a state sequence 1 , 2 , · · · , j−1 ∈ Qd satisfying

δd(i , ai+1) = i+1 , (0 ≤ i ≤ j − 1)

where j = . Furthermore, from δ(q0 , sk ) = qk ,and LP(q0 , N

0 , sk ) = k , (k = 1, 2), we have that(q1 , 1), (q2 , 2) ∈ by Definition 9.

Definition 10: Let Gd be the diagnoser with respect to σ ∈Efail . A state ∈ Qd is said to be F -certain if = F for all(q, ) ∈ , or = F for all (q, ) ∈ ; otherwise, if there are(q1 , 1), (q2 , 2) ∈ such that 1 = F and 2 = F , then issaid to be F -uncertain.



Property 2: Let Gd = (Qd , Ed , δd , 0) be the diagnoser withrespect to σ ∈ Efail and δd(0 , u) = .

1) is F -uncertain if and only if there exist s1 , s2 ∈E∗ such that (s1)l = (s2)l ∈ Ed , P

σ(s1) = P

σ(s2),

δd(0 , Pσ(s1)) = , and

Σf (s1) ≥ Σf (σ) > Σf (s2). (35)

2) is F -certain if and only if Σf (s) ≥ Σf (σ) for all s ∈P−1

σ(u), or Σf (s) < Σf (σ) for all s ∈ P−1

σ(u), where

sl ∈ Ed .Proof: 1) Sufficiency: Suppose there exists s2 ∈ P−1

σ(P

σ(s1))

such that Σf (s1) ≥ Σf (σ) > Σf (s2), where (s1)l = (s2)l ∈Ed and δd(0 , Pσ

(s1)) = . Denote

LP(q0 , N0 , s1) = 1 LP(q0 , N

0 , s2) = 2 .

From Definition 8, we have 1 = F but 2 = F . By Property 1,we obtain (q1 , 1), (q2 , 2) ∈ , where δ(q0 , s1) = q1 andδ(q0 , s2) = q2 . That is, is F -uncertain.

Necessity: It is clearly obtained from Definitions 9 and 10.2) It holds obviously since it is the inverse and negative propo-

sition of (1). Property 3: Let Gd = (Qd , Ed , δd , 0) be the diagnoser with

respect to σ ∈ Efail . If δd(1 , a) = 2 , where a ∈ Ed , 1 and2 are F -uncertain, and (q1 , F ), (q′1 , N

µ) ∈ 1 , then there ex-ist sa ∈ L(q1 , σ), ta ∈ L(q′1 , σ), and (q2 , F ), (q′2 , N

µ ′) ∈ 2

satisfying

Pσ(sa) = P

σ(ta) = a δ(q1 , sa) = q2 δ(q′1 , ta) = q′2 .

Moreover, if Σf (ta) ≤ µ, then µ′ = µ; if Σf (ta) > µ, then µ′ =Σf (ta).

Proof: It is clearly obtained from Definitions 8 and 9. Property 4: Let Gd be the diagnoser with respect to σ ∈ Efail .

If the set of states forms a cycle in Gd , then all states in the cyclehave the same failure label.

Proof: By Definition 8, it is easy to prove since any two statesin a cycle of Gd are reachable from each other.

B. Necessary and Sufficient Condition of Diagnosabilityfor FDESs

Definition 11: Let Gd = (Qd , Ed , δd , 0) be the diag-noser of G with respect to σ ∈ Efail . A set 1 , σ1 ,2 , σ2 , . . . , k , σk , 1 is said to form a µ-F -indeterminatecycle in Gd , if

1) 1 , 2 , . . . , k ∈ Qd are F -uncertain states, and the set1 , σ1 , 2 , σ2 , . . . , k , σk , 1 forms a cycle in Gd , i.e.,δd(j , σj ) = j+1 for j = 1, . . . , k − 1, and δd(k , σk ) =1 .

2) For each above j , there exist (xhj , h

j ), (yrj , dr

j ) ∈ j (j =1, . . . , k; h = 1, . . . ,m; r = 1, . . . , n) satisfyinga) h

j = F and drj = Nµ for all j, h, r; and

b) the sequences of statesxh

j

and

yr

j

form cycles in

G with

δ(xhj , sh

j σj ) = xhj+1 ,

(j = 1, . . . , k − 1;h = 1, . . . , m)

δ(xhk , sh

k σk ) = xh+11 ,

(h = 1, . . . , m − 1), δ(xmk , sm

k σk ) = x11

and

δ(yrj , trj σj ) = yr

j+1 ,

(j = 1, . . . , k − 1; r = 1, . . . , n)

δ(yrk , trk σk ) = yr+1

1 ,

(r = 1, . . . , n − 1), δ(ynk , tnk σk ) = y1

1

where shj σj ∈ L(xh

j , σ), trj σj ∈ L(yrj , σ).

Intuitively, a µ-F -indeterminate cycle in Gd is a cycle com-posed of F -uncertain states, and corresponding to this cycle,there exist two sequences

xh

j

and

yr

j

forming cycles of G,

in which one carries the failure label F and the other carries thefailure label Nµ . We call the above two state sequences

xh

j

and

yr

j

to be “failed-state” cycle and “normal-state” cycle of

G, respectively. All of the µ-F -indeterminate cycles are calledby a joint name of F -indeterminate cycle.

Theorem 1: Let L be a language generated by an FDESG = (Q, E, δ, q0) and σ ∈ Efail . If the diagnoser Gd with re-spect to σ contains a µ-F -indeterminate cycle, then the diag-nosability degree of G with respect to σ, denoted as λ, satisfiesthe following inequality:

λ ≤ µ

Σf (σ). (36)

Proof: Assume that the diagnosability degree of G withrespect to σ is λ, and Gd contains a µ-F -indeterminate cy-cle 1 , 2 , . . . , k. By Definition 11, there are two se-quences of states xh

j and yrj , which form two cycles in

G, and the corresponding strings shj σj and trj σj satisfy

condition (b) of Definition 11, where (xhj , h

j ), (yrj , dr

j ) ∈ j ,and h

j = F but drj = Nµ for all j = 1, . . . , k; h = 1, . . . ,m;

r = 1, . . . , n.Since (x1

1 , 11), (y

11 , d1

1) ∈ 1 , by Property 1, there exists0 , t0 ∈ E∗ such that P

σ(s0) = P

σ(t0), δ(q0 , s0) = x1

1 , and

δ(q0 , t0) = y11 . Notice that 1

1 = F , and drj = Nµ for all

j, r. Therefore, Σf (s0) ≥ Σf (σ), µ = Σf (t0) < Σf (σ), andΣf (trj σj ) ≤ µ for all j, r, i.e.

Σf (s0) ≥ Σf (σ) > Σf (t0) ≥ Σf (trj σj ) (37)

for all j, r, where Σf (t0) = µ.



Let l be a sufficiently large integer. We consider the followingtwo traces:

ω1 = s0(s11 σ1 . . . s1

k σk s21 σ1 . . . s2

k σk . . . sm1 σ1 . . . sm

k σk )ln

(38)

ω2 = t0(t11 σ1 . . . t1k σk t21 σ1 . . . t2k σk . . . tn1 σ1 . . . tnk σk )lm .

(39)

Then

Pσ(ω1) = Pσ (ω2) = P

σ(t0)(σ1 σ2 . . . σk )lmn . (40)

Notice that Σf (s0) ≥ Σf (σ). There is a prefix s of s0 suchthat s ∈ Ψ

σ(Σf ). Take t ∈ L/s satisfying ω1 = st. From (40),

we have ω2 ∈ P−1σ

(Pσ(st)). Furthermore, from (37) and (39),

we know that

Σf (ω2) = maxΣf (t0), Σf (trj σj ) :

j = 1, . . . , k; r = 1, . . . , n (41)

i.e., Σf (ω2) = µ. Therefore, from (21), the diagnosability de-gree λ of G with respect to σ satisfies

λ = FDσ(st) ≤ Σf (ω2)

Σf (σ)=

µ

Σf (σ). (42)

Definition 12: A µ-F -indeterminate cycle in diagnoser Gd is

called to be minimal, if for each µ′-F -indeterminate cycle inGd , the inequality µ′ ≥ µ always holds.

Now we are ready to present the necessary and sufficientcondition of the diagnosability for FDESs.

Theorem 2: Let L be a language generated by an FDES G

and Gd = (Qd , Ed , δd , 0) be the diagnoser of G with respectto σ ∈ Efail . G is f -diagnosable with degree λ with respect toσ, if and only if, there is a minimal µ-F -indeterminate cycle inGd , where µ = λΣf (σ).

Proof: Necessity: Assume that G is f -diagnosable with degreeλ with respect to σ. By Definition 5, for any n0 ∈ N, there ares ∈ Ψ

σ(Σf ) and t ∈ L/s that |t| ≥ n0 , we have

FDσ(st) =


(Pσ(st))

Σf (σ)= λ.

(43)Hence, there exists ω ∈ P−1

σ(P

σ(st)) such that

Σf (ω) = λΣf (σ) < Σf (σ) (44)

and

Σf (ω) = minΣf (σ), Σf (ω) : ω ∈ P−1σ

(Pσ(st)) (45)

i.e., for any ω′ ∈ P−1σ

(Pσ(st)), we have Σf (ω) ≤ Σf (ω′).

Denote δ(q0 , st) = q and δ(q0 , ω) = q′. It is obvious that

δd(0 , Pσ(st)) is F -uncertain since (q, F ), (q′, N Σf (ω )) are its

two elements. Note that the number of states of G is finite, andδd(0 , Pσ

(st)) and its all successor states form a cycle of Gd

when n0 is sufficiently large. Moreover, all of the states in the

cycle are F -uncertain because they have the same failure labelby Property 4. We denote the cycle of the F -uncertain states as1 , 2 , . . . , k , where

δd(j , σj ) = j+1 , (j = 1, . . . , k − 1) δd(k , σk ) = 1(46)

and σj ∈ Ed for each j = 1, . . . , k.

Since δd(1 , σ1) = 2 , and (q1 , F ), (q′1 , NΣf (ω )) ∈ 1 , by

Property 3, there are (q2 , F ), (q′2 , NΣf (ω )) ∈ 2 and s1 σ1 ∈

L(q1 , σ) and t1 σ1 ∈ L(q′1 , σ), such that

δ(q1 , s1 σ1) = q2 δ(q′1 , t1 σ1) = q′2 . (47)

Similarly, from δd(2 , σ2) = 3 , and (q2 , F ), (q′2 , NΣf (ω )) ∈

2 , we know that there exist (q3 , F ), (q′3 , NΣf (ω )) ∈ 3 and

s2 σ2 ∈ L(q2 , σ) and t2 σ2 ∈ L(q′2 , σ), such that

δ(q2 , s2 σ2) = q3 δ(q′2 , t2 σ2) = q′3 . (48)

With the above analogous process, we can obtain two sequencesof states q1 , q2 , q3 , . . . and q′1 , q′2 , q′3 · · ·. Furthermore, thetwo sequences form two cycles in G and satisfy the condi-tions of Definition 11 because 1 , 2 , . . . , k is a cycle ofF -uncertain states in Gd . That is, 1 , 2 , . . . , k is a µ-F -indeterminate cycle in Gd , where µ = Σf (ω). From (44), wehave µ = λΣf (σ).

In the following, we further prove that the above F -indeterminate cycle is minimal. Suppose that there is anotherµ′-F -indeterminate cycle in Gd . From Theorem 1, the diagnos-ability degree λ of G with respect to σ satisfies the followinginequality:

λ ≤ µ′

Σf (σ)(49)

i.e., µ′ ≥ λΣf (σ) = µ, which shows that 1 , 2 , . . . , k is aminimal µ-F -indeterminate cycle.

Sufficiency: Assume that there is a minimal µ-F -indeterminate cycle in Gd , where µ = λΣf (σ). We denote thediagnosability degree of G with respect to σ as x. Then by theabove proof of Necessity, there is a minimal µ′-F -indeterminatecycle in Gd , where µ′ = xΣf (σ). From the minimality of µ-F -indeterminate cycle, we have µ′ ≥ µ. Likewise, from the mini-mality of µ′-F -indeterminate cycle, we have µ ≥ µ′. That is

λΣf (σ) = µ = µ′ = xΣf (σ) (50)

i.e., x = λ. Therefore, the diagnosability degree of G with re-spect to σ is λ.

Theorem 3: Let L be a language generated by an FDES G.1) G is f -diagnosable with degree λ, if and only if the fol-

lowing two conditions hold:a) there is σ ∈ Efail such that the diagnoser Gd with re-

spect to σ contains a minimal µ-F -indeterminate cycle,where µ = λΣf (σ);

b) for any σ′ ∈ Efail , if the diagnoser with respect toσ′ contains a minimal µ′-F -indeterminate cycle, thenµ′ ≥ λΣf (σ′).



2) G is completely f -undiagnosable, if and only if there isσ ∈ Efail such that the diagnoser Gd with respect to σcontains a 0-F -indeterminate cycle.

3) G is completely f -diagnosable, if and only if for anyσ ∈ Efail , the diagnoser Gd with respect to σ does notcontain any F -indeterminate cycle.

Proof: (1) Necessity: Suppose that G is f -diagnosable withdegree λ. By Definition 6, there exists σ ∈ Efail such that Gis f -diagnosable with respect to σ with degree λ, and there isnot σ′ ∈ Efail that the diagnosability degree with respect to σ′

is less than λ. From Theorem 2, we know that the diagnoserwith respect to σ contains a minimal µ-F -indeterminate cycle,where µ = λΣf (σ); and for any σ′ ∈ Efail , if the diagnoser withrespect to σ′ contains a minimal µ′-F -indeterminate cycle, thenλ ≤ λ′. From Theorem 2 again, we obtain that

λ =µ

Σf (σ)λ′ =

µ′

Σf (σ′). (51)

Therefore, µ′ ≥ λΣf (σ′).Sufficiency: Assume that there is σ ∈ Efail such that the diag-

noser with respect to σ contains a minimal µ-F -indeterminatecycle, where µ = λΣf (σ); and for any σ′ ∈ Efail , if the diag-noser with respect to σ′ contains a minimal µ′-F -indeterminatecycle, then µ′ ≥ λΣf (σ′). By Theorem 2, G is f -diagnosablewith respect to σ with degree λ, and the diagnosability degreeof G with respect to each other σ′ ∈ Efail is not less than λ. ByDefinition 6, G is f -diagnosable with degree λ.

2) It is a special case of (1) when λ = 0.3) It holds clearly since it is the inverse and negative propo-

sition of (1).

C. Computing Process to Check the Diagnosability of FDESs

According to Theorems 2 and 3, we give the following methodto check the diagnosability for FDESs.

Step 1: Construct the diagnosers for each σ ∈ Efail .Let L be a language generated by an FDES G. For each

σ ∈ Efail , we construct the diagnoser Gd = (Qd , Ed , δd , 0) bymeans of Definitions 7 and 9. This specific procedure can beseen in Section IV for the details.

Step 2: Check whether there exists F -indeterminate cycles inthese diagnosers.

If the diagnoser Gd with respect to each σ ∈ Efail doesnot contain F -indeterminate cycle, then G is completely f -diagnosable; otherwise, we perform the next step.

Step 3: Collect all of the minimal F -indeterminate cycles andcompute the diagnosability degree of G.

For σ ∈ Efail , denote the minimal F -indeterminate cycle asµ-F -indeterminate cycle. Let

λ = min

µ

Σf (σ): σ ∈ Efail

. (52)

If λ = 0, then G is completely f -undiagnosable; if 0 < λ < 1,then G is f -diagnosable with degree λ.

Remark 5: The main complexity of the above computingprocess of checking the diagnosability for FDESs lies in the

construction of the diagnosers. In the following, we roughlyanalyze the complexity of constructing these diagnosers. Sup-pose that |Q| and |E| are the numbers of fuzzy states andfuzzy events of fuzzy finite automaton G = (Q, E, δ, q0), re-spectively. Let Gd = (Qd , Ed , δd , 0) be the diagnoser withrespect to σ ∈ Efail . There are at most |E| + 1 possible fail-ure labels for the set of possible failure labels = F ∪Nµ : µ = Σf (a), a ∈ E. Due to Qd ⊆ 2Q

σ×

, the number

of states in Gd is at most 2|Q |×(|E |+1) . Notice that Ed ⊆ E,so the maximum number of transitions in the diagnoser Gd

is 2|Q |×(|E |+1) × |E|. That is, the complexity of constructing

one diagnoser is O(2|Q |×(|E |+1) × |E|). Since Efail ⊆ E, theoverall complexity of the above computing process of testing

diagnosability of FDESs is, at most, O(2|Q |×(|E |+1) × |E|2).

D. Diagnosability for Multiple Failure Types

In this section, we extend the fuzzy diagnosability of FDESswith one single failure type to the case of multiple failure types.

Assume that a fuzzy system has events of multiple failuretypes f1 , f2 , . . . , fm , and Efaili = σ ∈ E : Σfi

(σ) > 0. Foreach σ ∈ Efaili , we define the fuzzy fi-diagnosability functionas

FDi

σ: E∗ → [0, 1] (53)

where i = 1, 2, . . . ,m, and, for s ∈ Ψσ(Σfi

) and t ∈ L/s,

FDi

σ(st) =

minΣfi(σ), Σfi

(ω) : ω ∈ P−1σ

(Pσ(st))

Σfi(σ)

. (54)

Definition 13: Given an FDES G = (Q, E, δ, q0) with multi-ple failure types f1 , f2 , . . . , fm . Let L be a language generatedby G. If

(∃n0 ∈ N)(∀s ∈ Ψσ(Σfi

))(∀t ∈ L/s)

(|t| ≥ n0 ⇒ FDi

σ(st) = 1)

(55)

then G is said to be completely fi-diagnosable with respect toσ; if for any n0 ∈ N, the following holds:

minFDi

σ(st) : s ∈ Ψ

σ(Σfi

), t ∈ L/s, |t| ≥ n0 = λ (56)

where 0 ≤ λ < 1, then G is said to be fi-diagnosable withdegree λ with respect to σ; if for any n0 ∈ N, there are s ∈Ψ

σ(Σfi

) and t ∈ L/s that |t| ≥ n0 , we have

FDi

σ(st) = 0 (57)

then G is said to be completely fi-undiagnosable with respectto σ.

The diagnosers with multiple failure types can be similarlyconstructed, and we only need to use the following tensor prod-uct to represent the possible failure labels

= 1 ×2 × · · · × m (58)



Fig. 1. Crisp DES G in Example 2.

in whichi = Fi ∪ Nµi : µ = Σfi

(a), a ∈ E for each i =1, 2, · · · ,m. A state ∈ Qd of the diagnoser is of the form

= (q1 , (11 , 12 , . . . , 1m )), . . . , (qn , (n1 , n2 , . . . , nm ))

in which qj ∈ Qσ, ji = Fi or ji = Nµ

i (µ ∈ [0, 1); j =1, 2, . . . , n; i = 1, 2, . . . , m), and each pair (qj , ji) is calledto be a component of state .

Remark 6: Each component of the states of the diagnoser isresponsible for detecting one type of failure, and the failure oftype fi is reported when a component of a certain state is labeledby Fi . Therefore, the approach to check the diagnosability with asingle failure type is also suitable for the case of multiple failuretypes. Since the complexity of testing the diagnosability with

a single failure type is O(2|Q |×(|E |+1) × |E|2), the complexityof testing the diagnosability with m failure types is at most

O(2|Q |×(m (|E |+1)) × |E|2).

VI. APPLICATIONS TO DIAGNOSIS OF CRISP DESS AND

FDESS: SOME EXAMPLES

In this section, we give some examples to illustrate that thefuzzy diagnosability for FDESs presented before may be appliedto the failure diagnosis for crisp DESs and the failure diagnosisfor FDESs.

A. Application to Diagnosis of Crisp DESs

As we know, for a crisp DES with n crisp states, if we regardthe crisp states as n-D unit state vectors, and the events aredenoted as n × n matrices that every entry takes two possiblevalues 0 or 1, then the crisp DES can be viewed as a specialFDES. Therefore, the fuzzy diagnosability for FDESs presentedabove may be used to cope with the problem of failure diagnosisfor crisp DESs. The following example verifies the view.

Example 2. Consider the crisp DES G described in Fig. 1, inwhich the set of observable events Eo = α, β, and the twofailure event sets are Ef1 = θ and Ef2 = τ, respectively.

In the following, we prove that L is f1-diagnosable but notf2-diagnosable by means of two methods: one is the classicaldiagnosability approach for crisp DESs proposed in [24] (i.e.,Definition 1), another is the fuzzy diagnosability for FDESsproposed in Section V.

1) Classical Diagnosability Approach:First, we prove the result using the classical diagnosabil-

ity approach for crisp DESs. For the failure type f1 , due toΨ(Ef1 ) = αθ, if we take n0 = 2, then for any s1 ∈ Ψ(Ef1 )

Fig. 2. Diagnosers with respect to θ and τ in Example 2.

and any t1 ∈ L/s1 , i.e., s1 = αθ and t1 = βn (n ≥ 2), we have

P−1(P (s1t1)) = P−1(P (αθβn )) = αθβn.Therefore, D(s1t1) = 1. Similarly, for the failure type f2 ,Ψ(Ef2 ) = ατ, for any n0 ∈ N, if we take s2 = ατ andt2 = αn ∈ L/s2 , then

P−1(P (s2t2)) = P−1(P (αταn )) = αταn , αγαn.So, D(s2t2) = 0. By Definition 1, L is f1-diagnosable but notf2-diagnosable.

2) Fuzzy Diagnosability Approach:Next, we verify the above result by means of the fuzzy diag-

nosability approach for FDESs presented in Section V.The crisp DES G can be viewed as a special FDES with the

fuzzy states: q0 = [1, 0, 0, 0], q1 = [0, 1, 0, 0], q2 = [0, 0, 1, 0],q3 = [0, 0, 0, 1]; and fuzzy events

α =

0 1 0 00 0 0 00 0 1 00 0 0 0

β =

0 0 0 00 0 0 00 0 0 00 0 0 1

γ =

0 0 0 00 0 1 00 0 0 00 0 0 0

θ =

0 0 0 00 0 0 10 0 0 00 0 0 0

τ =

0 0 0 00 0 1 00 0 0 00 0 0 0

.

From Eo = α, β, Ef1 = θ, and Ef2 = τ, we knowthat the observable event fuzzy subset Σo are determined as

Σo(γ) = Σo(θ) = Σo(τ) = 0, Σo(α) = Σo(β) = 1

and the failure event fuzzy subsets Σf1 and Σf2 are determinedas follows:

Σf1 (α) = Σf1 (β) = Σf1 (γ) = Σf1 (τ) = 0, Σf1 (θ) = 1

Σf2 (α) = Σf2 (β) = Σf2 (γ) = Σf2 (θ) = 0, Σf2 (τ) = 1.

For event θ and event τ , from Definition 7, we know that thediagnoser with respect to θ and that with respect to τ are thesame, which are shown as Fig. 2.

For the failure type f1 , notice that both of the diagnosersin Fig. 2 do not contain F1-indeterminate cycle since all ofthe states are F1-certain. However, for the failure type f2 , thediagnosers contain a 0-F -indeterminate cycle 3 , α, 3 , α,where

3 = (q2 , (N 01 , N 0

2 )), (q2 , (N 01 , F2)).



Therefore, by Theorem 3, we obtain the result that G is com-pletely f1-diagnosable but G is completely f2-undiagnosable,which coincides with the result obtained by the classical diag-nosability approach for crisp DESs.

B. Application to Diagnosis of FDESs

FDESs combines fuzzy set theory [31] with crisp DESs [4],and it has been successfully applied to the real-world complexsystems such as biomedical systems [11]–[13], [29], [30] androbotic control [8], [9], in which vagueness, impreciseness, andsubjectivity are typical features. Therefore, the fuzzy diagnos-ability for FDESs presented above may better cope with theproblem of uncertainty emerging from the issue of failure diag-nosis. Next, we apply our framework formalized in Sections IIIand IV to the following two examples of diagnosis for FDESs(Example 3 and Example 4). In particular, Example 3 arisingfrom medical diagnosis and treatment may be viewed as anapplicable background of the diagnosability for FDESs.

Example 3. Suppose that there is an animal growing sick froma new disease. For the new disease, the doctor has no completeknowledge about it, but he (or she) believes, by experience,that these drugs, such as theophylline, ipratropium bromide,erythromycin ethylsuccinate, and dopamine (denoted as α, β, γand θ, respectively) may be useful to the disease.

For simplicity, it is assumed that the doctor considers roughlythe animal’s condition to be three states, say “good,” “fair,” and“poor.” As mentioned in Section I and [11], [12], and [21], it isvague when the animal’s condition is said to be “good,” “fair,” or“poor,” since the animal’s condition can simultaneously belongto “good,” “fair,” and “poor” with respective memberships inthe real-life situation [12], [20], [21]. Therefore, when an FDESis used to model the treatment process of the animal, a fuzzystate is naturally denoted as a 3-D vector

good fair poorq = [a1 a2 a3 ]

(59)

which is represented as the possibility distribution of the ani-mal’s condition over states “good,” “fair,” and “poor.”

Similarly, it is imprecise to say at what point exactly theanimal has changed from one state to another state after a drugtreatment (i.e., event), because each drug event occurring maylead a state to multistates with respective memberships [20,p. 82]. Therefore, in the treatment process modeled by an FDES,a fuzzy event is represented as a 3 × 3 matrix

good fair poor

σ =goodfairpoor

a11 a12 a13a21 a22 a23a31 a32 a33

.(60)

Suppose that the treatment process of the animal is modeledby the following FDES G = (Q, E, δ, q0), shown as Fig. 3, inwhich the initial state vector is q0 = [0.9, 0.1, 0], and the drugevents are evaluated as follows by means of medical theory andthe doctor’s experience (there are many methods for estimatingthe memberships in the literature, for example, [6, pp. 256–260],

Fig. 3. Treatment process of the animal modeled by FDES G.

[7, p. 323], [29, pp. 668–669], [30, pp. 967-970])

α =

0.4 0.9 0.40 0.4 0.40 0 0.4

β =

0.4 0 00.9 0.4 00.4 0.4 0.4

γ =

0.9 0.9 0.40 0.4 0.40 0 0.4

θ =

0.5 0 00.1 0.1 00.1 0.1 0.1

.

By the max–min operation (i.e., Definition 2), we can calcu-late the other states as

q1 = [0.4, 0.9, 0.4] q2 = [0.9, 0.4, 0.4] q5 = [0.4, 0.5, 0.4]

q3 = [0.9, 0.9, 0.4] q4 = [0.5, 0.1, 0] q5 = [0.4, 0.5, 0.4]

q6 = [0.5, 0.4, 0.4] q7 = [0.5, 0.5, 0.4].

As the continuation of our work [16], [21], each fuzzy event isobservable with a certain membership degree in the frameworkof FDESs. For example, some symptoms of the drugs such asskin turning pale can be clearly seen by the doctor, but somesymptoms may be observable only by means of special medi-cal instruments, even some potential side effects such as highwhite blood cell count are undesired failures. Assume that highwhite blood cell count is considered as undesired failure in thismodel, and the observability degree and the possibility of failureoccurring are evaluated as follows by the doctor:

Σo(α) = 0.5, Σo(β) = 0.4, Σo(γ) = 0.6, Σo(θ) = 0.3

Σf (α) = 0.1, Σf (β) = 0.2, Σf (γ) = 0.3, Σf (θ) = 0.4.

The problem of the diagnosis for FDES G is how to detectthe failure (i.e., high white blood cell count) in time and furthercompute the possibility of failure occurring during the treatmentprocess.

In order to explain the notion of fuzzy diagnosability forFDESs formalized in Section III and illustrate the necessary andsufficient condition for diagnosis of FDESs (i.e., Theorems 2and 3), we present the following two methods: one is the defini-tion approach (i.e., Definitions 5 and 6), another is the diagnoserapproach proposed in Sections IV and V.

1) Definition Approach: First, we investigate the diagnosisproblem for the treatment process G by means of the definitionof diagnosability for FDESs.

For fuzzy event α, notice that Σf (α) is the least amongΣf (σ) : σ ∈ Σ, and therefore, by (21), for any s ∈ Ψ

α(Σf )



and t ∈ L/s that |t| ≥ n0 (n0 ∈ N), we have

FDα(st) =

minΣf (α), Σf (ω) : ω ∈ P−1α

(Pα(st))

Σf (α)= 1

which means that the possibility of failure occurring (i.e., highwhite blood cell count) on each treatment strategy ω is not lessthan that on α (i.e., drug theophylline treatment).

For fuzzy event β, take n0 = 3. Then for any s ∈ Ψβ(Σf )

and t ∈ L/s that |t| ≥ n0 , we have FDβ(st) = 1 because all

of the strings ω ∈ P−1β

(Pβ(st)) contain γ or θ. For simplicity,

we only calculate the case that s = αβ and t = γαβ ∈ L/s asfollows (the other cases can be similarly calculated):

P−1β

(Pβ(st)) = αβγα, αβγαβ, θαβγα, θαβγαβ.

Therefore, Σf (ω) = 0.3 or Σf (ω) = 0.4 for each ω ∈P−1

β(P

β(st)). By (21), we obtain

FDβ(st) =

minΣf (β), Σf (ω) : ω ∈ P−1β

(Pβ(st))

Σf (β)

≥ min0.2, 0.30.2

= 1

which indicates that we can detect the occurrence of eventswhose possibilities of failure occurring are not less than that onβ within delay of three steps (i.e., n0 = 3).

The case of fuzzy event γ is similar to that of β, and all ofthe strings ω ∈ P−1

γ(P

γ(st)) contain γ or θ. By (21), we have

FDγ(st) = 1.

For fuzzy event θ, if we take s = θ and t = α(βγα)n0 ∈ L/s(n0 ∈ N), then

P−1θ

(Pθ(st)) = P−1

θ(α(βγα)n0 ) = α(βγα)n0 , θα(βγα)n0 .

By (21), the diagnosability degree of G equals to 0.75 withrespect to θ for

FDθ(st) =

minΣf (θ), Σf (α(βγα)n0 ), Σf (θα(βγα)n0 )Σf (θ)

=min0.4, 0.3, 0.4

0.4= 0.75.

By Definition 6, G is f -diagnosable with degree 0.75. Theresult shows that we cannot detect the failure occurring on thetreatment strategy ω whose possibility of failure occurring isnot less than Σf (θ), but we may ensure that the possibilities offailure occurring on all of the treatment strategies are at least0.75 times that on θ.

2) Diagnoser Approach: In the following, we further verifythe above results by using the necessary and sufficient conditionfor the diagnosability of FDESs (i.e., Theorem 3).

First, we construct the diagnosers for each fuzzy event inEfail = α, β, γ, θ.

Fig. 4. Diagnosers with respect to α and γ .

Fig. 5. Diagnoser Gd with respect to β .

Fig. 6. Diagnoser Gd with respect to θ.

For α, the event set of the diagnoser with respect to α is

Ed = γ.The states of the diagnoser with respect to α are 0 =(q0 , N

0), and 1 = (q3 , F ), (q7 , F ). By Definition 7, thediagnoser with respect to α is constructed as in Fig. 4.

For β, the event set of the diagnoser with respect to β is

Ed = α, γ

and the states of the diagnoser with respect to β are as follows:

0 = (q0 , N0) 1 = (q1 , N

0.1), (q5 , F )2 = (q3 , F ), (q7 , F ) 3 = (q1 , F ), (q5 , F ).

By Definition 7, the diagnoser with respect to β is constructedas in Fig. 5.

For γ, it is not difficult to know from Definition 7 that thediagnoser with respect to γ is the same as the diagnoser withrespect to α as shown in Fig. 4.

For θ, the event set of the diagnoser is

Ed = α, β, γ

and the states of the diagnoser with respect to θ are as follows:

0 = (q0 , N0) 1 = (q1 , N

0.1), (q5 , F )2 = (q2 , N

0.2), (q6 , F ) 3 = (q3 , N0.3), (q7 , F )

4 = (q1 , N0.3), (q5 , F ) 6 = (q2 , N

0.3), (q6 , F ).

By Definition 7, the diagnoser with respect to θ is constructedas in Fig. 6.

Although there is not any F -indeterminate cycle in Figs. 4and 5, but there is a minimal 0.3-F -indeterminate cycle3 , α, 4 , β, 5 , γ, 3 in Fig. 6, where

3 = (q3 , N0.3), (q7 , F ) 4 = (q1 , N

0.3), (q5 , F )5 = (q2 , N

0.3), (q6 , F ).

That is, there is an F -indeterminate cycle only for θ ∈ Efail ,and the diagnoser Gd with respect to θ contains a minimal



Fig. 7. Fuzzy automaton G in Example 4.

µ-F -indeterminate cycle, where µ = λΣf (θ). Noticing that µ =0.3 (see Fig. 6) and Σf (θ) = 0.4, we obtain that λ = 0.75.Therefore, by Theorem 3, G is f -diagnosable with degree 0.75.

In fact, from Fig. 6, we know that corresponding to theF -indeterminate cycle 3 , α, 4 , β, 5 , γ, 3, the “failed-state” cycle (q7 , F ), (q5 , F ), (q6 , F ), (q7 , F ) carries the fail-ure label F , which shows that the possibility of failure occurringon the corresponding treatment strategy exceeds 0.4; however,the “normal-state” cycle

(q3 , N0.3)(q1 , N

0.3)(q2 , N0.3)(q3 , N

0.3)

carries the label N 0.3 , which means that the possibility of fail-ure occurring on the corresponding treatment strategy does notexceed 0.4. Therefore, we cannot detect the failure occurring onthe treatment strategies whose possibility of failure occurringis not less than Σf (θ), but we may ensure that the possibili-ties of failure occurring on all of the treatment strategies are atleast 0.3, which coincides with the results obtained above by theDefinition Approach.

Example 4. Consider the FDES G represented in Fig. 7, wherethe fuzzy states and fuzzy events are, respectively, given asfollows:

q0 = [0.9, 0.1, 0] q1 = [0.4, 0.9, 0.4] q2 = [0.9, 0.4, 0.4]

q3 = [0.9, 0.9, 0.4] q4 = [0.4, 0.1, 0] q5 = [0.4, 0.4, 0.4]

α =

0.4 0.9 0.40 0.4 0.40 0 0.4

β =

0.4 0 00.9 0.4 00.4 0.4 0.4

γ =

0.9 0.9 0.40 0.4 0.40 0 0.4

.

Suppose that the observability degree of the events and thepossibility of failure occurring are defined as

Σo(α) = 0.6 Σo(β) = 0.4 Σo(γ) = 0.6

Σf (α) = 0.1 Σf (β) = 0.2 Σf (γ) = 0.3.

Then the diagnosers with respect to each σ ∈ Efail are con-structed as in Fig. 8.

From Fig. 8, we know that all of the diagnosers do not con-tain any F -indeterminate cycle. Therefore, by Theorem 3, G iscompletely f -diagnosable.

Fig. 8. Diagnosers with respect to each σ ∈ Efa il . (a) Diagnosers with

respect to α and γ . (b) Diagnoser with respect to β .

VII. CONCLUDING REMARK

In this paper, we have dealt with the failure diagnosis ofFDESs. We have presented a fuzzy diagnosability approach forFDESs. A fuzzy diagnosability function that takes values in theinterval [0, 1] rather than 0, 1 has been introduced to charac-terize the diagnosability degree of the fuzzy systems. We haveformalized the construction of the diagnosers that are used toperform fuzzy diagnosis. In particular, we have proposed a nec-essary and sufficient condition for the diagnosability of FDESs,and a method for checking the diagnosability condition has beengiven. The newly proposed diagnosability approach allows us todeal with the problem of failure diagnosis for both crisp DESsand FDESs, which may better deal with the problems of fuzzi-ness, impreciseness, and subjectivity in the failure diagnosis.

With the results obtained in this paper, a further issueworthy of consideration is the I-diagnosability and the AA-diagnosability of FDESs, as those investigated in the frame-works of crisp DESs [24] and stochastic DESs [27]. Of course,since we have focused on the centralized diagnosis of FDESsin this paper, the decentralized diagnosis together with the safediagnosis of FDESs is another interesting issue, just as we havedealt with them in stochastic DESs [15], [17]. We would like toconsider these issues in the subsequent work.

ACKNOWLEDGMENT

The authors would like to thank Professor N. R. Pal, Editor-in-Chief, the Associate Editor, and the anonymous reviewers fortheir invaluable suggestions and comments that greatly helpedto improve the quality of the paper.

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[29] H. Ying, F. Lin, R. D. MacArthur, J. A. Cohn, D. C. Barth-Jones, H. Ye,and L. R. Crane, “A fuzzy discrete event system approach to determin-ing optimal HIV/AIDS treatment regimens,” IEEE Trans. Inf. Technol.Biomed., vol. 10, no. 4, pp. 663–676, Oct. 2006.

[30] H. Ying, F. Lin, R. D. MacArthur, J. A. Cohn, D. C. Barth-Jones, H. Ye, andL. R. Crane, “A self-learning fuzzy discrete event system for HIV/AIDStreatment regimen selection,” IEEE Trans. Syst., Man, Cybern. B, Cybern.,vol. 37, no. 4, pp. 966–979, Aug. 2007.

[31] L. A. Zadeh, “Fuzzy logic=computing with words,” IEEE Trans. FuzzySyst., vol. 4, no. 2, pp. 103–111, May 1996.

Fuchun Liu received the M.S. degree in mathematicsin 1997 from Jiangxi Normal University, Nanchang,China, and the Ph.D. degree in computer science fromSun Yat-Sen University, Guangzhou, China, in 2008.

He is an Associate Professor in the Facultyof Computer, Guangdong University of Technol-ogy, Guangzhou, and a Research Fellow in theDepartment of Electrical and Computer Engineer-ing, National University of Singapore, Kent Ridge,Singapore.

His current research interests include automatatheory and discrete event systems, mathematical logic, and rough set theory. Hehas authored or coauthored over 20 peer-reviewed papers in academic journals,including two papers about stochastic discrete-event systems published by theIEEE TRANSACTIONS ON AUTOMATIC CONTROl, and some papers about fuzzydiscrete-event systems published by the European Journal of Control and ac-cepted by the IEEE TRANSACTIONS ON FUZZY SYSTEMS.

Daowen Qiu received the M.S. degree in mathemat-ics in 1993 from Jiangxi Normal University, Nachang,China, and the Ph.D. degree in mathematics from SunYat-Sen University, Guangzhou, China, in 2000.

He completed the postdoctoral research in com-puter science at Tsinghua University, Beijing, China,in 2002. Since May, 2004, he has been a Profes-sor of computer science at Sun Yat-Sen University.His current research interests include automata the-ory and discrete-event systems, fuzzy logic, modelsof quantum computation, quantum information, and

the theory of computation based on nonclassical logic. He has authored orcoauthored over 40 peer-reviewed papers in academic journals, including In-formation and Computation, Artificial Intelligence, the Journal of Computerand System Sciences, Theoretical Computer Science, THE IEEE TRANSACTIONS

ON AUTOMATIC CONTROL, THE IEEE TRANSACTIONS ON SYSTEMS, MAN, AND

CYBERNETIC—PART B: CYBERNETICS, THE IEEE TRANSACTIONS ON FUZZY

SYSTEMS, Physical Review A, The Journal of Physics A, and Science in China.



On Generalized Fuzzy Belief Functionsin Infinite Spaces

Wei-Zhi Wu, Yee Leung, and Ju-Sheng Mi

Abstract—Determined by a fuzzy implication operator, a gen-eral type of fuzzy belief structure and its induced dual pair of fuzzybelief and plausibility functions in infinite universes of discourseare first defined. Relationship between the belief-structure-basedand the belief-space-based fuzzy Dempster–Shafer models is thenestablished. It is shown that the lower and upper fuzzy probabil-ities induced by the fuzzy belief space yield a dual pair of fuzzybelief and plausibility functions. For any fuzzy belief structure,there must exist a fuzzy belief space such that the fuzzy beliefand plausibility functions defined by the given fuzzy belief struc-ture are just the lower and upper fuzzy probabilities induced bythe fuzzy belief space, respectively. Essential properties of the fuzzybelief and plausibility functions are also examined. The fuzzy beliefand plausibility functions are, respectively, a fuzzy monotone Cho-quet capacity and a fuzzy alternating Choquet capacity of infiniteorder.

Index Terms—Belief functions, fuzzy implication operators,fuzzy rough sets, monotone capacity, rough sets.

I. INTRODUCTION

A S A GENERALIZATION of Bayesian theory of subjec-tive judgment, the Dempster–Shafer theory of evidence

(also called the theory of belief functions) is a method devel-oped to model and manipulate uncertain, imprecise, incomplete,and even vague information. It was originated by Dempster’sconcept of lower and upper probabilities [8] and extended byShafer [32] as a theory. The basic representational structure inthis theory is a belief structure, which consists of a family ofsubsets, called focal elements, with associated individual posi-tive weights summing to 1. The fundamental numeric measuresderived from the belief structure are a dual pair of belief andplausibility functions. Since its inception, evidential reasoninghas emerged as a powerful methodology for pattern recognition,image analysis, diagnosis, knowledge discovery, information fu-sion, and decision making.

The original concepts of belief and plausibility functions inthe Dempster–Shafer theory of evidence come from the lower

Manuscript received April 11, 2008; accepted December 4, 2008. First pub-lished January 27, 2009; current version published April 1, 2009. This workwas supported by the Hong Kong Research Grants Council under the Ear-marked Grant CUHK 4126/04H.

W.-Z. Wu is with the School of Mathematics, Physics and Informa-tion Science, Zhejiang Ocean University, Zhoushan 316004, China (e-mail:[email protected]).

Y. Leung is with the Department of Geography and Resource Management,Center for Environmental Policy and Resource Management, Institute of Spaceand Earth Information Science, The Chinese University of Hong Kong, Shatin,Hong Kong (e-mail: [email protected]).

J.-S. Mi is with the College of Mathematics and Information Science, HebeiNormal University, Hebei 050016, China (e-mail: [email protected]).


and upper probabilities induced by a multivalued mapping car-rying a probability measure defined over subsets of the do-main of the mapping [32]. The belief (respectively, plausibility)function is a monotone Choquet capacity of infinite order (re-spectively, alternating Choquet capacity of infinite order) [4]satisfying the subadditive (respectively, superadditive) propertyat any order [32]. Subadditivity and superadditivity at any orderform the essential properties of belief and plausibility functions,respectively.

One of the main directions of research in the Dempster–Shafertheory of evidence is naturally the generalization of the beliefand plausibility functions. There are mainly two approachesfor the definitions of belief and plausibility functions, namely,the belief-space-based and belief-structure-based approaches. Inthe belief-space-based approach, a belief space that consists of amultivalued mapping (or a binary relation) carrying a probabilitydistribution on the domain of the mapping is a primitive notion.The lower and upper probabilities are constructed by means ofthe belief space. Unlike the belief-space-based approach, thebelief-structure-based approach does not take belief spaces asprimitive notion. It regards the abstract belief structure (definedby axioms) as primitive notion. The belief and plausibility func-tions are derived from the belief structure. In the finite universesof discourse, it has been proved that a belief space can yielda belief structure such that the lower and upper probabilitiesin the belief space are, respectively, the belief and plausibilityfunctions defined by the belief structure. Conversely, for a beliefstructure defined by axioms, there must exist a belief space suchthat the belief and plausibility functions defined by the beliefstructure are, respectively, the the lower and upper probabilitiesinduced by the belief space [7], [32], [43], [51].

Another important method used to deal with uncertainty inintelligent systems characterized by insufficient and incom-plete information is the theory of rough sets originated byPawlak [24], [25]. The basic structure of rough set theory is anapproximation space consisting of a universe of discourse and abinary relation imposed on it. Using the concepts of lower andupper approximations in rough set theory, knowledge hiddenin information systems may be unravelled and expressed in theform of decision rules (see, e.g., the literature cited in [25]–[27]).The lower and upper probabilities in the Dempster–Shafer the-ory of evidence seem to have some natural correspondenceswith the lower and upper approximations in rough set theory.The relationships between the Dempster–Shafer theory of evi-dence and rough set theory have received wide attention in theresearch community [33]–[35], [43], [51]. In finite universes ofdiscourse, it has been demonstrated that for each type of beliefstructure and its induced belief and plausibility functions, theremust exist an approximation space such that its derived dual

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pair of lower and upper approximation operators may be usedto interpret the belief and plausibility functions [43], [51]. Itcan be observed that the belief and plausibility functions in theDempster–Shafer theory of evidence and the lower and upperapproximations in rough set theory, respectively, capture themechanisms of numeric and nonnumeric aspects of uncertainknowledge.

Shafer’s belief and plausibility functions are constructed un-der the assumption that the focal elements in the belief structureare all crisp. In some situations, it seems to be quite naturalthat the evidence mass may be assigned to a fuzzy subset of theuniverse of discourse. In fact, combining the Dempster–Shafertheory and fuzzy set theory has been suggested to be a way todeal with different kinds of uncertain information in intelligentsystems in a number of studies. Zadeh was the first to generalizethe Dempster–Shafer theory to the fuzzy environment. He de-fined the concepts of expected certainty and expected possibilityas a generalization of Dempster–Shafer belief and plausibilityfunctions. The generalization is based on the concept of in-formation granularity [56] and the theory of possibility [55].Following Zadeh’s work, the Dempster–Shafer theory has beenenriched in slightly different ways in order to evaluate the de-gree of belief in fuzzy events (see, e.g., [9], [11], [13], [15], [16],[18], [20], [23], [29], [30], [36]–[39], [46], [47], [52], and [53]).For example, based on the upper and lower expectations of amembership function, Smets [36] defined the probability of afuzzy event within the framework of belief functions and showedthat the fuzzy belief function satisfies the subadditive propertyat any order in the fuzzy case. Like Zadeh’s expected certaintythat is constructed by a fuzzy belief structure and a measureof fuzzy inclusion, Ishizuka et al. [15] and Yager [46], respec-tively, employed the Łukasiewicz implicator (fuzzy implicationoperator) and the Kleene–Dienes implicator to define fuzzy in-clusions. As a result, they developed two types of fuzzy beliefand plausibility functions. Analogous in spirit to conditionalprobability, Ogawa and Fu [23] employed relative sigma countto compute the degree of inclusion for defining fuzzy belieffunction. Although these extensions to the Dempster–Shafertheory of evidence arrive at frameworks in which both proba-bilistic information and vague information can be handled, theuniverses of discourse are all finite. On the other hand, whetheror not the belief functions defined in [15], [23], and [46] satisfythe subadditivity property at any order have not been exam-ined. Biacino [2] recently used the Kleene–Dienes implicator tocalculate the degree of fuzzy inclusion in infinite universe of dis-course. A fuzzy belief function on infinite universe of discoursewas then defined and the belief function was shown to satisfythe subadditivity property at any order. However, almost all ofthese extensions have not shown if the belief and plausibilityfunctions can be represented as lower and upper probabilitiessimilar to the case of crisp sets. Biacino [2] considered fuzzybelief functions induced by an infinitely monotone inclusionand she proved that they are lower probabilities. More recently,with reference to the R-implicator and S-implicator in fuzzy settheory, Chen et al. [3] explored two types of fuzzy belief andplausibility functions, which are, respectively, the lower and up-per probabilities defined by a fuzzy belief space. They showed

that belief functions satisfy the subadditive property at anyorder.

It is well known that there are a lot of implicators [31] thathave been widely used in fuzzy sets research. It should be notedthat fuzzy inference results often depend upon the choice ofthe implicator and choice of the triangular norm. For analyzinguncertainty in complicated fuzzy systems, lower and upper fuzzyrough approximations defined by arbitrary implicators in roughset theory were examined in a number of studies [5], [21], [22],[28], [42], [54]. With reference to different requirements, moregeneral fuzzy belief functions associated with various fuzzyimplicators need to be developed to evaluate fuzzy events.

II. BRIEF NOTE ON RESULTS AND CONTRIBUTIONS

The purpose of this paper is to develop a general type offuzzy belief structure induced by a fuzzy implication operatorin an infinite universe of discourse and the fuzzy belief andplausibility functions through the generalization of the Shaferapproach to the fuzzy environment.

In the next section, we review some basic notions of fuzzylogical operators and fuzzy relations. In Section IV, we showsome useful properties of a fuzzy probability measure on an in-finite universe of discourse defined by Chen et al. [3]. We show,in particular, that Zadeh’s probability measures of fuzzy eventscan be extended to infinite countable universes of discourse[see (11)]. In Section V, we define generalized fuzzy roughsets determined by fuzzy implicators and examine propertiesof the fuzzy rough approximation operators. In Section VI, byusing an arbitrary implicator, we define a measure of inclusionon fuzzy sets (Definition 11). We further define a dual pair ofgeneralized fuzzy belief and plausibility functions constructedby a fuzzy belief structure and the measure of fuzzy inclusion(Definition 12). We then introduce the main novel contributions(Theorems 7, 8, and 10) of the present paper. In Theorem 7, weconsider a fuzzy belief space arising from a pair of lower andupper fuzzy rough approximation operators and prove that as-sociated with every fuzzy belief space, it is possible to obtain afuzzy belief structure. In Theorem 8, by employing generalizedfuzzy rough set theory, we show that for any belief structure,there must exist a countable approximation space and a proba-bility measure imposed on it such that the probabilities of lowerand upper approximations of a fuzzy set with respect to the ap-proximation space are just the belief and plausibility measuresof the fuzzy set in the given belief structure. That is to say,for every belief structure, there exists a fuzzy belief space suchthat the arising belief structure coincides with the given one. InTheorem 10, we prove that every belief or plausibility function isinfinitely monotone, i.e., fuzzy belief and plausibility functions,respectively, satisfy the essential properties of subadditivity andsuperadditivity at any order.

III. PRELIMINARIES

A. Fuzzy Logical Operators

A triangular norm, or t-norm in short, is an increasing, as-sociative, and commutative mapping T : [0, 1] × [0, 1] →


WU et al.: ON GENERALIZED FUZZY BELIEF FUNCTIONS IN INFINITE SPACES 387

[0, 1], satisfying the boundary condition: for all α ∈ [0, 1],T (α, 1) = α.

A triangular conorm (t-conorm in short) is an increasing,associative, and commutative mapping S: [0, 1] × [0, 1] →[0, 1], satisfying the boundary condition: for all α ∈ [0, 1],S(α, 0) = α.

A negator N is a decreasing [0, 1] → [0, 1] mapping sat-isfying N (0) = 1 and N (1) = 0. The negator Ns(α) = 1 − αis usually referred to as the standard negator. A negator N issaid to be involutive if and only if (iff) N (N (α)) = α for allα ∈ [0, 1]. It is well known that every involutive negator iscontinuous [17].

Given a negator N , a t-norm T and a t-conorm S are said bedual w.r.t. N iff the De Morgan’s laws are satisfied, i.e.

S(N (α),N (β)) = N (T (α, β)) ∀α, β ∈ [0, 1]

T (N (α),N (β)) = N (S(α, β)) ∀α, β ∈ [0, 1].

It is well known [17] that for an involutive negator Nand a t-conorm S, the function TS(α, β) = N (S(N (α),N (β))), α, β ∈ [0, 1], is a t-norm such that TS and S are dualw.r.t. N . In what follows, it will be referred to as a t-norm dualto S w.r.t. N .

Let X be a nonempty set called the universe of discourse. Bya fuzzy set A in X , we mean a mapping A : X → [0, 1]. Theclass of all subsets of X (respectively, all fuzzy sets in X) willbe denoted by P(X) [respectively, F(X)]. Zadeh’s fuzzy unionand fuzzy intersection will be denoted by ∪ and ∩, respectively,and∼N will be used to denote the fuzzy complement determinedby a negator N , i.e., for every A ∈ F(X) and every x ∈ X ,(∼N A)(x) = N (A(x)). IfN = Ns , we then write∼ A insteadof ∼N A.

By an implicator (fuzzy implication operator), we mean afunction I: [0, 1] × [0, 1] → [0, 1] satisfying I(1, 0) = 0and I(1, 1) = I(0, 1) = I(0, 0) = 1. An implicator I is calledleft monotonic (respectively, right monotonic) iff for every α ∈[0, 1], I(·, α) is decreasing (respectively, I(α, ·) is increasing).If I is both left monotonic and right monotonic, then it is calledhybrid monotonic. I is semicontinuous if

I(∨

j

aj ,∧k

bk

)=

∧j,k

I(aj , bk ) (1)

for all indexed families aj : j ∈ J and bk : k ∈ K of realnumbers in [0, 1].

It is easy to see that I(α, 1) = 1 for all α ∈ [0, 1], when Iis a left monotonic implicator, and if I is right monotonic, thenI(0, α) = 1 for all α ∈ [0, 1].

For a left monotonic implicator I, the function N (α) =I(α, 0), α ∈ [0, 1], is a negator, called a negator inducedby I. For example, the Łukasiewicz implicator IL(α, β) =min1, 1 − α + β induces the standard negator Ns .

Given a t-norm T , a t-conorm S, an implicator I, and twofuzzy sets A and B of a set X , we, respectively, define thecorresponding generalized union, intersection, and implication

of A and B as follows:

(A ∩T B)(x) = T (A(x), B(x)), x ∈ X

(A ∪S B)(x) = S(A(x), B(x)), x ∈ X

(A ⇒I B)(x) = I(A(x), B(x)), x ∈ X.

An implicator I is called a border implicator (or it satisfiesthe neutrality principle [6]) iff for every α ∈ [0, 1], I(1, α) = α.

An implicator I is called an exchange principle (EP) impli-cator ([1], [31]) if it satisfies for all α, β, γ ∈ [0, 1]

I(α, I(β, γ)) = I(β, I(α, γ)). (2)

An implicator I is called a confinement principle (CP) impli-cator ([6]) if it satisfies for all α, β ∈ [0, 1]

α ≤ β ⇐⇒ I(α, β) = 1. (3)

Several classes of implicators have been studied in the lit-erature. We recall here the definitions of two main classes ofoperators [6].

Let T , S, and N be a t-norm, a t-conorm, and a negator,respectively. An implicator I is called

1) an S-implicator based on S and N iff

I(α, β) = S(N (α), β) for all α, β ∈ [0, 1]. (4)

2) an R-implicator (residual implicator) based on a left-continuous t-norm T iff for all α, β ∈ [0, 1]

I(α, β) = supλ ∈ [0, 1] : T (α, λ) ≤ β. (5)

We will use θT to denote the R-implicator, i.e., θT (α, β) =supλ ∈ [0, 1] : T (α, λ) ≤ β.

From a t-conorm S, we define a binary operation σS on [0, 1]as follows: for all α, β ∈ [0, 1]

σS(α, β) = infλ ∈ [0, 1] : S(α, λ) ≥ β. (6)

If S is the dual t-conorm of a t-norm T w.r.t. Ns , then σS isdual to θT w.r.t. Ns [21], i.e., for all α, β ∈ [0, 1]

σS(1 − α, 1 − β) = 1 − θT (α, β). (7)

Theorem 1 [1], [28]: Every S-implicator and every R-implicator is a hybrid monotonic, border, and EP implicator.Every R-implicator is a CP implicator.

B. Fuzzy Relations

Let X be a nonempty universe of discourse. For any fuzzy setA ∈ F(X), the α-level set of A, denoted by Aα , is Aα = x ∈X : A(x) ≥ α, where α ∈ [0, 1]. In what follows, α denotesthe constant fuzzy set: α(x) = α for all x ∈ X .

Definition 1: Let U and W be two nonempty universes ofdiscourse. A subset R ∈ P(U × W ) is referred to as a (crisp)binary relation from U to W . The relation R is serial if for eachx ∈ U , there exists a y ∈ W such that (x, y) ∈ R. If U = W ,then R is called binary relation on U . A fuzzy subset R ∈F(U × W ) is referred to as a fuzzy binary relation from U toW , R(x, y) is the degree of relation between x and y, where(x, y) ∈ U × W . If for each x ∈ U , there exists a y ∈ W such



that R(x, y) = 1, then R is a serial fuzzy relation from U to W .If U = W , then R is called a fuzzy relation on U .

For a fuzzy relation R ∈ F(U × W ), we can define afuzzy-set-valued mapping F : U → F(W ) by F (u)(w) =R(u,w), (u,w) ∈ U × W. Conversely, for a given fuzzy-set-valued mapping F : U → F(W ), we can derive a fuzzy rela-tion R from U to W by setting R(u,w) = F (u)(w), (u,w) ∈U × W . In such a case, F can be viewed as a fuzzy compatibilityrelation between U and W . Similar to the crisp case [52], the im-age F (u) of an element u in U under the fuzzy-set-valued map-ping is called the fuzzy granule of u. It can be easily observedthat the fuzzy relation R is serial if and only if F : U → F(W )is a normal fuzzy-set-valued mapping, i.e., for each u ∈ U ,there exists w ∈ W such that F (u)(w) = 1. For a given fuzzy-set-valued mapping F : U → F(W ) (or a fuzzy binary relationfrom U to W ), we define an operator j : F(W ) → P(U)

j(A) = u ∈ U : F (u) = A, A ∈ F(W ). (8)

It can be easily shown that j satisfies the following.

J1) A = B =⇒ j(A) ∩ j(B) = ∅.

J2)⋃

A∈F(W )

j(A) = U.

IV. PROBABILITIES OF FUZZY SETS

The notion of a fuzzy event and its probability are first definedby Zadeh [57]. A fuzzy event is a fuzzy set whose membershipfunction is Borel measurable. Its probability of occurrence is de-fined by the expectation of the membership function. Recently,Chen et al. [3] defined probabilities of fuzzy sets in infiniteuniverses of discourse. In this section, we present the formaldefinition of probabilities of fuzzy sets in infinite universes ofdiscourse and give some basic properties of the fuzzy probabilitymeasure.

Throughout this section, we always assume that (U,A, P ) isa probability space, i.e., U is a nonempty set and may not befinite, A ⊆ P(U) is a σ-algebra on U , and P is a probabilitymeasure on U . The pair (U,A) is called a measurable space.

Definition 2: A fuzzy set A ∈ F(U) is said to be measurablew.r.t. (U,A) if A : U → [0, 1] is a measurable function w.r.t.A− B([0, 1]), where B([0, 1]) is the family of Borel sets on[0, 1]. We denote by F(U,A) the family of all measurable fuzzysets of U w.r.t. A− B([0, 1]).

For any measurable fuzzy set A ∈ F(U,A),Aα , clearlyAα ∈A for all α ∈ [0, 1], is a measurable set on the probabilityspace (U,A, P ) and P (Aα ) ∈ [0, 1]. Since f(α) = P (Aα ) ismonotone decreasing and left continuous, f(α) is integrable,we denote the integrand as

∫ 10 P (Aα )dα.

Definition 3: If a fuzzy set A is measurable w.r.t. (U,A) andP is a probability measure on (U,A). Denote

P (A) =∫ 1

0P (Aα )dα (9)

where P (A) is called the probability of A.For a singleton set x, we will write P (x) instead of P (x)

for short.

Theorem 2: The fuzzy probability measure P in Definition 3satisfies the following properties.

1) P (A) ∈ [0, 1] and P (A) + P (∼ A) = 1 for all A ∈F(U,A).

2) P is countably additive, i.e., for Ai ∈ F(U,A), i =1, 2, . . . , Ai ∩ Aj = ∅ ∀i = j, then

P

( ∞⋃i=1

Ai

)=

∞∑i=1

P (Ai). (10)

3) A,B ∈ F(U,A), A ⊆ B =⇒ P (A) ≤ P (B).4) If U = ui : i = 1, 2, . . . is an infinite countable set and

A = P(U), then for all A ∈ F(U)

P (A) =∫ 1

0P (Aα )dα =

∑x∈U

A(x)P (x). (11)

5) IfU is a finite set with |U | = n (where |U | is the cardinalityof the set U ), A = P(U), and P (u) = 1/n, then P (A) =∫ 1

0 P (Aα )dα = |A|/n for all A ∈ P(U).Proof: Properties 1)–3) are given in [3].4) For any A ∈ F(U), since U = ui : i = 1, 2, . . . is count-

able, the valued set A(x) : x ∈ U is also countable. Withoutloss of generality, we assume that A(x) : x ∈ U = αi : i =1, 2, . . . such that

0 ≤ α1 < α2 < · · · < αk < · · · ≤ 1.

Denote

α∞ =∞∨

k=1

αk .

Obviously, α∞ ≤ 1. Denote

Aαj = = x ∈ U : A(x) = αj.

Let P (Aαj =) = pj , then

Aαj= x ∈ U : A(x) ≥ αj =

∞⋃k=j

Aαk = (12)

and by (10), P is countably additive. Hence

P (Aαj) = P

( ∞⋃k=j

Aαk =

)=

∞∑k=j

P (Aαk =) =∞∑

k=j

pk . (13)

For any α ∈ (0, 1], if αk < α ≤ αk+1 , then

Aα = x ∈ U : A(x) ≥ α

=∞⋃

j=k+1

x ∈ U : A(x) = αj

=∞⋃

j=k+1

Aαj =

and hence

P (Aα ) =∞∑

j=k+1

P (Aαj =) =∞∑

j=k+1

pj = P (Aαk + 1). (14)



It follows that∫ αk + 1

αk

P (Aα )dα =∫ αk + 1

αk

P (Aαk + 1 )dα

= P (Aαk + 1 )(αk+1 − αk ). (15)

If α1 = 0, then of course∫ α1

0P (Aα )dα = 0.

If α1 > 0, when α ∈ (0, α1 ], it is clear that

Aα = x ∈ U : A(x) ≥ α1 = U

then P (Aα ) = P (U) = 1. Consequently∫ α1

0P (Aα )dα = α1P (U) = α1 = α1P (Aα1 ). (16)

If α∞ < 1, then for any α ∈ (α∞, 1], we have Aα = x ∈ U :A(x) ≥ α = ∅. Hence, P (Aα ) = 0, and as a result∫ 1

α∞

P (Aα )dα = 0. (17)

If α∞ = 1, then the aforementioned equation obviously holds.Since the integral function is a measure and the series∑x∈U A(x)P (x) is absolutely convergent, by employing (15)

and (16), we can prove that∞∑

j=1

αjpj = limk→∞

∫ αk

0P (Aα )dα =

∫ α∞

0P (Aα )dα. (18)

In terms of (17) and (18), we conclude that∫ 1

0P (Aα )dα =

∫ α∞

0P (Aα )dα +

∫ 1

α∞

P (Aα )dα

=∞∑

j=1

αjpj =∑x∈U

A(x)P (x).

5) It follows immediately from 4).It should be noted that (11) is very important and is used

throughout the rest of the paper. It also implies that Zadeh’sprobability measures [57] of fuzzy events can be extended toinfinite countable universes of discourse.

V. ROUGH SETS ON TWO UNIVERSES OF DISCOURSE

In this section, we introduce concepts related to rough sets ontwo universes of discourse in both the crisp and fuzzy environ-ments, and examine their basic properties.

A. Generalized Crisp Rough Sets on TwoUniverses of Discourse

Definition 4: Let U and W be two nonempty universes ofdiscourse. Suppose that R is an arbitrary crisp relation from Uto W . We define a set-valued function Rs : U → P(W ) by

Rs(x) = y ∈ W : (x, y) ∈ R, x ∈ U (19)

where Rs(x) is referred to as the successor neighborhoodof x w.r.t. R. Obviously, any set-valued function F fromU to W defines a binary relation from U to W by setting

R = (x, y) ∈ U × W : y ∈ F (x). The triple (U,W,R) is re-ferred to as a generalized crisp approximation space. For anyset A ∈ P(W ), the lower and upper approximations, denoted asR(A) and R(A), w.r.t. (U,W,R) are defined, respectively, by

R(A) =x ∈ U : Rs(x) ⊆ AR(A) =x ∈ U : Rs(x) ∩ A = ∅. (20)

The pair (R(A), R(A)) is referred to as a generalized crisp roughset, and R and R : P(W ) → P(U) are referred to as lower andupper generalized crisp approximation operators, respectively.

From the definition, many interesting properties of the lowerand upper approximation operators can be derived (see, e.g.,[25], and [48]–[50]).

B. Generalized Fuzzy Rough Sets Determinedby Fuzzy Implicators

Definition 5: Let U and W be two nonempty universes of dis-course and R a fuzzy relation from U to W . The triple (U,W,R)is called a fuzzy approximation space. When U = W and R is afuzzy relation on U , we also call (U,R) a fuzzy approximationspace.

Definition 6: Let (U,W,R) be a fuzzy approximation space, Ian implicator on [0, 1], and N a negator on [0, 1]. For a fuzzy setA ∈ F(W ), the I-lower approximation of A w.r.t. (U,W,R),denoted as RI(A), is a fuzzy set of U whose membership func-tion is defined by

RI(A)(x) =∧

y∈W

I(R(x, y), A(y)), x ∈ U. (21)

The I-upper approximation of A w.r.t. (U,W,R), denoted asRI(A), is defined by

RI(A) = ∼N RI(∼N A). (22)

The operators RI and RI from F(W ) to F(U) are referredto as I-upper and I-lower fuzzy rough approximation opera-tors of (U,W,R), respectively, and the pair (RI(A), RI(A)) iscalled the I-fuzzy rough set of A w.r.t. (U,W,R). Specially,if I is an S-implicator based on a t-conorm S and an involu-tive negator N , and T and S are dual w.r.t. N , then we call(RI(A), RI(A)) the S-fuzzy rough set of A w.r.t. (U,W,R).If I is an R-implicator based on a t-norm T , then we call(RI(A), RI(A)) the R-fuzzy rough set of A w.r.t. (U,W,R).

For simplicity, in the discussion to follow, we always takeN = Ns .

Theorem 3: Let (U,W,R) be a fuzzy approximation spaceand I a semicontinuous, hybrid monotonic implicator on[0, 1]. Then, the I-fuzzy rough approximation operators RIand RI in Definition 6 have the following properties: for allA,B ∈ F(W ), Aj ∈ F(W )(∀j ∈ J, J is an index set) and thefollowing hold.

FL1) RI(W ) = U.

FU1) RI(∅) = ∅.

FL2) RI

( ⋂j∈J

Aj

)=

⋂j∈J

RI(Aj ).



FU2) RI

( ⋃j∈J

Aj

)=

⋃j∈J

RI(Aj ).

FL3) A ⊆ B =⇒ RI(A) ⊆ RI(B).

FU3) A ⊆ B =⇒ RI(A) ⊆ RI(B).

FL4) RI

( ⋃j∈J

Aj

)⊇

⋃j∈J

RI(Aj ).

FU4) RI

( ⋂j∈J

Aj

)⊆

⋂j∈J

RI(Aj ).

Proof: The proofs of FL1)–FL4) are given in [42]. PropertiesFU1)–FU4) can be deduced from (22) and properties FL1)–FL4), respectively.

Theorem 4: Let (U,W,R) be a fuzzy approximation spaceand I a semicontinuous border implicator. If R is a serial fuzzyrelation from U to W , then the I-fuzzy rough approximationoperators in Definition 6 satisfy the following properties.

FLU0) RI(A) ⊆ RI(A)∀A ∈ F(W ).

FL0) RI(α) = α ∀α ∈ [0, 1].

FU0) RI(α) = α ∀α ∈ [0, 1].

FL0)′ RI(∅) = ∅.FU0)′ RI(W ) = U.

Proof: FLU0) Let A ∈ F(W ), for any x ∈ U , since R is aserial fuzzy relation from U to W , there exists y0 ∈ W such thatR(x, y0) = 1. It should be noted that I is a border implicator.Then, we have

RI(A)(x) + RI(∼ A)(x)

=∧

y∈W

I(R(x, y), A(y)) +∧

y∈W

I(R(x, y), 1 − A(y))

≤ I(R(x, y0), A(y0)) + I(R(x, y0), 1 − A(y0))

= I(1, A(y0)) + I(1, 1 − A(y0))

= A(y0) + 1 − A(y0) = 1.

It follows that

RI(A)(x) ≤ 1 − RI(∼ A)(x) = RI(A)(x).

Consequently, FLU0) holds.FL0) Let α ∈ [0, 1]. Since I is a semicontinuous border

implicator, for any x ∈ U , we have

RI(α)(x) =∧

y∈W

I(R(x, y), α)

= I( ∨

y∈W

R(x, y), α)

= I(1, α) = α = α(x).

Thus, FL0) holds.

FU0) Combining FL0) and (22), we have

RI(α) =∼ RI(∼ α) =∼ RI( 1 − α)

=∼ 1 − α =∼∼ α = α.

FL0)′ By taking α = 0 in FL0).FU0)′ By taking α = 1 in FU0).

VI. FUZZY BELIEF STRUCTURES AND FUZZY

BELIEF FUNCTIONS

In this section, we first review the main results of the originalDempster–Shafer theory of evidence with its existing extensionin the fuzzy environment. We then extend Dempster–Shafertheory of evidence to fuzzy sets in infinite universes of discourseby employing arbitrary fuzzy implication operators.

A. Belief Structures and Belief Functions in Crisp Environmentin Finite Spaces

The original Dempster–Shafer theory of evidence is based ona set function called the basic probability assignment on a finiteuniverse of discourse.

Definition 7 [32]: Let W be a nonempty finite set. A setfunction m : P(W ) → [0, 1] is referred to as a basic probabilityassignment on W if it satisfies the following axioms:

M1) m(∅) = 0 M2)∑

X⊆W

m(X) = 1.

The value m(X) represents the degree of belief that a specificelement of W belongs to set X , but not to any particular subsetof X . It is a relative level of confidence in X . It reflects theprobability that this information is properly and completely de-scribed by the set X [12]. A set X ∈ P(W ) with m(X) > 0 isreferred to as a focal element of m. We denote by M the familyof all focal elements of m. The pair (M,m) is called a beliefstructure [13], [46] or a body of evidence [16]. Obviously, mdefines a probability measure on P(W ), but not on W . Associ-ated with each belief structure, a pair of belief and plausibilityfunctions can be derived [32].

Definition 8: Let (M,m) be a belief structure on W . A setfunction Bel : P(W ) → [0, 1] is referred to as a belief functionon W if

Bel(X) =∑

A∈P(W ):A⊆X m(A) ∀X ∈ P(W ). (23)

A set function Pl : P(W ) → [0, 1] is referred to as a plausibilityfunction on W if

Pl(X) =∑

A∈P(W ):A∩X =∅m(A) ∀X ∈ P(W ). (24)

Bel(X) reflects the weight of evidence, which focuses onthe subsets of X . Belief and plausibility functions based on thesame belief structure are connected by the dual property

Pl(X) = 1 − Bel(∼ X). (25)

Thus, Pl(X) reflects the weight of evidence, which does notfocus on ∼ X , the opposite of X . Alternatively, Pl(X) quan-tifies the amount of our belief that might be allocated to X .



Moreover, Bel(X) + Bel(∼ X) ≤ 1 and Bel(X) ≤ Pl(X) forall X ∈ P(W ).

The belief function Bel on a finite set W can be equivalentlydefined as a set function from P(W ) to [0, 1] satisfying thefollowing properties [32].

MC1) Bel(∅) = 0.MC2) Bel(W ) = 1.MC3) Bel satisfies subadditive property, i.e., for any n ∈ N

(N is the set of positive integer numbers) and Xi ∈P(W ), i = 1, 2, . . . , n

Bel

( n⋃i=1

Xi

)≥

∑∅=I⊆1,2,...,n

(−1)|I |+1Bel

( ⋂i∈I

Xi

).

Likewise, Pl can be equivalently defined as a set function fromP(W ) to [0, 1] satisfying the following properties.

AC1) Pl(∅) = 0.AC2) Pl(W ) = 1.AC3) Pl satisfies superadditive property, i.e., for any n ∈ N

and Xi ∈ P(W ), i = 1, 2, . . . , n

Pl

( n⋂i=1

Xi

)≤

∑∅=I⊆1,2,...,n

(−1)|I |+1Pl

( ⋃i∈I

Xi

).

The corresponding basic probability assignment m is linkedwith Bel and Pl as follows:

m(A) =∑B⊆A

(−1)|A−B |Bel(B), A ∈ P(W )

Pl(A) = 1 − Bel(∼ A), A ∈ P(W ).

In the literature, a set function satisfying properties MC1)–MC3) is called a monotone Choquet capacity of infinite order,and it is said to be an alternating Choquet capacity of infiniteorder if it satisfies properties AC1)–AC3) [4]. Thus, MC1)–MC3) and AC1)–AC3) are, respectively, the essential propertiesof the belief and plausibility functions.

Definition 9: If U and W are two finite sets, R is a serialbinary relation from U to W , and P is a probability measure onU , then we call ((U,P ),W,R) a crisp belief space.

The original concepts of belief and plausibility functions inthe Dempster–Shafer theory of evidence come from the lowerand upper probabilities induced by a multivalued mapping [32].The following theorem shows that any belief structure with itsinducing belief and plausibility functions can be represented asprobabilities of lower and upper approximations derived froman approximation space in rough set theory.

Theorem 5 [43], [51]: Assume that ((U,P ),W,R) is a crispbelief space, and R and R are the lower and upper approximationoperators defined by (20). Denote

Bel(X) = P (R(X)), X ∈ P(W )

Pl(X) = P (R(X)), X ∈ P(W ). (26)

Then, Bel, Pl : P(W ) → [0, 1] are a dual pair of belief andplausibility functions on W , and the basic probability as-signment is defined by m(X) = P (j(X)),X ∈ P(W ), wherej(X) = u ∈ U : Rs(u) = X. Conversely, if (M,m) is a be-lief structure on W and Bel, Pl : P(W ) → [0, 1] are the belief

and plausibility functions derived by (M,m), then there existsa crisp belief space ((U,P ),W,R) (i.e., there exists a finite setU , a probability measure P on U , and a serial binary relation Rfrom U to W ) such that

Bel(X) = P (R(X)), X ∈ P(W )

Pl(X) = P (R(X)), X ∈ P(W )

m(X) = P (j(X)), X ∈ P(W ) (27)

where j(X) = u ∈ U : Rs(u) = X.

B. Belief Structures and Belief Functionsin the Fuzzy Environment

Based on the concept of information granularity and thetheory of possibility [55], [56], Zadeh first generalized theDempster–Shafer theory to fuzzy sets. First of all, the beliefstructure should be generalized to the fuzzy environment.

Definition 10: Let W be a nonempty set that may be infinite.A set function m : F(W ) → [0, 1] is referred to as a fuzzy basicprobability assignment if it satisfies the following axioms.

FM1) m(∅) = 0.

FM2)∑

X∈F(W )

m(X) = 1.

A fuzzy set X ∈ F(W ) with m(X) > 0 is referred to as afocal element of m. We denote by M the family of all focal ele-ments of m. The pair (M,m) is called a fuzzy belief structure.In the discussion to follow, all the focal elements are supposedto be normal, i.e., for any A ∈ M, there exists an x ∈ W suchthat A(x) = 1.

We recall that a crisp belief function Bel : P(W ) → [0, 1]defined from a crisp belief structure can be represented as

Bel(X) =∑A⊆X

m(A) =∑

A∈P(W )

m(A)I(A,X) (28)

where I(A,X) has to be understood as the inclusion degree ofA in X , which is 1 if A ⊆ X , and 0 otherwise. Based on a fuzzybelief structure (M,m) on a finite universe of discourse W ,Zadeh defined the expected certainty, denoted by EC(X), andthe expected possibility, denoted by EΠ(X), as a generalizationof Dempster–Shafer belief and plausibility functions: for allX ∈ F(W )

EC(X) =∑

A∈Minf(A ⇒ X) = 1 − EΠ(∼ X) (29)

EΠ(X) =∑A∈M

m(A) sup(X ∩ A) (30)

where inf(A ⇒ X) measures the degree to which A is includedin X and sup(X ∩ A) measures the degree that X intersectswith A. It is easy to verify that the expected certainty and theexpected possibility degenerate into the crisp belief and plau-sibility functions when the belief structure (M,m) and X arecrisp. In order to define the belief degree of the elements ofF(W ), (28) is very useful because it is enough to substitutein it a suitable extension of the inclusion degree I(A,X) to



F(W ) ×F(W ). So, the crucial point of the problem becomesthe search for an extension of the inclusion relation to F(W ),which allows us to express within the fuzzy framework the well-known theory of belief measures [2].

Following Zadeh’s work, Ishizuka et al. [15], Ogawa andFu [23], Yager [46], and recently Biacino [2] have extended theDempster–Shafer theory to fuzzy sets by defining a measure ofinclusion I(A ⊆ X), the degree to which a set A is included inthe set X by using the following formula, which is similar toZadeh’s expected certainty EC(X)

Bel(X) =∑

A∈Mm(A)I(A ⊆ X), X ∈ F(W ). (31)

Ishizuka et al. [15] and Yager [46], respectively, used theŁukasiewicz implicator IL(x, y) = min1, 1 − x + y and theKleene–Dienes implicator IKD(x, y) = max1 − x, y (bothare S-implicators) in fuzzy set theory to define the inclu-sion measure. Analogous to conditional probability, Ogawa andFu [23] used relative sigma count to compute the degree of inclu-sion. Although these extensions to the Dempster–Shafer theoryof evidence arrive at frameworks within which both probabilis-tic information and vague information can be managed, theuniverses of discourse are all finite. Moreover, we are not surewhether or not these belief and plausibility functions satisfyaxioms similar to MC3) and AC3) in the fuzzy environment.Biacino [2] recently used the Kleene–Dienes implicator to cal-culate the degree of inclusion in an infinite universe of discourse.As a result, a fuzzy belief function on the infinite universe ofdiscourse was defined. However, all these extensions have notshown whether or not the belief and plausibility functions canbe presented as lower and upper probabilities similar to the caseof crisp sets.

By using a general type of implicator, we will define a gener-alized fuzzy belief structure and its induced dual pair of fuzzybelief and plausibility functions in an infinite universe of dis-course. We first introduce a general type of inclusion degree infuzzy sets.

Definition 11: Let W be a nonempty universe of discourse thatmay be infinite and I an implicator on [0, 1], for A,B ∈ F(W ).We define

I(A ⊆ B) =∧

x∈W

I(A(x), B(x))

=∧

x∈W

(A ⇒I B)(x). (32)

Theorem 6: If I is a hybrid monotonic, semicontinuous,and border implicator on [0, 1], and A,B,C ∈ F(W ), then thefollowing hold.

1) B ⊆ C =⇒ I(A ⊆ B) ≤ I(A ⊆ C).2) B ⊆ C =⇒ I(C ⊆ A) ≤ I(B ⊆ A).3) If I is a CP implicator, then

I(A ⊆ B) = 1 ⇐⇒ A ⊆ B.

4) If A is a normal fuzzy set, then the function NI,A :F(W ) → [0, 1] defined by

NI,A (B) = I(A ⊆ B), B ∈ F(W ) (33)

is an extended necessity measure on W in the senseof Abdel–Hamid and Morsi [1], i.e., it satisfies axiomsENM1) and ENM2).

ENM1) N(⋂

j∈J

Bj ) =∧

j∈J

N(Bj ), for every index family

Bj : j ∈ J ⊆ F(W ).

ENM2) N(α) = α for all α ∈ [0, 1].Proof: 1) and 2) can be concluded immediately by the hybrid

monotonicity of I.3) Recall from [42] that I is a CP implicator iff for all (a, b) ∈

[0, 1] × [0, 1]

a ≤ b ⇐⇒ I(a, b) = 1.

Then

I(A ⊆ B) =∧

x∈W

I(A(x), B(x)) = 1

⇐⇒ I(A(x), B(x)) = 1, ∀x ∈ W

⇐⇒ A(x) ≤ B(x), ∀x ∈ W

⇐⇒ A ⊆ B.

4) is obtained in [1] and [10].In the discussion to follow, we always assume that I is a

hybrid monotonic, semicontinuous, and border implicator on[0, 1], and we take N = Ns .

Lemma 1 [2]: Let (M,m) be a fuzzy belief structure on W .Then, the focal elements of m constitute a countable set.

Lemma 1 is quite important and instrumental to achieve themain results of this paper. It is possible for us to define beliefand plausibility functions on an arbitrary infinite universe ofdiscourse.

Definition 12: Let W be a nonempty universe of discoursethat may be infinite, (M,m) a fuzzy belief structure on W , andI an implicator on [0, 1]. A fuzzy set function Bel : F(W ) →[0, 1] is referred to as a generalized fuzzy belief function on Wif for all X ∈ F(W )

Bel(X) =∑

A∈F(W )

m(A)I(A ⊆ X)

=∑

A∈F(W )

m(A)∧

x∈W

I(A(x),X(x)). (34)

The fuzzy set function Pl : F(W ) → [0, 1] is referred to as ageneralized fuzzy plausibility function on W

Pl(X) = 1 − Bel(∼ X), X ∈ F(W ). (35)

It is worthy of note that, by Lemma 1, the not null elementsthat appear under the sign of sum are, for every fixed X ∈F(W ), finite or countable, and since I(A ⊆ X) ≤ 1 for everyA and X ∈ F(U), the series that defines Bel(X) in (34) isconvergent. So, Definition 12 is reasonable.

By taking different implicators and belief structures, we canobtain various classes of dual pairs of fuzzy belief and plausi-bility functions.

Example 1: When I is an S-implicator based on a t-conorm Sand T is a t-norm dual to S w.r.t. Ns , it can be checked that [46]



for X ∈ F(W )

Bel(X) =∑

A∈F(W )

m(A)∧

x∈W

S(1 − A(x),X(x)) (36)

Pl(X) =∑

A∈F(W )

m(A)∨

x∈W

T (A(x),X(x)). (37)

Example 2: When I = θT is an R-implicator determined bya t-norm T and its dual t-conorm S w.r.t. Ns , then it can bechecked that for X ∈ F(W )

Bel(X) =∑

A∈F(W )

m(A)∧

x∈W

θT (A(x),X(x)) (38)

Pl(X) =∑

A∈F(W )

m(A)∨

x∈W

σS(A(x),X(x)). (39)

Example 3: When W is a finite universe of discourse, (M,m)is a crisp belief structure, and I is the Kleene–Dienes implicator,then [9], [11], [36] for X ∈ F(W )

Bel(X) =∑

A :A∈Mm(A)

∧x∈W

X(x) (40)

Pl(X) =∑

A :A∈Mm(A)

∨x∈W

X(x). (41)

Example 4: When W is a finite universe of discourse and(M,m) is a crisp belief structure, then

Bel(X) =∑

A∈M:A⊆X m(A), X ∈ P(W ) (42)

Pl(X) =∑

A∈M:A∩X =∅m(A), X ∈ P(W ). (43)

Definition 13: If U is a countable set, P is a probabilitymeasure on U , W is a nonempty set that may be infinite, andR is a serial fuzzy binary relation from U to W , then we call((U,P ),W,R) a fuzzy belief space.

Theorem 7 next shows that any fuzzy belief space can be asso-ciated with a fuzzy belief structure such that the probabilities ofthe lower and upper fuzzy rough approximations induced fromthe fuzzy belief space produce, respectively, the correspondingfuzzy belief and plausibility functions derived from the fuzzybelief structure.

Theorem 7: Let ((U,P ),W,R) be a fuzzy belief space inwhich U is a countable set and I an implicator. If RI and RIare the I-fuzzy rough approximation operators in Definition 6,denote

Bel(X) = P (RI(X)), X ∈ F(W )

Pl(X) = P (RI(X)), X ∈ F(W ). (44)

Then, Bel : F(W ) → [0, 1] and Pl : F(W ) → [0, 1] are a dualpair of fuzzy belief and plausibility functions.

Proof: Let

j(A) = x ∈ U : R(x, y) = A(y) ∀y ∈ W, A ∈ F(W ).

It can be easily checked that j satisfies properties J1) and J2).

J1) A = B =⇒ j(A) ∩ j(B) = ∅.

J2)⋃

A∈F(W )

j(A) = U.

Since R is serial, we can observe that j(∅) = ∅. Consequently,P (j(∅)) = P (∅) = 0 and∑

A∈F(W )

P (j(A)) = P

( ⋃A∈F(W )

j(A))

= P (U) = 1.

Define a fuzzy set function m : F(W ) → [0, 1] as follows:

m(A) = P (j(A)), A ∈ F(W ).

Obviously, m(∅) = 0 and∑A∈F(W )

m(A) =∑

A∈F(W )

P (j(A))

= P

( ⋃A∈F(W )

j(A))

= P (U) = 1.

Hence, m is a fuzzy basic probability assignment on W . Andfor any X ∈ F(W ), by (11), J1), and J2), we have

Bel(X) = P (RI(X)) =∑x∈U

RI(X)(x)P (x)

=∑x∈U

∧y∈W

I(R(x, y),X(y))P (x)

=∑

A∈F(W )

∑x∈j (A)

∧y∈W


=∑

A∈F(W )

∧y∈W

I(A(y),X(y))P (j(A))

=∑

A∈F(W )

[m(A)

∧y∈W

I(A(y),X(y))].

Therefore, we have proved that Bel is a fuzzy belief function.By the duality, we can conclude that Pl is a fuzzy plausibilityfunction dual to Bel.

Theorem 7 provides us a potential tool for the reasoning andknowledge acquisition in fuzzy information systems.

Example 5: Given a fuzzy information system (U,AT, F ),where U = u1 , u2 , . . . , un is the object set, AT =a1 , a2 , . . . , am is the attribute set, F = fa : a ∈ AT is aset of information functions such that fa : U → F(Va), and Va

is called the domain of attribute a. Let P be a probability mea-sure on U (if there is no probability measure on U , we cantake P (X) = |X|/|U | = |X|/n). Each attribute set B ⊆ ATcorresponds to a fuzzy binary relation RB ∈ F(U × U) onthe object set. Such binary relation is often reflexive (see,e.g., [14], [40], and [44]), and therefore, each attribute set Bcan be associated with a belief structure (MB ,mB ) on U , andmB (X) = P (jB (X)) for all X ∈ F(U), where

jB (X) = x ∈ U : RB (x, y) = X(y),∀y ∈ U.



If we write RB (x, y) = RB (x)(y), then jB can be equivalentlywritten as

jB (X) = x ∈ U : RB (x) = X.

Obviously, RB (x) is a normal fuzzy subset of U for all x(RB (x)(x) = 1). Thus, each focal element of mB is nor-mal. Moreover, if B = B1 ∪ B2 , then RB = RB1 ∩ RB2 andmBi

(X) = P (jBi(X)) for all X ∈ F(U), i = 1, 2. It should

be noted that RB (x) = RB1 (x) ∩ RB2 (x). Then

mB (X) = mB1 ⊕ mB2 (X)

= P (x ∈ U : RB1 (x) ∩ RB2 (x) = X).

Thus, the combination of two fuzzy basic probability assign-ments (which is one of the main issues in the study of theDempster–Shafer theory of evidence) can be transformed intothe calculation of the intersection of the corresponding fuzzybinary relations. On the other hand, just as crisp belief and plau-sibility functions can be used to study knowledge reduction incomplete and incomplete information systems [19], [41], [45],[58], we believe that the Dempster–Shafer theory of evidencefor fuzzy sets might be used to analyze knowledge reduction infuzzy information systems and fuzzy decision systems.

Theorem 8 next shows that for any fuzzy belief structure,there exists a fuzzy belief space such that fuzzy belief andplausibility functions induced from the fuzzy belief structurecan be represented by the probabilities of lower and upper fuzzyrough approximations determined by the fuzzy belief space.

Theorem 8: Let (M,m) be a fuzzy belief structure on Wand I an implicator on [0, 1]. If Bel : F(W ) → [0, 1] and Pl :F(W ) → [0, 1] are the dual pair of fuzzy belief and plausibilityfunctions defined in Definition 12, then there exists a countableset U , a serial fuzzy relation R from U to W , and a probabilitymeasure P on U such that for all X ∈ F(W )


RI(X)(x)P (x) (45)

Pl(X) = P (RI(X)) =∑x∈U

RI(X)(x)P (x). (46)

Proof: Since∑

A∈F(W ) m(A) = 1, by Lemma 1, we knowthat the focal elements of m constitute a countable set. Withoutloss of generality, we assume that M has infinite countableelements and we denote

M = Ai ∈ F(W ) : i ∈ N

where∑

i∈N m(Ai) = 1. Let U = ui : i ∈ N be a set hav-ing infinite countable elements. We define a set function P :P(U) → [0, 1] as follows:

P (ui) = m(Ai), i ∈ N

P (X) =∑u∈X

P (u), X ∈ P(U).

Obviously, P is a probability measure on U .We further define a fuzzy relation R from U to W as follows:

R(ui, w) = Ai(w), i ∈ N, w ∈ W.

From R, we can obtain a mapping j : F(W ) → P(U) asfollows:

j(A)= u ∈ U : R(u,w) = A(w) ∀w ∈ W, A ∈ F(W ).

It is easy to see that j(A) = ui for A = Ai , and ∅ otherwise.Consequently, m(A) = P (j(A)) > 0 for A ∈ M, and 0 other-wise. It should be noted that j(A) ∩ j(B) = ∅ for A = B and⋃

A∈F(W ) j(A) = U. Then, by Theorem 2, we can conclude thatfor any X ∈ F(W )

P (RI(X)) =∑x∈U

RI(X)(x)P (x)

=∑x∈U

∧y∈W


=∑

A∈F(W )

∑x∈j (A)

∧y∈W


=∑

A∈F(W )

∧y∈W

I(A(y),X(y))P (j(A))

=∑

A∈F(W )

[m(A)

∧y∈W

I(A(y),X(y))]

= Bel(X).

By the duality property, we have, on the other hand

Pl(X) = 1 − Bel(∼ X) = 1 − P (RI(∼ X))

= 1 − P (∼ RI(X)) = P (RI(X)).

Similar to Definitions 10 and 13, the definitions of belief struc-ture and belief space in the crisp environment can be extendedto infinite spaces (we omit the definitions here). By regarding acrisp set as special fuzzy set, according to Theorems 7 and 8,we can arrive at Corollary 1, which is a generalization to infinitecases of Theorem 5.

Corollary 1: Assume that ((U,P ),W,R) is a crisp beliefspace in which U is a countable set and W may be infinite,and R and R are the lower and upper approximation operatorsdefined by (20). Denote

Bel(X) = P (R(X)), X ∈ P(W )

Pl(X) = P (R(X)), X ∈ P(W ). (47)

Then, Bel, Pl : P(W ) → [0, 1] are a dual pair of belief and plau-sibility functions on W , and the basic probability assignmentis defined by m(X) = P (j(X)),X ∈ P(W ), where j(X) =u ∈ U : Rs(u) = X. Conversely, if (M,m) is a belief struc-ture on W in which W may be infinite, and Bel, Pl : P(W ) →[0, 1] the belief and plausibility functions derived by (M,m),then there exists a crisp belief space ((U,P ),W,R) (i.e., thereexists a countable set U , a probability measure P on U , and aserial binary relation R from U to W ) such that (47) holds andthe corresponding basic probability assignment is defined as

m(X) = P (j(X)), X ∈ P(W )

where j(X) = u ∈ U : Rs(u) = X.



Theorem 9: Let (M,m) be a fuzzy belief structure on W , Ia border implicator on [0, 1]. Then

1) Bel(X) ≤ Pl(X) ∀X ∈ F(W ).2) Bel(X) + Bel(∼ X) ≤ 1 ∀X ∈ F(W ).Proof: 1) By Theorem 8, we can find a fuzzy belief space

((U,P ),W,R) such that Bel(X) = P (RI(X)) and Pl(X) =P (RI(X)) for all X ∈ F(W ). Since R is a serial fuzzy re-lation, by Theorem 4, we know that RI(X) ⊆ RI(X), thenP (RI(X)) ≤ P (RI(X)), i.e., Bel(X) ≤ Pl(X).

2) can be directly concluded from 1) and (35).Definition 14: Let W be a nonempty set that may be infinite

and n ∈ N. A fuzzy set function F : F(W ) → [0, 1] is called afuzzy monotone Choquet capacity of order n on W if it satisfiesthe following axioms.

FMC1) F (∅) = 0.

FMC2) F (W ) = 1.

FMC3) F

( n⋃i=1

Ai

)≥

∑∅=J⊆1,2,...,n

(−1)|J |+1F

( ⋂i∈J

Ai

)for all Ai ∈ F(W ), i = 1, 2, . . . , n.

A fuzzy set function F : F(W ) → [0, 1] is called a fuzzy mono-tone Choquet capacity of infinite order if F is a fuzzy monotoneChoquet capacity of order n for all n ∈ N.

A fuzzy set function F : F(W ) → [0, 1] is called a fuzzyalternating Choquet capacity of order n on W if it satisfies thefollowing axioms.

FAC1) F (∅) = 0.

FAC2) F (W ) = 1.

FAC3) F

( n⋂i=1

Ai

)≤

∑∅=J⊆1,2,...,n

(−1)|J |+1F

( ⋃i∈J

Ai

)for all Ai ∈ F(W ), i = 1, 2, . . . , n.

A fuzzy set function F : F(W ) → [0, 1] is called a fuzzy al-ternating Choquet capacity of infinite order if F is a fuzzyalternating Choquet capacity of order n for all n ∈ N.

The following lemma can be deduced directly from the defi-nition of the α-level set of a fuzzy set.

Lemma 2: Let W be a nonempty set and Ai ∈ F(W ), i =1, 2, . . . , n. Then, for all α ∈ [0, 1], the following hold.

1)( n⋃

i=1Ai

)α

=n⋃

i=1(Ai)α .

2)( n⋂

i=1Ai

)α

=n⋂

i=1(Ai)α .

Theorem 10 next shows that fuzzy belief and plausibilityfunctions determined by an implicator, respectively, satisfy thesubadditive and superadditive properties at any order.

Theorem 10: Let W be a nonempty set that may be infinite andI a semicontinuous implicator on [0, 1]. If Bel, Pl : F(W ) →[0, 1] are the fuzzy belief and plausibility functions inducedfrom a fuzzy belief structure (M,m), then Bel is a fuzzy mono-tone Choquet capacity of infinite order on W and Pl is a fuzzyalternating Choquet capacity of infinite order on W .

Proof: By Theorem 8, there exists a countable set U , a serialfuzzy relation R from U to W , and a probability measure P onU such that for all X ∈ F(W )


RI(X)(x)P (x)

Pl(X) = P (RI(X)) =∑x∈U

RI(X)(x)P (x).

Then, by Theorems 3 and 4, we have the following.

FMC1) Bel(∅) = P (RI(∅)) = P (∅) = 0.

FMC2) Bel(W ) = P (RI(W )) = P (U) = 1.

FMC3) For any setsXi ∈ F(W ), i = 1, 2, . . . , n, n ∈ N,

we have

Bel

( n⋃i=1

Xi

)= P

(RI

( n⋃i=1

Xi

))

≥ P

( n⋃i=1

RI(Xi))

(by FL4) in Theorem 3)

=∫ 1

0P

(( n⋃i=1

RI(Xi))

α

)dα (by Definition 3)

=∫ 1

0P

( n⋃i=1

(RI(Xi))α

)dα (by Lemma 2)

=∫ 1

0

∑∅=J⊆1,2,...,n

(−1)|J |+1P

( ⋂j∈J

(RI(Xj ))α

)dα

(by property of crisp probability measure)

=∑

∅=J⊆1,2,...,n(−1)|J |+1

∫ 1

0P

(( ⋂j∈J

RI(Xj ))

α

)dα

(by Lemma 2)

=∑

∅=J⊆1,2,...,n(−1)|J |+1

∫ 1

0P

((RI

( ⋂j∈J

Xj

))α

)dα

(by FL2) in Theorem 3)

=∑

∅=J⊆1,2,...,n(−1)|J |+1P

(RI

( ⋂j∈J

Xj

))(by Definition 3)

=∑

∅=J⊆1,2,...,n(−1)|J |+1Bel

( ⋂j∈J

Xj

)(by Theorem 8).

Thus, we have proved that Bel is a fuzzy monotone Choquetcapacity of infinite order on W .

Similarly, we can prove that Pl is a fuzzy alternating Choquetcapacity of infinite order on W .

In [2, Corollary 4.4], it was proved that, in general, everyinfinitely monotone map is a lower envelope, so by Theorem 10,the same can be said of every fuzzy belief function defined by asystem of fuzzy focal events and a fuzzy implicator.



VII. CONCLUSION

We have defined a general type of fuzzy belief and plausibilityfunctions in infinite spaces by constructing a fuzzy belief struc-ture and a measure of fuzzy inclusion determined by a fuzzyimplication operator. Many existing fuzzy belief and plausibil-ity functions in the literature can be treated as special cases ofour definitions by taking specific fuzzy implication operators.We have shown that the fuzzy belief and plausibility functionsdefined by a fuzzy belief structure can be represented as thelower and upper probabilities in a countable set induced bythe fuzzy belief space. We have also investigated properties ofthe fuzzy belief and plausibility functions. The essential prop-erties are that the fuzzy belief and plausibility functions are,respectively, the fuzzy monotone Choquet capacity and fuzzyalternating Choquet capacity of infinite order.

Our results offer several advantages over previous work. First,semantics of the original Dempster–Shafer theory of evidenceis still maintained. Of greater significance, the fuzzy belief andplausibility functions on infinite universes of discourse can berepresented as the lower and upper fuzzy probabilities in a count-able set. Second, we have provided a useful class of fuzzy beliefand plausibility functions for real applications. One can selectdifferent fuzzy implication operators with reference to variousrequirements to define fuzzy belief and plausibility functions inthe analysis of uncertain information in complicated intelligentsystems. Third, since the fuzzy belief and plausibility functionscan be interpreted as the lower and upper approximations inrough set theory, on one hand, rough set theory may be regardedas the basis of the Dempster–Shafer theory of evidence. On theother hand, just as crisp belief and plausibility functions can beemployed in knowledge reduction in complete and incompleteinformation systems [41], [45], [58], the Dempster–Shafer the-ory of evidence for fuzzy sets might provide a potentially usefultool for reasoning and knowledge acquisition in fuzzy systemsand fuzzy decision systems.

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Wei-Zhi Wu received the B.Sc. degree in mathemat-ics from Zhejiang Normal University, Jinhua, China,in 1986, the M.Sc. degree in mathematics from EastChina Normal University, Shanghai, China, in 1992,and the Ph.D. degree in applied mathematics fromXi’an Jiaotong University, Xi’an, China, in 2002.

He is currently a Professor of mathematics inthe School of Mathematics, Physics, and Informa-tion Science, Zhejiang Ocean University, Zhoushan,China. His current research interests include approxi-mate reasoning, rough sets, random sets, formal con-

cept analysis, and granular computing. He has authored or coauthored twomonographs and more than 60 articles in international journals and book chap-ters. He serves in the editorial boards of several international journals.

Yee Leung received the B.S.Sc. degree in geographyfrom The Chinese University of Hong Kong, Shatin,Hong Kong, in 1972, the M.Sc. and Ph.D. degrees ingeography, in 1974 and 1977, respectively, and theM.S. degree in engineering in 1977 from the Univer-sity of Colorado, Boulder.

He is currently a Professor of geography in the De-partment of Geography and Resource Management,The Chinese University of Hong Kong. His currentresearch interests include the development and ap-plications of intelligent spatial decision support sys-

tems, spatial data mining, fuzzy sets and logic, rough sets, concept lattices,neural networks, and evolutionary computation. He has authored or coauthoredfive monographs and more than 150 articles in international journals and bookchapters. He serves in the editorial boards of several international journals.

Ju-Sheng Mi received the B.Sc. degree in mathe-matics from Hebei Normal University, Hebei, China,in 1986, the M.Sc. degree in mathematics from EastChina Normal University, Shanghai, China, in 1992,and the Ph.D. degree in applied mathematics fromXi’an Jiaotong University, Xi’an, China, in 2003.

He is currently a Professor in the College of Math-ematics and Information Science, Hebei Normal Uni-versity. His current research interests include approx-imate reasoning, rough sets, concept lattices and ran-dom sets. He has authored or coauthored more than

50 articles in international journals and book chapters.



Fault Detection for Fuzzy Systems With IntermittentMeasurements

Yan Zhao, James Lam, Senior Member, IEEE, and Huijun Gao, Member, IEEE

Abstract—This paper investigates the problem of fault detectionfor Takagi–Sugeno (T–S) fuzzy systems with intermittent measure-ments. The communication links between the plant and the fault de-tection filter are assumed to be imperfect (i.e., data packet dropoutsoccur intermittently, which appear typically in a network environ-ment), and a stochastic variable satisfying the Bernoulli randombinary distribution is utilized to model the unreliable communica-tion links. The aim is to design a fuzzy fault detection filter suchthat, for all data missing conditions, the residual system is stochasti-cally stable and preserves a guaranteed performance. The problemis solved through a basis-dependent Lyapunov function method,which is less conservative than the quadratic approach. The re-sults are also extended to T–S fuzzy systems with time-varyingparameter uncertainties. All the results are formulated in the formof linear matrix inequalities, which can be readily solved via stan-dard numerical software. Two examples are provided to illustratethe usefulness and applicability of the developed theoretical results.

Index Terms—Basis-dependent Lyapunov functions, fault de-tection, intermittent measurements, Takagi–Sugeno (T–S) fuzzysystems, uncertainties.

I. INTRODUCTION

IN CONTROL systems, due to the unexpected variationsin external surroundings, normal wear in components, or

sudden changes in signals, there may appear different kindsof malfunction or imperfect behavior in normal operations, andpeople call them faults. Since a fault can degrade a system’s per-formance and even cause catastrophic accidents, it is of greatsignificance to detect it in time for the safety and reliability ofcontrol systems. The objective of fault detection is to detect thefault signal accurately whenever it appears. Many researchershave devoted themselves to investigating this problem, and a lotof methods have been established, mainly including the model-based fault detection approach [2], [4], the parameter estima-tion approach [24], and the generalized likelihood method [30].Among these methods, the model-based one is very popular,which is to design a fault detection filter or observer generating aresidual including a threshold to detect the fault signal. In virtue

Manuscript received April 15, 2008; revised August 26, 2008 and November5, 2008; accepted December 24, 2008. First published February 10, 2009;current version published April 1, 2009. This work was supported in part bythe National Natural Science Foundation of China (60825303, 60834003), inpart by the 973 Project (2009CB320600), in part by the Research Fund forthe Doctoral Programme of Higher Education of China (20070213084), in partby the Heilongjiang Outstanding Youth Science Fund (JC200809), and in partby Postdoctoral Science Foundation of China (200801282), and in part by theUniversity of Hong Kong Research and Conference Grants (CRCG) under Grant200707176077.

Y. Zhao and H. Gao are with the Space Control and Inertial TechnologyResearch Center, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]; [email protected]).

J. Lam is with the Department of Mechanical Engineering, University ofHong Kong, Hong Kong (e-mail: [email protected]).


of the advancement of modeling and state estimation techniques[6], [7], [19], [20], model-based fault detection has been welldeveloped [26]. To mention a few, fault detection problems havebeen investigated for sampled-data systems in [9] and [32], un-certain systems in [1], [11], and [33], systems with time delaysin [10], and Markovian jump linear systems in [14].

Most of the aforementioned results are concerned with linearmodels. But in reality, most physical systems are nonlinear, andthus, how to develop effective fault detection methods for non-linear systems is an important and practical problem. However,the difficulty in modeling nonlinearities makes fault detectiona hard task. To solve this problem, some researchers model thenonlinear plants as differential equations and solve the fault de-tection problem based on the conventional nonlinear system the-ory [16], whose limitations often do not generalize the obtainedresults. Other researchers take the advantage of artificial intel-ligence techniques [23], and use the conventional fuzzy modelsto represent the nonlinear systems by applying the inferenceengine. It is worth mentioning that there are no systematic andconsistent approaches for the stability and performance analysisof those conventional fuzzy systems, and hence, the applicabil-ity of those results is also limited.

In recent years, Takagi–Sugeno (T–S) fuzzy models are play-ing more and more important roles in dealing with problemsconcerning nonlinear systems [3]. It has been proven that T–Sfuzzy systems with affine terms can smoothly approximate anynonlinear functions to any specified accuracy within any com-pact set, which provides a theoretical foundation for using T–Sfuzzy models to represent complex nonlinear systems approx-imately. Meanwhile, T–S fuzzy models formulate the complexnonlinear systems into a framework that interpolates some affinelocal models by a set of fuzzy membership functions. Based onthis framework, a systematic analysis and design procedure forcomplex nonlinear systems can be possibly developed in viewof the powerful control theories and techniques in linear sys-tems. The T–S fuzzy model has attracted great interests fromresearchers, and a number of results have been reported in lit-eratures, including stability analysis [13], [27], stabilizing andH∞ control design [12], [21], [34], and state estimation [35].Since T–S fuzzy models have provided a convenient way tostudy nonlinear systems, a feasible solution of the fault detec-tion problem for nonlinear systems can be converted to that offault detection for T–S fuzzy systems [17].

On the other hand, data packet dropout phenomena may oftenappear in many practical situations, i.e., some measurementsor control inputs may be lost during the transmission. Thisproblem has attracted more and more attention as the insertionof networked control systems (NCSs) in the control loopsbecomes popular [15]. Compared with the traditional point-to-point communication bus, NCSs have several advantages such

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ZHAO et al.: FAULT DETECTION FOR FUZZY SYSTEMS WITH INTERMITTENT MEASUREMENTS 399

as low cost, reduced weight and power requirements, simpleinstallation and maintenance, and high reliability. However,in an NCS, since several components communicate over ashared network, information flows are prone to the curse oftime sharing, and data loss always inevitably occurs from theplant to the filter or controller. Since data packet dropout candegrade a system’s performance and even cause instability,it has been regarded as an important issue in the analysisand synthesis of network-based control systems, and someresearchers have begun to study various problems of controlsystems, simultaneously considering this communicationissue [28]. It is noted that most investigations concerning thedata loss phenomenon are focused on the stability analysis andsynthesis, and the plants are mostly linear. To the best of theour knowledge, there are no results about the fault detectionproblem for nonlinear systems with intermittent measurements.

Motivated by the aforementioned observations, in this pa-per, we investigate the problem of fault detection for T–S fuzzysystems with intermittent measurements. The measurements be-tween the plant and the fault detection filter are assumed to beintermittent, and a stochastic variable is utilized to describethe imperfect communication links. Attention is focused on thefuzzy fault detection filter design such that the residual sys-tem is stochastically stable with the prescribed performance. Abasis-dependent Lyapunov function is utilized in the derivativeprocess, which renders the results to be potentially less con-servative. Furthermore, the results are extended to T–S fuzzysystems with time-varying uncertainties. All the results are for-mulated in the form of linear matrix inequalities (LMIs). Twoexamples are illustrated to show the usefulness and applicabilityof the obtained results.

The remainder of the paper is organized as follows. Section IIformulates the problem under consideration. Section III presentsthe fault detection filter design for the nominal fuzzy system,and the results are extended to the fuzzy system with time-varying uncertainties in Section IV. Two examples are illustratedin Section V to show the usefulness and applicability of theproposed approaches, and the paper is concluded in Section VI.

The notation used throughout the paper is fairly stan-dard. The superscript “T ” stands for matrix transposition, R

n

denotes the n-dimensional Euclidean space, 0 represents thezero matrix with appropriate dimensions, the notation P > 0(≥ 0) means that P is real symmetric and positive definite(semidefinite), l2 [0,∞) is the space of square-integrable vectorfunctions over [0,∞), and ‖ · ‖2 stands for the usual l2 [0,∞)norm. In symmetric block matrices or complex matrix expres-sions, we use an asterisk (∗) to represent a term that is induced bysymmetry and diag. . . stands for a block-diagonal ma-trix. In addition, Ex and Ex| y will, respectively, meanexpectation of x and expectation of x conditional on y. Matri-ces, if their dimensions are not explicitly stated, are assumed tobe compatible for algebraic operations.


Consider the fault detection problem for T–S fuzzy systemswith intermittent measurements. The physical plant is repre-

sented by a T–S fuzzy model, and the signal transmissions ex-isting between the plant and the fault detection filter are inter-mittent.

A. Physical Plant

The nonlinear discrete-time system whose faults are to bedetected is represented by the following T–S fuzzy model.

Plant rule i: IF θ1(k) is Mi1 and θ2(k) is Mi2 and · · · andθp(k) is Mip , THEN

xk+1 = (Ai + ∆Ai(k)) xk + (Bi + ∆Bi(k)) uk

+ E1iwk + E2ifk

yk = Cixk + Diuk + F1iwk + F2ifk ,

i = 1, . . . , r (1)

where Mij is the fuzzy set, xk ∈ Rnp is the state vector; r

is the number of IF–THEN rules, θ(k) = [θ1(k), θ2(k), . . . ,θp(k)] is the premise variable vector, uk ∈ R

m is the deter-ministic input vector, wk ∈ R

p is the exogenous disturbanceinput that belongs to l2 [0,∞), and fk ∈ R

q is the fault vectorthat is also deterministic. Without the loss of generality, we as-sume that the l2 norms of uk and fk exist and are bounded.Ai,Bi, E1i , E2i , Ci,Di, F1i , and F2i are known constant ma-trices with appropriate dimensions, ∆Ai(k) and ∆Bi(k) denotethe uncertainties in the model and are of the form

∆Ai(k) = NiZ(k)Qai ∆Bi(k) = NiZ(k)Qbi,

i = 1, . . . , r (2)

where Ni ∈ Rnp ×nz , Qai ∈ R

nz ×np , and Qbi ∈ Rnz ×m are

known constant matrices, and Z(k) ∈ Rnz ×nz is an un-

known time-varying matrix with Lebesgue measurable elementsbounded by

ZT (k)Z(k) ≤ I. (3)

Given a pair of (xk , uk ), the overall fuzzy system is inferredas

xk+1 =r∑

i=1

hi(θ(k))[(Ai + ∆Ai(k))xk + (Bi + ∆Bi(k))uk

+ E1iwk + E2ifk ]

yk =r∑

i=1

hi(θ(k))[Cixk + Diuk + F1iwk + F2ifk ] (4)

where hi(θ(k)) = ωi(θ(k))/∑r

i=1 ωi(θ(k)) and ωi(θ(k)) =∏pj=1 Mij (θj (k)), with Mij (θj (k)) representing the grade of

membership of θj (k) in Mij . Then, it can be seen that

ωi(θk ) ≥ 0, i = 1, 2, . . . , r,

r∑i=1

ωi(θk ) > 0

for all k.

B. Fault Detection Filter

One key step of fault detection is the generation of a residualsignal, which must be sensitive to faults. This is often realized



by utilizing fault detection observers [25], [26], [33] or filters[10], [14], [20], [33]. Since disturbances often inevitably appearin many systems, the residual signal must also be capable ofdistinguishing faults from exogenous signals. H∞ filter can notonly describe the estimated signal accurately but also suppressthe disturbance effectively. Thus, for the physical plant withdisturbance in (1), we adopt the following fuzzy fault detectionfilter form, whose role is to generate residual signal based onthe input yf k .

Filter Rule i: IF θ1(k) is Mi1 and θ2(k) is Mi2 and · · · andθp(k) is Mip , THEN

xk+1 = Af ixk + Bf iyf k

rk = Cf ixk + Df iyf k

i = 1, . . . , r. (5)

Here, xk ∈ Rnf and rk ∈ Rq , and Af i, Bf i, Cf i , and Df i are

to be determined. Thus, the filter can be represented by thefollowing form:

xk+1 =r∑

i=1

hi(θ(k)) (Af ixk + Bf iyf k )

rk =r∑

i=1

hi(θ(k)) (Cf ixk + Df iyf k ) . (6)


In this paper, we assume that a communication medium existsbetween the physical plant and the fault detection filter, andthe data packet dropout phenomenon happens intermittently.Therefore, the measurement of the plant is no longer equivalentto the input of the filter (i.e., yk = yf k ). A stochastic process isutilized to model the data loss phenomenon, i.e.

yf k = αkyk (7)

where αk is a Bernoulli process. When the link fails (i.e., dataare lost), αk = 0, and when the transmission is perfect, αk = 1.A natural assumption on αk can be made as

Prob αk = 1 = E αk = α Prob αk = 0 = 1 − α

where α is assumed to be known. Based on this, we have

xk+1 =r∑

i=1

hi(θ(k)) (Af ixk + αkBf iyk )

rk =r∑

i=1

hi(θ(k)) (Cf ixk + αkDf iyk ) . (8)

Remark 1: The description of imperfect communication linksexisting between the plant and the fault detection filter followsthat in the previous literature [28], [29]. The process of missingdata considered is assumed to satisfy the Bernoulli distributedprocess. The probability distribution of the process is estimatedbased on experimental measurements of data transmitting fromoutput of the plant to the input of the fault detection filter. Thiscan be achieved by sending a sequence of indexed data throughthe communication medium and measuring the data dropout

characteristics. The inferred statistics of the Bernoulli processwill then be used for designing the fault detection filter.

D. Fault Weighting System

In fault detection, a reference residual model is usually neededto describe the desired behavior of the residual vector rk . In thispaper, the reference model is chosen as f (z) = W (z) f (z)[33], where W (z) is given a priori. The choice of Wz is toimpose frequency weighting on the spectrum of the fault signalfor detection. Here, we choose a stable matrix W (z) to weightthe fault signal fk [33], whose state-space realization is

xk+1 = AW xk + BW fk

fk = CW xk + DW fk (9)

where xk ∈ RnW and AW,BW,CW , and DW are priorly chosen.

E. Residual Evaluation

The residual evaluation function is to evaluate the generatedresidual. After the residual signal being constructed, a residualevaluation value will be computed through a prescribed evalua-tion function, and it will be compared with a predefined thresh-old. When the evaluation value is larger than the threshold, analarm of fault is generated. Here, we consider the followingevaluation function:

‖r‖T=

1T

√√√√ t2∑k=t1

rTk rk T = t2 − t1 + 1.

Choose a threshold Jth > 0, and for the detailed discussion ofthe threshold Jth , readers are referred to [1] and [5]. The residualevaluation function value and the threshold satisfy the followingrelationship: ‖r‖T > Jth =⇒ with faults =⇒ alarm

‖r‖T ≤ Jth =⇒ no faults.

F. Residual System

From (4), (5), and (9), the residual system can be obtained as

ξk+1 =r∑

i=1

r∑j=1

hi(θ(k))hj (θ(k))

× [Aij ξk + Bijϑk + αk A1ij ξk + αk B1ij ϑk ]

ek =r∑

i=1

r∑j=1

hi(θ(k))hj (θ(k))

×[Cij ξk + Dijϑk + αk C1ij ξk + αk D1ij ϑk

](10)

where

ξk = [ xTk xT

k xTk ]T ϑk = [ uT

k wTk fT

k ]T

ek = rk − fk , Aij =

AW 0 00 Ai + ∆Ai(k) 00 αBf iCj Af i



Bij =

0 0 BW

Bi + ∆Bi(k) E1i E2i

αBf iDj αBf iF1j αBf iF2j

A1ij =

0 0 00 0 00 Bf iCj 0

B1ij =

0 0 00 0 0

Bf iDj Bf iF1j Bf iF2j

Cij = [−CW αDf iCj Cf i ]

Dij = [ αDf iDj αDf iF1j αDf iF2j − DW ]

C1ij = [ 0 Df iCj 0 ]

D1ij = [Df iDj Df iF1j Df iF2j ]

αk = αk − α, Eαk = 0 and Eαk αk = α(1 − α).The residual system presents the difference between the gen-

erated residual and the idealized reference residual signal. Byminimizing the H∞ norm of the difference, the effect of the dis-turbance can be minimized and the sensitivity of the residual tofault can be maximized [1], [18], [33]. Therefore, design of thefault detection filter can be converted as an H∞ model matchingproblem [33].


Problem: Fuzzy fault detection with intermittent measure-ments (FFDIMs): Consider the fuzzy system in (4), and supposethat the intermittent transmission parameter α is known. Givena scalar γ > 0, design a fuzzy fault detection filter in the formof (5) such that:

1) the residual system in (10) is stochastically stable; and2) under zero initial conditions, the residual error ek satisfies

‖e‖E ≤ γ‖ϑ‖2 (11)

where ‖e‖E= E

√∑∞k=0 eT

k ek and ‖ϑ‖2= (

∑∞k=0 ϑT

k

ϑk )12 . It is noted that the l2 norm of ϑk exists and is

bounded since its constituent variables are all l2 normbounded.


Definition 1: The residual system in (10) is said to be stochas-tically stable in the mean square (ϑk ≡ 0) if there exists a finiteV > 0 independent of ξ0 , such that for any initial conditionξ0E

∑∞k=0 ξT

k ξk |ξ0 < ξT0 V ξ0 .

Remark 2: Different statements of stochastic stability defini-tions are presented in [22], which serve for different systems.They have the same criterion that the expectation values of allthe solutions of the system must be energy-bounded when theoperation time is infinite. The solutions are generally dependenton initial conditions or other elements, and thus, the condition

expectation is often used. The definition proposed here alsoobeys the criterion.

III. FUZZY FAULT DETECTION FILTER DESIGN

FOR NOMINAL SYSTEMS

In this section, the FFDIM problem is solved for the nominalfuzzy system in (12). The fault detection analysis problem is firstsolved, and then, based on that, a full-rank fault detection filteris designed (i.e., nf = np + nW ). The fuzzy basis-dependenttechnique is utilized, which potentially reduces the conservatismof the obtained results.

The nominal system of (4) takes the following form:

xk+1 =r∑

i=1

hi(θ(k)) [Aixk + Biuk + E1iwk + E2ifk ]

yk =r∑

i=1

hi(θ(k)) [Cixk + Diuk + F1iwk + F2ifk ] (12)

and the nominal residual fuzzy system is given by

ξk+1 =r∑

i=1

r∑j=1

hi(θ(k))hj (θ(k))

×[Aij ξk + Bijϑk + αk A1ij ξk + αk B1ij ϑk

]ek =

r∑i=1

r∑j=1

hi(θ(k))hj (θ(k))

×[Cij ξk + Dijϑk + αk C1ij ξk + αk D1ij ϑk

](13)

where

Aij =

AW 0 0

0 Ai 0

0 αBf iCj Af i

Bij =

0 0 BW

Bi E1i E2i

αBf iDj αBf iF1j αBf iF2j

.

A. Fault Detection Analysis

In this section, we assume that the fault detection filter matri-ces in (5) are known, and the conditions are investigated underwhich the residual system is stochastically stable and guaranteesthe performance defined in (11). The following theorem tells usthat the performance of the residual system can be guaranteedif there exist some matrices satisfying certain LMIs.

Theorem 1: Consider the fuzzy system in (12) and suppose thefault detection filter matrices Af i, Bf i, Cf i , and Df i in (6) areknown. The residual system in (13) is stochastically stable witha guaranteed performance γ if there exist n-dimensional matri-ces Pl > 0 for l = 1, . . . , r, where n = nf + nW + np , and an



n-dimensional matrix G satisfying the following inequality:

Πl 0 0 0 GT Aij GT Bij

∗ Πl 0 0 fGT A1ij fGT B1ij

∗ ∗ −I 0 Cij Dij

∗ ∗ ∗ −I fC1ij fD1ij

∗ ∗ ∗ ∗ −Pi 0∗ ∗ ∗ ∗ ∗ −γ2I

< 0

i, j, l = 1, . . . , r (14)

where f =√

α (1 − α) and Πl = Pl − G − GT .Proof: Suppose there exist real symmetric positive definite

matrices Pl for l = 1, . . . , r and a nonsingular matrix G satis-fying (14). Noting the inequality (Pl − G)T P−1

l (Pl − G) ≥ 0implies Pl − G − GT ≥ −GT P−1

l G, which together with (14)yields

Πi 0 0 0 GT Aij GT Bij

∗ Πi 0 0 fGT A1ij fGT B1ij



∗ ∗ ∗ ∗ −Pi 0∗ ∗ ∗ ∗ ∗ −γ2I

< 0 (15)

where Πl = −GT P−1l G.

Pre- and postmultiplying diagG−T ,G−T , I, I, I, I

and

diagG−1 , G−1 , I, I, I, I

to (15) and by Schur complement,

we have[AT

ij fAT1ij

BTij fBT

1ij

][Pl 00 Pl

][Aij Bij

fA1ij fB1ij

]+[

CTij fCT

1ij

DTij fDT

1ij

]

×[

Cij Dij

fC1ij fD1ij

]−

[Pi 0∗ γ2I

]< 0. (16)

Now, we first prove the stochastic stability of the residual systemin (13). Define an index as

J =E

ξTk+1

r∑l=1

hl(θ(k+1))Plξk+1

∣∣∣∣∣ξk

− ξT

k

r∑i=1

hi(θ(k))Piξk .

(17)When ϑk ≡ 0, along the nominal system in (13), J gives

J = E

r∑

l=1

hl(θ(k + 1))r∑

i=1

r∑j=1

r∑s=1

r∑t=1

hi(θ(k))hj (θ(k))

×hs(θ(k))ht(θ(k))[Aij ξk + αk A1ij ξk

]TPl

×[Astξk + αk A1stξk

]∣∣ ξk

− ξT

k

r∑i=1

hi(θ(k))Piξk

≤ ξTk

r∑l=1

hl(θ(k + 1))r∑

i=1

r∑j=1

hi(θ(k))hj (θ(k))

×AT

ijPlAij + α (1 − α) AT1ijPlA1ij − Pi

ξk . (18)

Define

Ψij l = ATijPlAij + α (1 − α) AT

1ijPlA1ij − Pi

Ψ(k) =r∑

l=1

hl(θ(k + 1))r∑

i=1

r∑j=1

hi(θ(k))hj (θ(k))Ψij l .

Then, the derivative process in (18) concludes J ≤ ξTk Ψ(k)ξk ,

i.e.

E

ξTk+1

r∑l=1

hl(θ(k + 1))Plξk+1

∣∣∣∣∣ ξk

− ξTk

r∑i=1

hi(θ(k))Piξk ≤ ξTk Ψ(k)ξk . (19)

Taking mathematical expectation and summing up the terms onboth sides of (19) for k = 0, . . . , β, for any β > 1, we have

E

ξTβ+1

r∑l=1

hl(θ(β + 1))Plξβ+1

− ξT

0

r∑i=1

hi(θ(0))Piξ0

≤ E

β∑

k=0

r∑l=1

hl(θ(β+1))r∑

i=1

r∑j=1

hi(θ(k))hj (θ(k))ξTk Ψij lξk

≤ E

β∑

k=0

r∑l=1

hl (θ (β + 1))r∑

i=1

r∑j=1

hi(θ(k))hj (θ(k))

×(λmax

(Ψij l

)ξTk ξk

)

= E

β∑

k=0

maxi,j,l=1,...,r

(λmax

(Ψij l

))ξTk ξk

.

From the aforementioned inequalities, it is not difficult to con-clude that, for i, j, l = 1, . . . , r and β, the following inequalityis true:

EξTβ+1Plξβ+1

− ξT

0 Piξ0

≤ maxi,j,l=1,...,r

(λmax

(Ψij l

))E

β∑

k=0

ξTk ξk

.

When β → ∞, we obtain

EξT∞Plξ∞

− ξT

0 Piξ0

≤ maxi,j,l=1,...,r

(λmax

(Ψij l

))E

∞∑k=0

ξTk ξk

.

Considering nonzero initial condition and EξT∞Plξ∞

≥ 0,

we have

E

∞∑k=0

ξTk ξk

∣∣∣∣∣ ξ0

≤

(− max

i,j,l=1,...,r(λmax(Ψij l))

)−1ξT0 Piξ0

≤ ξT0

(−(

maxi,j,l=1,...,r

(λmax(Ψij l)))−1

maxi=1,...,r

(λmax(Pi)))ξ0

= σξT0 ξ0



where

σ= −

(max

i,j,l=1,...,r(λmax(Ψij l))

)−1max

i=1,...,r(λmax(Pi))

and ξ0 is the initial condition. From (1, 1) block in the left sideof (16), Ψij l < 0 is obtained, and thus, σ > 0. According toDefinition 1, the residual system is stochastically stable in themean square.

Next, we prove that the performance defined in (11) is guar-anteed. To this end, assume zero initial condition and ϑk = 0.An index is introduced as

J = E

ξTk+1

r∑l=1


∣∣∣∣∣ ξk

− ξTk

r∑i=1

hi(θ(k))Piξk + E

eTk ek

∣∣ ξk

− γ2ϑT

k ϑk . (20)

Along the nominal system in (13), we have

E

ξTk+1

r∑l=1


∣∣∣∣∣ ξk

= E

r∑

l=1

hl(θ(k + 1))r∑

i=1

r∑j=1

r∑s=1

r∑t=1

hi(θ(k))hj (θ(k))

×hs(θ(k))ht(θ(k))ηTk

([AT

ij

BTij

]+ αk

[AT

1ij

BT1ij

])

×Pl ([ Ast Bst ] + αk [ A1st B1st ]) ηk

∣∣∣∣∣ ξk

≤r∑

l=1

hl(θ(k + 1))r∑

i=1

r∑j=1

hi(θ(k))hj (θ(k))ηTk

×[

ATij

BTij

]Pl [ Aij Bij ] + f 2

[AT

1ij

BT1ij

]Pl [ A1ij B1ij ]

ηk

E

eTk ek

∣∣ ξk

= E

r∑

i=1

r∑j=1

r∑s=1

r∑t=1

hi(θ(k))hj (θ(k))

×hs(θ(k))ht(θ(k))ηTk

([CT

ij

DTij

]+ αk

[CT

1ij

DT1ij

])

× ([ Cst Dst ] + αk [ C1st D1st ]) ηk

∣∣∣∣∣ ξk

≤ ηTk

r∑i=1

r∑j=1

hi(θ(k))hj (θ(k))

×([

CTij

DTij

][ Cij Dij ] + f 2

[CT

1ij

DT1ij

][ C1ij D1ij ]

)ηk

(21)

where ηk = [ ξTk ϑT

k ]T . By substituting (21) into (20), thefollowing holds:

J ≤r∑

l=1

hl(θ(k + 1))r∑

i=1

r∑j=1

hi(θ(k))hj (θ(k))

× ηTk

[AT

ij

BTij

]Pl [ Aij Bij ] + f 2

[AT

1ij

BT1ij

]Pl

× [ A1ij B1ij ] +

[CT

ij

DTij

][ Cij Dij ]

+ f 2

[CT

1ij

DT1ij

][ C1ij D1ij ]

ηk − ηT

k

[Pi 00 γ2I

]ηk

which leads to J ≤ 0 by consideration of (16), i.e.

E

ξTk+1

r∑l=1


∣∣∣∣∣ ξk

− ξT

k

r∑i=1

hi(θ(k))Piξk

+ E

eTk ek

∣∣ ξk

− γ2ϑT

k ϑk ≤ 0. (22)

Taking mathematical expectation on both sides of (22), weobtain

E

ξTk+1

r∑l=1


− E

ξTk

r∑i=1

hi(θ(k))Piξk

+ EeT

k ek − γ2ϑTk ϑk ≤ 0.

For k = 0, 1, 2, . . . , summing up both sides of the afore-mentioned inequality, considering zero initial condition andEξT

∞∑r

i=1 hi(θ(∞))Piξ∞ > 0, we have

E

∞∑k=0

eTk ek

− γ2

∞∑k=0

ϑTk ϑk ≤ 0

which is equivalent to the inequality in (11), and thus, the proofis completed.

Remark 3: Results of fault detection for fuzzy systems providefeasible solutions to the problem of fault detection for nonlinearsystems [17], which are useful in practice since most physicalsystems in the real world are nonlinear. Previous results aremostly concerned with the perfect communication links. Ac-tually, in practice, the transmission is often imperfect betweenthe plant and the filter, i.e., data packet dropout may occurintermittently, especially in systems based on the network com-munication links. In this paper, data missing is considered in thefuzzy model, which makes the obtained results more generaland practical.

B. Fault Detection Filter Design

In this section, the fault detection filter design problem willbe investigated based on Theorem 1, i.e., a method will bedeveloped to determine the fault detection filter matrices in (5),such that the residual system in (13) is stochastically stable andthe performance defined in (11) is guaranteed.



Theorem 2: Consider the fuzzy system in (12). For a givenpositive constant γ, if there exist n-dimensional matrices

Pl =[

P1l P2l

P T2l P3l

]> 0,

where n = nf + nW + np , matrices Af i , Bf i , Cf i , and Df i ,for any i, l = 1, . . . , r, and (np + nW )-dimensional matricesU,X , and W satisfy the following inequality:

Φij l =

Θl 0 0 0 Θ15ij Θ16ij

∗ Θl 0 0 Θ25ij Θ26ij

∗ ∗ −I 0 Θ35ij Θ36ij

∗ ∗ ∗ −I Θ45ij Θ46ij

∗ ∗ ∗ ∗ Θ55i 0∗ ∗ ∗ ∗ ∗ −γ2I

< 0

i, j, l = 1, . . . , r (23)

where

Θl = Pl − Ω − ΩT Ω =[

U XWT WT

]Θ15ij =

[UT Ai + αBf iCj Af i

XT Ai + αBf iCj Af i

]Θ25ij =

[fBf iCj 0fBf iCj 0

]Θ45ij = [ fDf iCj 0 ] Θ35ij = [ αDf iCj − CW Cf i ]

Θ55i =[−P1i −P2i

−P T2i −P3i

]Θ16ij =

[UT B1i + αBf iDj

XT B1i + αBf iDj

]Θ26ij =

[fBf iDj

fBf iDj

]Θ36ij =

[αDf iDj − DW

]Θ46ij = fDf iDj Ai =

[AW 00 Ai

]B1i =

[0 0 BW

Bi E1i E2i

]Df i = Df i Ci = [ 0 Ci ]

Di = [ Di F1i F2i ] DW = [ 0 0 DW ] . (24)

then there exists a fuzzy fault detection filter in the form of(6), such that the residual system in (13) is stochastically stablewith the performance γ defined in (11). Moreover, if the afore-mentioned conditions are satisfied, the matrices for the faultdetection filter in (6) are given by[

Af i Bf i

Cf i Df i

]=

[G−T

4 00 I

] [Af i Bf i

Cf i Df i

] [V −1 00 I

](25)

with G4 and V being nonsingular matrices and satisfying W =GT

4 V .Proof: Suppose there exist real symmetric positive definite

matrices Pl for l = 1, . . . r, matrices U,X , and W satisfy (23).

From (23), we know that

Pl < Ω + ΩT

which implies Ω and W are nonsingular. One can always findsquare and nonsingular matrices G3 and G4 such that W =GT

4 G−13 G4 . Let

G1 = U X = GT2 G−1

3 G4 V = G−13 G4 G =

[G1 G2G4 G3

](26)

and define a transposition matrix

T =[

I 00 G−1

3 G4

]. (27)

Without loss of generality, we can assume[P1i P2i

P T2i P3i

]= TT PiT. (28)

From (25) and (26), we know that[Af i Bf i

Cf i Df i

]=

[GT

4 00 I

] [Af i Bf i

Cf i Df i

] [G−1

3 G4 00 I

].

(29)Substituting (26)–(29) into (23), we have

Θl = TT(Pl − G − GT

)T = Pl − Ω − ΩT

Θ15ij =

[GT

1 Ai + αGT4 Bf iCj GT

4 Af iG−13 G4

GT4 G−T

3 G2Ai + αGT4 Bf iCj GT

4 Af iG−13 G4

]

Θ25ij =

[fGT

4 Bf iCj 0

fGT4 Bf iCj 0

]Θ45ij = [ fDf iCj 0 ]

Θ35ij = [ αDf iCj − CW Cf iG−13 G4 ]

Θ55ij = TT

[−P1i −P2i

−PT2i −P3i

]T

Θ16ij =

[UT B1i + αGT

4 Bf iDj

GT4 G−T

3 G2B1i + αGT4 Bf iDj

]

Θ26ij =

[fGT

4 Bf iDj

fGT4 Bf iDj

]Θ36ij =

[αDf iDj − DW

]Θ46ij = fDf iDj .

Then, one can conclude that the inequality in (23) is equiv-alent to (30), as shown at the bottom of this page, whichclearly guarantees the inequality in (14). Thus, the proof iscompleted.

Remark 4: Without loss of generality, we assume the numberof rows in CW is nCW

. Then, the two identity matrices in (3,3)

TT 0 0 0 0 00 TT 0 0 0 00 0 I 0 0 00 0 0 I 0 00 0 0 0 TT 00 0 0 0 0 I

Πl 0 0 0 GT Aij GT Bij

∗ Πl 0 0 fGT A1ij fGT B1ij



∗ ∗ ∗ ∗ −Pi 0∗ ∗ ∗ ∗ ∗ −γ2I

T 0 0 0 0 00 T 0 0 0 00 0 I 0 0 00 0 0 I 0 00 0 0 0 T 00 0 0 0 0 I

< 0 (30)



and (4,4) blocks in the left side of (14) are nCW-dimensional, and

the identity matrix in (6,6) block is (m + p + q)-dimensional,which are same as those of the identity matrices in (15) and (23).

Remark 5: After the variables Af i , Bf i , Cf i , and Df i areobtained from the LMI in (23), we have to perform the decom-position on the matrix W to obtain the solution in (25). Sinceby most matrix decomposition methods G4 and V cannot bedetermined uniquely, the solution of (25) is not unique.

Remark 6: Theorem 2 presents the conditions under which theresidual system is stochastically stable and satisfies the guaran-teed performance γ. The result also covers the case of the per-fect communication links existing between the physical plantand the fault detection filter, i.e., there is no data packet dropoutand α = 1, with Φij l modified as

Φij l =

Θl 0 Θ15ij Θ16ij

∗ −I Θ35ij Θ36ij

∗ ∗ Θ55i 0∗ ∗ ∗ −γ2I

where Θl ,Θ55i , Ai , B1i , Ci , Di , and DW are defined in (24)

Θ15ij =[

UT Ai + Bf iCj Af i

XT Ai + Bf iCj Af i

]Θ16ij =

[UT B1i + Bf iDj

XT B1i + Bf iDj

]Θ35ij = [Df iCj − CW Cf i ]

Θ36ij =[Df iDj − DW

].

Remark 7: It is noted that (23) is an LMI over both thematrix variables and the scalar γ. Among those feasible so-lutions, the best performance scalar γ can be found by solvingan optimization problem in which γ is included as an opti-mization variable. The minimum [in terms of the feasibilityof (23)] attenuation level of the fault detection filter can bereadily obtained by solving the following convex optimizationproblem using LMI Toolbox: minimize γ subject to (23) overP1l , P2l , P3l , U,X,W, Af i , Bf i , Cf i , and Df i .

IV. FUZZY FAULT DETECTION FILTER DESIGN

FOR UNCERTAIN SYSTEMS

In this section, the results obtained previously for nominalsystems will be extended to fuzzy systems with uncertaintiesdescribed in (2), i.e., the fuzzy fault detection filter is designedfor the uncertain fuzzy system in (4), such that the residualsystem in (10) is stochastically stable with the performancedefined in (11).

Before proceeding further, we first give the following lemmathat is needed for our subsequent derivation.

Lemma 1 [31]: Given matrices Φ = ΦT , N, Q, and R =RT > 0 of appropriate dimensions

Φ + NFQ + QT FT NT < 0

for all F satisfying FT F ≤ R, if and only if there exists a scalarε > 0, such that

Φ + ε−1NNT + εQT RQ < 0.

Then, we are in a position to give the fault detection filterdesign for the fuzzy system with norm bounded uncertainties.

Theorem 3: Consider the fuzzy system in (4). For a givenpositive constant γ, if there exist n-dimensional matrices

Pl =[

P1l P2l

P T2l P3l

]> 0

where n = nf + nW + np , matrices Af i , Bf i , Cf i , and Df i ,scalars εij l for any i, j, l = 1, . . . , r, and n-dimensional matricesU,X , and W satisfying the following inequality:[

Φij l + εij lQTi Qi Ni

NTi −εij lI

]< 0 (31)

where

Ni =[

0 Ni

0 0

]Ni =

[UT

XT

] [ [0Ni

]0

[0Ni

] ]

Qi =[

0 00 Qi

]Qi =

[ 0 Qai ] 0 0∗ 0 0∗ ∗ [ Qbi 0 0 ]

and Φij l is defined in Theorem 2, then there exists a fuzzyfault detection filter in the form of (6), such that the residualsystem is stochastically stable with the performance γ definedin (11). Moreover, if the aforementioned condition is satisfied,the matrices for the fault detection filter in (6) are given by[

Af i Bf i

Cf i Df i

]=

[U−T 0

0 I

] [Af i Bf i

Cf i Df i

] [V −1 00 I

](32)

with G4 and V being nonsingular matrices and satisfying W =GT

4 V .Proof: Replacing Ai and Bi in (23) with Ai + NiZ(k)Qai

and Bi + NiZ(k)Qbi , respectively, we have

Φij l + NiZ(k)Qi + (NiZ(k)Qi)T < 0

where Z(k) is an appropriate dimensioned block-diagonal ma-trix with entries Z(k). According to Lemma 1, the aforemen-tioned inequality holds if

Φij l + ε−1ij l NiN

Ti + εij lQ

Ti Qi < 0

which, by Schur complement, is equivalent to the inequality in(31). The proof is completed.

Remark 8: The conditions derived here are based on the basis-dependent Lyapunov function method, which can potentiallyreduce the conservatism of the results. But the computation costwill be increased at the same time, especially when the numberof fuzzy rules of the plant is large. One way to solve this problemis to try to reduce the rules in modeling of the physical plant.However, when the number of fuzzy rules for some complexnonlinear systems is large and cannot be reduced in fuzzy mod-eling, we can adopt the quadratic Lyapunov function approachin solving the fault detection problem. The quadratic approachcan be found in [3] and [27].

In this paper, we utilize the flexible and powerful LMI tool tosolve the fault detection problem, which is preferred by manyresearchers [1], [33]. The Schur complement and congruenttransformation are used to convert the H∞ norm constraints



into LMIs feasibility conditions, which are actually convex op-timization problems. Solutions can be determined by solvingthose optimization problems via MATLAB toolbox.

V. ILLUSTRATIVE EXAMPLES

In this section, two examples are presented to illustrate theusefulness and applicability of the fault detection filter designapproaches developed previously.

A. Example 1

In this example, we will illustrate the applicability of the faultdetection filter design method for the nominal fuzzy system.

Consider a tunnel diode circuit system whose model is estab-lished in [8]. With a sampling time T = 0.02 s, the discrete-timemodel is obtained as

xk+1 =2∑

i=1

hi(x1k ) (Aixk + Eiwk )

yk =2∑

i=1

hi(x1k ) (Cixk + Fiwk ) (33)

where the state variables are chosen as x1 (t) = vC (t) andx2 (t) = iL (t), and vC (t) and iL (t) are the capacitor voltageand inductance current, respectively. The parameter matricesare given by

A1 =[

0.9887 0.9024−0.0180 0.8100

]E1 =

[0.00930.0181

]A2 =

[0.9033 0.8617−0.0172 0.8103

]E2 =

[0.00910.0181

]Ci = [ 1 0 ] Fi = 1.

We assume that there are faults on the capacitor voltage, and thefault matrices are given by

G1 =[

0.9887−0.0180

]G2 =

[0.9033−0.0172

].

The aim is to design a fuzzy fault detection filter such that theresidual system in the form of (13) is stochastically stable andthe performance defined in (11) is guaranteed.

The fault weighting system is in the form of (9) and thematrices are chosen as follows:

AW = 0.1 BW = 0.25 CW = 0.5 DW = 0.

We first consider the perfect communication case, i.e., thereis no data packet dropout between the physical plant and thefault detection filter, and thus, α = 1. By solving the LMI inTheorem 2, the matrix variables are obtained as

Af 1 =

0.0301 −0.0043 −0.0628−0.0028 0.0010 0.0273−0.0038 0.0210 0.7217

Df 1 = 0.0369

Bf 1 =

0.0164−0.00140.0063

Cf 1 = [−0.5023 0.0372 0.0066 ]

Af 2 =

0.0294 −0.0048 −0.0658−0.0027 0.0011 0.0273−0.0039 0.0222 0.7146

Df 2 = 0.0368

Bf 2 =

0.0143−0.0011−0.0104

Cf 2 = [−0.4995 0.0372 0.0041 ]

W =

0.2959 −0.0250 −0.0736−0.0248 0.0033 0.0327−0.0778 0.0330 0.8847

and the guaranteed performance defined in (11) is γ∗ = 0.1042.Applying a full-rank factorization on W , we get G4 and V

G4 =

0.1249 −0.0399 −0.99140.9893 −0.0711 0.1275−0.0755 −0.9967 0.0306

V =

0.1151 −0.0360 −0.88760.2845 −0.0207 0.0377−0.0000 −0.0004 0.0000

and thus, the fuzzy fault detection filter matrices can be calcu-lated by (25)

Af 1 =

0.8034 −0.2985 −3.1752−0.0266 0.1147 0.62100.0005 0.0011 0.0089

Df 1 = 0.0369

Bf 1 =

−0.00420.01710.0004

Cf 1 = [−0.0810 −1.7323 4.1194 ]

Af 2 =

0.7961 −0.2950 0.3848−0.0224 0.1104 0.17220.0005 0.0009 0.0245

Df 2 = 0.0368

Bf 2 =

−0.00850.01560.0003

Cf 2 = [−0.0779 −1.7237 3.4360 ] .

Fig. 1 shows the residual response rk and the response of theresidual evaluation function ‖ · ‖T varying as time k when wk =0, where the fault is supposed to be

fk =

1, 300 ≤ k ≤ 6000, else. (34)

From the figure, we can see that the designed filter can detectthe fault effectively when it occurs.

Then, we assume the disturbance

wk =

rand [0, 1] , 200 ≤ k ≤ 7000, else. (35)

Fig. 2 shows the residual response and the residual evaluationfunction response with the disturbance wk , respectively, whichindicate that the residual can not only reflect the fault in time, butalso recognize the fault without confusing it with the disturbancewk .

Next, we consider the fault detection problem with imperfectcommunication links. The parameter is assumed to be α = 0.8,which means that there are 20% data lost during the transmission



Fig. 1. Residual response and evaluation function for the nominal system with zero uk and wk (α = 1).

Fig. 2. Residual response and evaluation function for the nominal system with wk (α = 1).

from the physical plant to the fault detection filter. By applyingTheorem 2, the fuzzy fault detection filter matrices are given by

Af 1 =

0.7880 −0.4444 10.9624−0.0734 0.1514 −8.8659−0.0007 −0.0010 0.6085

Df 1 = 0.0240

Bf 1 =

0.0007−0.00460.0002

Cf 1 = [ 0.1412 1.7282 11.2867 ]

Af 2 =

0.7986 −0.4315 8.3850−0.0754 0.1482 −5.3386−0.0005 −0.0008 0.3067

Df 2 = 0.0240

Bf 2 =

0.0022−0.00490.0003

Cf 2 = [ 0.1560 1.7534 7.8455 ]

and the guaranteed performance is γ∗ = 0.1216. By assumingthe same fault as in (34) and zeros disturbance wk , Fig. 3 showsthe residual response of rk and the evaluation function response,which clearly tell us that the fault can be detected effectivelywhen it appears.

Then, assume the same disturbance as that in (35). Fig. 4shows the residual response and the evaluation function re-

sponse, which clearly indicate that the residual can also detectthe fault without confusing it with the disturbance.

It is worth noting that the obtained minimum-guaranteed per-formance γ∗ will change as the different values of α, which isshown in Table I. From the table, we know that as the valuesof α become larger, the higher γ∗ can be obtained. This is truesince the larger α means the less missing measurements; there-fore, the better disturbance attenuation performance γ∗ can beobtained.

B. Example 2

In this example, we will design a fuzzy fault detection filterfor the fuzzy system with uncertainties. We still consider thefuzzy system in Example 1. For simulation, we assume thereare some uncertainties in the form of (2), and the parameters aregiven by

Ni =[

0.250.25

]Qai = [ 0.1 0 ] Qbi = 0.1.

First, we consider the perfect communication condition (α =1). By solving the inequalities in Theorem 3, the minimumperformance value is obtained as γ∗ = 0.1043, and the fuzzy



Fig. 3. Residual response and evaluation function for the nominal system with zero uk and wk (α = 0.8).

Fig. 4. Residual response and evaluation function for the nominal system with wk (α = 0.8).

TABLE IMINIMUM GUARANTEED PERFORMANCE γ FOR DIFFERENT VALUES OF α

detection filter matrices are

Af 1 =

0.7912 −0.2836 1.5684−0.1096 0.1483 −0.7187−0.0013 −0.0008 −0.0020

Df 1 = 0.0377

Bf 1 =

−0.0017−0.01660.0003

Cf 1 = [ 0.3017 1.6957 4.1770 ]

Af 2 =

0.7660 −0.2853 −0.9327−0.1011 0.1451 −0.9949−0.0013 −0.0005 0.0236

Df 2 = 0.0376

Bf 2 =

0.0014−0.01540.0003

Cf 2 = [ 0.2969 1.6880 3.5792 ] .

Next, we consider the robust case with intermittent measure-ments. Without loss of generality, we suppose α = 0.8. The min-imum performance defined in (11) is obtained as γ∗ = 0.1241,and the fuzzy fault detection filter matrices are obtained as

Af 1 =

0.2766 −0.3567 −5.0519−0.2921 0.6742 −6.2744−0.0018 0.0009 0.6778

Df 1 = 0.0203

Cf 1 = [ 1.4222 1.0240 10.7865 ] Bf 1 =

−0.0017−0.00290.0002

Af 2 =

0.2961 −0.3280 −1.9103−0.2847 0.6502 −6.2402−0.0013 −0.0003 0.4207

Df 2 = 0.0195

Bf 2 =

−0.0011−0.00370.0002

Cf 2 = [ 1.4503 0.9920 8.0067 ] .

Fig. 5 shows the residual response and the response of the resid-ual evaluation function. It can be seen that the designed fuzzy



Fig. 5. Residual response and evaluation function for the uncertain system with wk (α = 0.8).

fault detection filter is also effective for the uncertain fuzzysystem with missing measurements.


In this paper, the problem of fault detection for T–S fuzzysystems with intermittent measurements has been investigated.The communication links between the plant and the fault detec-tion filter are assumed to be imperfect, and a stochastic variablesatisfying the Bernoulli random binary distribution is utilizedto model the unreliable communication links. A fuzzy faultdetection filter has been designed such that, for all data miss-ing conditions, the residual system is stochastically stable andpreserves a guaranteed performance. The results have been ex-tended to the T–S fuzzy systems with time-varying parameteruncertainties. All the results are formulated in the form of LMIs.Two examples have been provided to illustrate the usefulnessand applicability of the results.

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[7] H. Gao and C. Wang, “Robust L2 – L∞ filtering for uncertain systemswith multiple time-varying state delays,” IEEE Trans. Circuits Syst. I,Fundam. Theory Appl., vol. 50, no. 4, pp. 594–599, Apr. 2003.

[8] W. Assawinchaichote and S. K. Nguang, “H∞ filtering for fuzzy dy-namic systems with D stability constraints,” IEEE Trans. Circuits Syst. I,Fundam. Theory Appl., vol. 50, no. 11, pp. 1503–1508, Nov. 2003.

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[11] M. Bask, A. Johansson, and T. Norlander, “Dynamic threshold generatorsfor robust fault detection in linear systems with parameter uncertainty,”Automatica, vol. 42, pp. 1095–1106, 2006.

[12] J. Lam and S. S. Zhou, “Dynamic output feedback H∞ control of discrete-time fuzzy systems: A fuzzy-basis-dependent Lyapunov function ap-proach,” Int. J. Syst. Sci., vol. 38, no. 1, pp. 25–37, 2007.

[13] C. Lin, Q. G. Wang, and T. H. Lee, “Stability and stabilization of a class offuzzy time-delay descriptor systems,” IEEE Trans. Fuzzy Syst., vol. 14,no. 4, pp. 542–551, Aug. 2006.

[14] Z. Mao, B. Jiang, and P. Shi, “H∞ fault detection filter design for net-worked control systems modelled by discrete Markovian jump systems,”IET Control Theory Appl., vol. 1, no. 5, pp. 1336–1343, 2007.

[15] Z. Mao and B. Jiang, “Fault identification and fault-tolerant control for aclass of network control systems,” Int. J. Innov. Comput., Inf. Control,vol. 3, no. 5, pp. 1121–1130, 2007.

[16] R. Mattone and A. D. Luca, “Relaxed fault detection and isolation: Anapplication to a nonlinear case study,” Automatica, vol. 42, no. 1, pp. 109–116, 2006.

[17] S. K. Nguang, P. Shi, and S. Ding, “Fault detection for uncertain fuzzysystems: An LMI approach,” IEEE Trans. Fuzzy Syst., vol. 15, no. 6,pp. 1251–1262, Dec. 2007.

[18] D. C. Oh and J. H. Kim, “A simple frequency weighted model reductionusing structurally balanced truncation: Existence of solutions,” Int. J.Control, vol. 75, pp. 1190–1195, 2002.

[19] P. Shi, “Filtering on sampled-data systems with parametric uncertainty,”IEEE Trans. Autom. Control, vol. 43, no. 7, pp. 1022–1027, Jul. 1998.

[20] P. Shi, E. K. Boukas, and R. K. Agarwal, “Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters,” IEEETrans. Autom. Control, vol. 44, no. 8, pp. 1592–1597, Aug. 1999.

[21] P. Shi and S. K. Nguang, “H∞ output feedback control of fuzzy systemmodels under sampled measurements,” Comput. Math. Appl., vol. 46,no. 5–6, pp. 705–717, 2003.

[22] P. Shi, Y. Xia, G. P. Liu, and D. Rees, “On designing of sliding-modecontrol for stochastic jump systems,” IEEE Trans. Autom. Control, vol. 51,no. 1, pp. 97–103, Jan. 2006.

[23] I. Skrjanc, S. Blazic, and O. Agamennoni, “Identification of dynamicalsystems with a robust interval fuzzy model,” Automatica, vol. 41, pp. 327–332, 2005.

[24] A. Stoorvogel, H. Niemann, A. Saberi, and P. Sannuti, “Optimal faultsignal estimation,” Int. J. Robust Nonlinear Control, vol. 12, pp. 697–727, 2002.

[25] H. Wang and J. Lam, “Robust fault detection for uncertain discrete-timesystems,” J. Guid. Control Dyn., vol. 25, no. 2, pp. 291–301, 2002.

[26] H. Wang, J. Wang, and J. Lam, “Robust fault detection observer design:Iterative LMI approaches,” J. Dyn. Syst. Meas. Control, vol. 129, no. 1,pp. 77–82, 2007.





[29] Z. Wang, F. Yang, D. W. C. Ho, and X. Liu, “Robust finite-horizon filteringfor stochastic systems with missing measurements,” IEEE Signal Process.Lett., vol. 12, no. 6, pp. 437–440, Jun. 2005.

[30] A. S. Willsky and H. L. Jones, “A generalized likelihood ratio approachto the detection and estimation of jumps in linear sytems,” IEEE Trans.Autom. Control, vol. AC-21, no. 1, pp. 108–112, Feb. 1976.

[31] L. Xie, “Output feedback H∞ control of systems with parameter uncer-tainty,” Int. J. Control, vol. 63, pp. 741–750, 1996.

[32] P. Zhang, S. X. Ding, G. Z. Wang, and D. H. Zhou, “A frequency domainapproach to fault detection in sampled-data systems,” Automatica, vol. 39,no. 7, pp. 1303–1307, 2003.

[33] M. Zhong, S. X. Ding, J. Lam, and H. Wang, “An LMI approachto design robust fault detection filter for uncertain LTI systems,” Au-tomatica, vol. 39, no. 3, pp. 543–550, 2003.


[35] S. Xu and J. Lam, “Exponential H∞ filter design for uncertain Takagi–Sugeno fuzzy systems with time delay,” Eng. Appl. Artif. Intell., vol. 17,pp. 645–659, 2004.

Yan Zhao received the B.S. degree in chemical engi-neering and equipment control and the M.S. degreein mechanical engineering from the Inner MongoliaUniversity of Technology, Hohhot, China, in 2002and 2005, respectively. She is currently working to-ward the Ph.D. degree in control science and engi-neering at Harbin Institute of Technology, Harbin,China.


James Lam (S’86–M’86–SM’99) received the B.Sc.degree (with first class) in mechanical engineer-ing from the University of Manchester, Manchester,U.K., and the M.Phil. and Ph.D. degrees in con-trol engineering from the University of Cambridge,Cambridge, U.K. He received the Ashbury Scholar-ship, the A.H. Gibson Prize, and the H. Wright BakerPrize for his academic performance.

He is currently a Professor in the Department ofMechanical Engineering, University of Hong Kong,Hong Kong. He is an Associate Editor of the Asian

Journal of Control, International Journal of Systems Science, Journal of Soundand Vibration, International Journal of Applied Mathematics and Computer Sci-ence, the IEEE TRANSACTIONS ON SIGNAL PROCESSING, Journal of the FranklinInstitute, Dynamics of Continuous, Discrete and Impulsive Systems (Series B:Applications and Algorithms), and Automatica. He is also a member of theEditorial Board of the Institution of Engineering and Technology (IET) ControlTheory and Applications, Open Electrical and Electronic Engineering Journal,Research Letters in Signal Processing, International Journal of Systems, Con-trol and Communications, and Journal of Electrical and Computer Engineering.His current research interests include reduced-order modeling, delay systems,descriptor systems, stochastic systems, multidimensional systems, robust con-trol, and filtering. He was an Editor-in-Chief of the Institution of ElectricalEngineers (IEE) Proceedings Control Theory and Applications.

Prof. Lam is a Chartered Mathematician and a Chartered Scientist. He is aFellow of the Institute of Mathematics and Its Applications, and the IET. He isa Scholar and a Fellow of the Croucher Foundation.


From November 2003 to August 2004, he was aResearch Associate in the Department of MechanicalEngineering, University of Hong Kong, Hong Kong.In November 2004, he joined Harbin Institute ofTechnology, where he is currently a Professor. From

October 2005 to October 2007, he was a Postdoctoral Researcher in the Depart-ment of Electrical and Computer Engineering, University of Alberta, Edmonton,AB, Canada. He is an Associate Editor of the Journal of Intelligent and RoboticsSystems, Circuits, System and Signal Processing, etc. His current research in-terests include network-based control, robust control, and time-delay systemsand their industrial applications.

Prof. Gao is an Associate Editor of the IEEE TRANSACTIONS ON SYSTEMS,MAN AND CYBERNETICS PART B: CYBERNETICS and the IEEE TRANSACTIONS

ON INDUSTRIAL ELECTRONICS. He was an outstanding reviewer for the IEEETRANSACTIONS ON AUTOMATIC CONTROL and Automica in 2008 and 2007, re-spectively, and an appreciated reviewer for the IEEE TRANSACTIONS ON SIGNAL

PROCESSING in 2006. He was the recipient of the University of Alberta DorothyJ. Killam Memorial Postdoctoral Fellow Prize in 2005, the National Outstand-ing Youth Science Fund in 2008, and the National Outstanding Doctoral ThesisAward in 2007. He was the corecipient of the National Natural Science Awardof China in 2008.



Robust Output Feedback Stabilization for UncertainDiscrete-Time Fuzzy Markovian Jump Systems

With Time-Varying DelaysYashun Zhang, Shengyuan Xu, and Baoyong Zhang

Abstract—This paper provides a delay-dependent approach tothe design of fuzzy dynamic output feedback controllers for un-certain discrete-time fuzzy Markovian jump systems with intervaltime-varying delays. First, by a fuzzy-basis-dependent and mode-dependent Lyapunov functional, a stochastic stability condition isderived by using the Finsler’s lemma. Second, in terms of linearmatrix inequalities (LMIs), a delay-dependent sufficient conditionis presented, under which there exists a fuzzy output feedbackcontroller such that the resulting closed-loop system is robustlystochastically stable. A desired controller can be constructed whenthese LMIs are feasible. Finally, the effectiveness of the proposeddesign method is demonstrated by a simulation example.

Index Terms—Delay-dependent stabilization, fuzzy systems, in-terval time-varying delays, Markovian jump systems, output feed-back controllers.

I. INTRODUCTION

MANY practical complex systems may experience abruptchanges such as random component failures or repairs,

sudden environmental disturbances, and changes in the intercon-nections of subsystems [15]. These systems can be representedas Markovian jump systems, which involve both time-evolvingand event-driven mechanisms. In Markovian jump systems, eachevent governed by a Markov process corresponds to the jump infinite operation modes of practical systems. Hence, considerableattention has been paid to the study of Markovian jump systemsover the past decades. Many significant results on estimationand control problems for Markovian jump linear systems havebeen reported in [9], [10], [15], [24], and [32], and the referencestherein.

However, few results on the problems of stability and sta-bilization for Markovian jump nonlinear systems have beengiven because of the difficulties inherent in the analysis ofnonlinear dynamics. Recently, the technique based on Takagi–Sugeno (T–S) fuzzy model has been proved to be a power-ful method for the control problem of complex nonlinear sys-tems. The problems of stability analysis and controller synthe-

Manuscript received June 4, 2008; revised October 31, 2008; acceptedDecember 23, 2008. Current version published April 1, 2009. This work wassupported in part by the National Science Foundation for Distinguished YoungScholars of P. R. China under Grant 60625303, in part by the SpecializedResearch Fund for the Doctoral Program of Higher Education under Grant20060288021, in part by the Natural Science Foundation of Jiangsu Provinceunder Grant BK2008047, and in part by the National Natural Science Founda-tion of P.R. China under Grant 60850005.

The authors are with the School of Automation, Nanjing University of Scienceand Technology, Nanjing 210094, China (e-mail: [email protected];[email protected]; [email protected]).


sis for T–S fuzzy systems have been extensively studied; see,e.g., [7], [12]–[14], [19], [33], [35], [36], and the referencestherein. Therefore, during the past few years, fuzzy Markovianjump systems (FMJSs) have received much attention. For ex-ample, on the basis of a new fuzzy model with two levels ofstructure, a fuzzy-model-based technique for Markovian jumpnonlinear systems was introduced in [1]. The design method ofrobust H∞ output feedback controllers for FMJSs was proposedin [20]. Recently, the problems of stabilization for continuous-and discrete-time FMJSs were investigated in [29] and [30], re-spectively. Very recently, the problems of stability analysis andcontroller design for FMJSs were studied in [11] by introduc-ing some slack variables to separate Lyapunov matrices fromsystem matrices. It should be pointed out that in [1], [11], [29],and [30] the controllers were designed under the assumptionthat all state variables are available.

On the other hand, time delays are inherent in a lot of phys-ical systems. The existence of time delays is often the mainsource of instability or poor performance of a control system.Hence, a great number of results on time-delay systems havebeen presented in [2], [6], [17], [18], [21], [28], and [37], andthe references therein. Recently, much attention has also beendevoted to the problems of stability analysis and stabilization forMarkovian jump linear systems with time delays. For instance,some sufficient conditions for stability analysis were proposedin [3], [26], and [34] for the continuous case, where stabilizingcontrollers were also designed. The corresponding results forthe discrete case can be found in [4], [5], [8], [25], and [31].It is shown that the delay-dependent results in [4], [8], [31],and [34] are less conservative than the delay-independent onesin [3], [5], [25], and [26], especially when the delays are small.

In this paper, we consider the problem of robust output feed-back stabilization for a class of discrete-time FMJSs with inter-val time-varying delays and parameter uncertainties. The pur-pose is to provide a delay-dependent condition for the exis-tence of a full-order fuzzy dynamic output feedback controllersuch that the resulting closed-loop system is robustly stochas-tically stable. We employ a fuzzy-basis-dependent and mode-dependent Lyapunov functional, which contains full informa-tion on the lower and upper bounds of interval time-varyingdelays. By using the Finsler’s lemma, a delay-dependent suffi-cient condition for the solvability of this problem is obtained interms of certain linear matrix inequalities (LMIs). By solvingthese LMIs, a desired fuzzy dynamic output feedback controllercan be constructed. A simulation example is given to show theeffectiveness of the proposed design method.

1063-6706/$25.00 © 2009 IEEE



Notations: For real symmetric matrices X and Y , the notationX ≥ Y and X > Y mean that the matrix X − Y is positivesemidefinite and positive definite, respectively. I denotes theidentity matrix with appropriate dimensions. The superscript“T ” represents the transpose. “∗” is used as an ellipsis for termsthat are induced by symmetry. Matrices, if explicitly stated, areassumed to have compatible dimensions for algebra operations.


Consider the following uncertain discrete-time FMJS withinterval time-varying delays:

Plant Rule i: IF s1(k) is µi1 and s2(k) is µi2 and . . . andsg (k) is µig , THEN

x (k + 1) = [Ai (rk ) + ∆Ai (rk , k)] x (k)

+ [Adi (rk ) + ∆Adi (rk , k)] x (k − τ (k))

+ [Bi (rk ) + ∆Bi (rk , k)] u (k) (1)

y (k) = Ci (rk ) x (k) + Cdi (rk ) x (k − τ (k)) (2)

x (k) = φ (k) , k = −τ2 , −τ2 + 1, . . . , 0 (3)

where i ∈ S = 1, 2, . . . , w, and w is the number of IF-THEN rules; µij is the fuzzy set; x (k) ∈ R

n is the state;u (k) ∈ R

p is the control input; y (k) ∈ Rq is the measured

output; s1 (k) , s2 (k) , . . . , sg (k) are the premise variablesthat do not depend on the input variables u (k) explicitly;rk is a discrete-time Markov process taking values in a fi-nite set T = 1, 2, . . . ,m with transition probability matrix

Π= πlf given by πlf

= Pr (rk+1 = f |rk = l), where πlf is

the transition probability from mode l at time k to mode f attime k + 1, and for all l, f ∈ T, πlf ≥ 0, and

∑mf =1 πlf = 1.

In (1) and (2), τ(k) is a positive integer denoting the intervaltime-varying delay and satisfies 0 < τ1 ≤ τ(k) ≤ τ2 , where τ1and τ2 are positive integers. Ai(rk ), Adi(rk ), Bi(rk ), Ci(rk ),and Cdi(rk ) are known real-valued matrix functions of rk .∆Ai(rk , k), ∆Adi(rk , k), and ∆Bi(rk , k) are unknown matrixfunctions of rk and are assumed to be of the following form:

[ ∆Ai(rk , k) ∆Adi(rk , k) ∆Bi(rk , k) ]

= Ei(rk )F (rk , k) [ H1i(rk ) H2i(rk ) H3i(rk ) ] (4)

where Ei(rk ), H1i(rk ), H2i(rk ), and H3i(rk ) are known real-valued matrices, and F (rk , k) is the unknown time-varying ma-trix satisfying FT (rk , k)F (rk , k) ≤ I for all rk ∈ T.

For notational simplicity, in the sequel, any matrix as Ω(rk )will be denoted by Ωl when rk = l, l ∈ T. For example, Ai(rk )is denoted by Ai,l , and ∆Ai(rk , k) is denoted by ∆Ai,l(k), andso on.

Given a pair (x(k), u(k)), we obtain the following final outputof the fuzzy system in (1) and (2) for any rk = l, l ∈ T:

x(k + 1) =w∑

i=1

hi(s(k))[Ai,l + ∆Ai,l(k)]x(k)

+ [Adi,l + ∆Adi,l(k)]x(k − τ(k))

+ [Bi,l + ∆Bi,l(k)]u(k) (5)

y(k) =w∑

i=1

hi(s(k)) [Ci,lx(k) + Cdi,lx(k − τ(k))] (6)

where

hi(s(k)) =i(s(k))∑w

j=1 j (s(k))i(s(k)) =

g∏j=1

µij (sj (k))

s(k) = [ s1(k) s2(k) · · · sg (k) ]

in which, hi(s(k)) is the fuzzy basis function, and µij (sj (k)) isthe grade of membership of sj (k) in µij . Then, it can be seen thati(s(k)) ≥ 0 and

∑wj=1 j (s(k)) > 0 for all k. Therefore, we

have that hi(s(k)) ≥ 0 and∑w

j=1 hj (s(k)) = 1 for all k. Forthe sake of simplicity, we shall use the following notations:

hki = hi(s(k)) hk+1

i = hi(s(k + 1))

hk−τi = hi(s(k − τ(k))).

Throughout this study, we shall use the following concept ofstochastic stability:

Definition 1: The uncertain FMJS in (1)–(3), with u(k) ≡ 0,is said to be robustly stochastically stable, if for any initialcondition (φ, r(0)), the following is satisfied

limN →∞

E

N∑

k=1

xT (k, φ, r(0))x(k, φ, r(0))

< ∞

where x(k, φ, r(0)) denotes the solution to the system at time kunder the initial conditions φ and r(0).

Now, we consider the following full-order fuzzy dynamicoutput feedback controller for any rk = l, l ∈ T:

Control Rule i: IF s1(k) is µi1 and s2(k) is µi2 and . . . andsg (k) is µig , THEN

x(k + 1) = Aci,l x(k) + Bci,ly(k)

u(k) = Cci,l x(k)

where x(k) ∈ Rn is the controller state, Aci,l , Bci,l , and Cci,l

are matrices to be determined later. Then, we obtain the overallfuzzy controller, for any rk = l, l ∈ T, as follows:

x(k + 1) =w∑

i=1

hki [Aci,l x(k) + Bci,ly(k)] (7)

u(k) =w∑

i=1

hki Cci,l x(k). (8)

Using the overall fuzzy output feedback controller, we havethe following closed-loop system for any rk = l, l ∈ T:

ξ(k + 1) =w∑

i=1

w∑j=1

hki hk

j

[Aij,lξ(k) + Adij,lξ(k − τ(k))

](9)

where

ξ(k) = [xT (k) xT (k) ]T

Aij,l =[

Ai,l + ∆Ai,l(k) [Bi,l + ∆Bi,l(k)] Ccj,l

Bcj,lCi,l Acj,l

]


ZHANG et al.: ROBUST OUTPUT FEEDBACK STABILIZATION FOR UNCERTAIN DISCRETE-TIME FUZZY MARKOVIAN JUMP SYSTEMS 413

Adij,l =[

Adi,l + ∆Adi,l(k) 0Bcj,lCdi,l 0

].

The purpose is to design a fuzzy dynamic output feedbackcontroller in the form of (7) and (8) for the uncertain FMJS withtime delays in (1)–(3) such that the resulting closed-loop systemin (9) is robustly stochastically stable.

III. MAIN RESULTS

We first give the following lemma that will be used in theproof of our main results.

Lemma 1 (Finsler’s lemma) [22]: Consider θ ∈ Rn , H =

HT ∈ Rn×n , and A ∈ R

m×n such that rank(A) = r < n. Thefollowing conditions are equivalent:

1) θT Hθ < 0, ∀θ such that Aθ = 0, θ = 0;2) ∃M ∈ Rn×m , H + MA + AT MT < 0.Now, we propose a delay-dependent stochastic stability con-

dition for the closed-loop system in (9).Theorem 1: Given integers τ2 and τ1 with τ2 > τ1 > 0,

then for any τ(k) satisfying 0 < τ1 ≤ τ(k) ≤ τ2 , the FMJSin (9) is robustly stochastically stable if there exist matricesM1i , M2i , N1i , N2i , S1i , S2i , Pi,l > 0, Q1i > 0, i ∈ S, l ∈ T,

Q2 > 0, Q3 > 0, R1 > 0, R2 > 0, X =[

X11 X12∗ X22

]≥ 0,

Y =[

Y11 Y12∗ Y22

]≥ 0, Z, and scalars εij,l > 0, i, j ∈ S, l ∈ T

such that (10) at the bottom of this page, and the followingmatrix inequalities hold for each l ∈ T: Giitv ,l εii,lZEi,l Hii,l

∗ −εii,lI 0∗ ∗ −εii,lI

< 0, i, t, v ∈ S (11)

Ψ1 =

X11 X12 M1i

∗ X22 M2i

∗ ∗ R1

> 0, i ∈ S (12)

Ψ2 =

Y11 Y12 S1i

∗ Y22 S2i

∗ ∗ R2

> 0, i ∈ S (13)

Ψ3 =

X11 + Y11 X12 + Y12 N1i

∗ X22 + Y22 N2i

∗ ∗ R1 + R2

>0, i ∈ S (14)

where

Gijtv ,l = Φij tv ,l + ZWij,l + WTij,lZ

T ,

Wij,l = [W1ij,l W2ij,l ]

W1ij,l =[−In×n 0 Ai,l

0 −In×n Bcj,lCi,l

]

W2ij,l =[

Bi,lCcj,l Adi,l 0n×5n

Acj,l Bcj,lCdi,l 0n×5n

]Ei,l = [ ET

i,l 0n×n ]T

Hij,l = [ 0n×2n H1i,l H3i,lCcj,l H2i,l 0n×5n ]T

σ = τ2 − τ1

Φij tv ,l =

Φ11 Φ12 0 0 0∗ Φ22 Φ23 S1i −N1i

∗ ∗ Φ33 S2i −N2i

∗ ∗ ∗ −Q2 0∗ ∗ ∗ ∗ −Q3

Φ11 =

m∑f =1

πlf Pt,f + τ2R1 + σR2

Φ12 = −τ2R1 − σR2

Φ22 = −Pi,l + (σ + 1)Q1i + Q2 + Q3 + M1i

+ MT1i + τ2X11 + σY11 + τ2R1 + σR2

Φ23 = −M1i + MT2i + N1i − S1i + τ2X12 + σY12

Φ33 = −Q1v − M2i − MT2i + N2i + NT

2i − S2i

− ST2i + τ2X22 + σY22 .

Proof: We first denote

Pl(hk ) =w∑

i=1

hki Pi,l Q1(hk ) =

w∑i=1

hki Q1i

M1(hk ) =w∑

i=1

hki M1i M2(hk ) =

w∑i=1

hki M2i

N1(hk ) =w∑

i=1

hki N1i N2(hk ) =

w∑i=1

hki N2i

S1(hk ) =w∑

i=1

hki S1i S2(hk ) =

w∑i=1

hki S2i

η(k) = ξ(k + 1) − ξ(k).

Now, we define ξk = (ξ(k), . . . , ξ(k − τ(k))) and choosethe following fuzzy-basis-dependent stochastic Lyapunovfunctional:

V (ξk , rk , k) =4∑

i=1

Vi(ξk , rk , k) (15)

where

V1(ξk , rk , k) = ξT (k)

(w∑

i=1

hki Pi(rk )

)ξ(k)

Gijtv ,l + Gjitv ,l εij,lZEi,l εji,lZEj,l Hij,l Hj i,l

∗ −εij,lI 0 0 0∗ ∗ −εji,lI 0 0∗ ∗ ∗ −εij,lI 0∗ ∗ ∗ ∗ −εji,lI

< 0, 1 ≤ i < j ≤ w, t, v ∈ S. (10)



V2(ξk , rk , k) =−τ1∑

j=−τ2

k−1∑i=k+j

ξT (i)Q1(hi)ξ(i)

V3(ξk , rk , k) =k−1∑

i=k−τ1

ξT (i)Q2ξ(i) +k−1∑

i=k−τ2

ξT (i)Q3ξ(i)

V4(ξk , rk , k) =−1∑

j=−τ2

k−1∑i=k+j

ηT (i)R1η(i)

+−τ1 −1∑j=−τ2

k−1∑i=k+j

ηT (i)R2η(i).

Then, define

ϕ1(k) = [ ξT (k + 1) ξT (k) ξT (k − τ(k))

ξT (k − τ1) ξT (k − τ2) ]T

ϕ2(k) = [ ξT (k) ξT (k − τ(k)) ]T

ϕ3(k, i) = [ ξT (k) ξT (k − τ(k)) ηT (i) ]T

and for each rk = l, l ∈ T, simple computation yields

E [V1(ξk+1 , rk+1)|ξk , rk = l] − V1(ξk , l)

= ξT (k + 1)

m∑f =1

πlf Pf (hk+1)

ξ(k + 1)

− ξT (k)Pl(hk )ξ(k) (16)


≤ (σ + 1)ξT (k)Q1(hk )ξ(k)

− ξT (k − τ(k))Q1(hk−τ )ξ(k − τ(k)) (17)


= ξT (k) (Q2 + Q3) ξ(k) − ξT (k − τ1)Q2ξ(k − τ1)

− ξT (k − τ2)Q3ξ(k − τ2) (18)


= ηT (k)(τ2R1 + σR2)η(k)

−k−1∑

i=k−τ (k)

ηT (i)R1η(i) −k−τ1 −1∑

i=k−τ (k)

ηT (i)R2η(i)

−k−τ (k)−1∑i=k−τ2

ηT (i)(R1 + R2)η(i). (19)

Now, we consider the following general case: τ(k) = τ2 andτ(k) = τ1 . Observe that

k−1∑i=k−τ2

ϕT2 (k)Xϕ2(k) =

k−τ (k)−1∑i=k−τ2

ϕT2 (k)Xϕ2(k)

+k−1∑

i=k−τ (k)

ϕT2 (k)Xϕ2(k) (20)

k−τ1 −1∑i=k−τ2

ϕT2 (k)Y ϕ2(k) =

k−τ (k)−1∑i=k−τ2

ϕT2 (k)Y ϕ2(k)

+k−τ1 −1∑

i=k−τ (k)

ϕT2 (k)Y ϕ2(k). (21)

On the other hand, it can be verified that

ψ1(k)= ξ(k) − ξ(k − τ(k)) −

k−1∑i=k−τ (k)

η(i) = 0 (22)

ψ2(k)= ξ(k − τ(k)) − ξ(k − τ2) −

k−τ (k)−1∑i=k−τ2

η(i) = 0

(23)

ψ3(k)= ξ(k − τ1) − ξ(k − τ(k)) −

k−τ1 −1∑i=k−τ (k)

η(i) = 0.

(24)

From (16)–(24), we have, for each rk = l, l ∈ T

E [V (ξk+1 , rk+1)|ξk , rk = l] − V (ξk , l)

=4∑

i=1

E [Vi(ξk+1 , rk+1)|ξk , rk = l] − Vi(ξk , l)

+ 2ϕT2 (k) [ MT

1 (hk ) MT2 (hk ) ]T ψ1(k)

+ 2ϕT2 (k) [ NT

1 (hk ) NT2 (hk ) ]T ψ2(k)

+ 2ϕT2 (k) [ ST

1 (hk ) ST2 (hk ) ]T ψ3(k)

≤ ϕT1 (k)Φl(k)ϕ1(k) −

k−1∑i=k−τ (k)

ϕT3 (k, i)Ψ1ϕ3(k, i)

−k−τ1 −1∑

i=k−τ (k)

ϕT3 (k, i)Ψ2ϕ3(k, i)

−k−τ (k)−1∑i=k−τ2

ϕT3 (k, i)Ψ3ϕ3(k, i) (25)

where

Φl(k) =

Φ11(k) Φ12 0 0 0

∗ Φ22(k) Φ23(k) S1(hk ) −N1(hk )

∗ ∗ Φ33(k) S2(hk ) −N2(hk )

∗ ∗ ∗ −Q2 0

∗ ∗ ∗ ∗ −Q3

Φ11(k) =

m∑f =1

πlf Pf (hk+1) + τ2R1 + σR2

Φ22(k) = −Pl(hk ) + (σ + 1)Q1(hk ) + Q2 + Q3 + M1(hk )

+ MT1 (hk ) + τ2X11 + σY11 + τ2R1 + σR2



Φ23(k) = −M1(hk ) + MT2 (hk ) + N1(hk ) − S1(hk )

+ τ2X12 + σY12

Φ33(k) = −Q1(hk−τ ) − M2(hk ) − MT2 (hk ) + N2(hk )

+ NT2 (hk ) − S2(hk ) − ST

2 (hk ) + τ2X22 + σY22 .

Then, for each l ∈ T

Φl(k) =w∑

i=1

w∑j=1

w∑t=1

w∑v=1

hki hk

j hk+1t hk−τ

v Φij tv ,l .

Now, we suppose that the conditions in (10)–(14) are satisfied.By using the Schur complements, it follows from (10) and (11)that there exists a sufficiently small scalar ε > 0 satisfying thefollowing inequalities for each l ∈ T and t, v ∈ S

Gijtv ,l + Gjitv ,l + εij,lZEi,l ETi,lZ

T

+ εji,lZEj,l ETj,lZ

T + ε−1ij,l Hij,l H

Tij,l

+ ε−1j i,l Hj i,l H

Tji,l < −2εI1 , 1 ≤ i < j ≤ w (26)

Giitv ,l + εii,lZEi,l ETi,lZ

T

+ ε−1ii,l Hii,l H

Tii,l < −εI1 , i ∈ S (27)

where

I1 =

02n×2n 0 0∗ I2n×2n 0∗ ∗ 06n×6n

.

From (26) and (27), it can be seen that, for each l ∈ T

w∑t=1

w∑v=1

hk+1t hk−τ

v

w∑

i=1

hki hk

i

[Giitv ,l + εii,lZEi,l E

Ti,lZ

T

+ ε−1ii,l Hii,l H

Tii,l + εI1

]+

w−1∑i=1

w∑j=i+1

hki hk

j

×[Gijtv ,l + Gjitv ,l + εij,lZEi,l E

Ti,lZ

T

+ εji,lZEj,l ETj,lZ


Tij,l

+ ε−1j i,l Hj i,l H

Tji,l + 2εI1

]

= Φl(k) +w∑

i=1

w∑j=1

hki hk

j

(ZWij,l + WT

ij,lZT

+ εij,lZEi,l ETi,lZ

T

+ ε−1ij,l Hij,l H

Tij,l

)+ εI1 < 0.

By Proposition 2.2 in [16], it is easy to see that

ZEi,lFl(k)HTij,l + Hij,lF

Tl (k)ET

i,lZT

≤ εij,lZEi,l ETi,lZ


Tij,l

for each l ∈ T and i, j ∈ S. Therefore, for each l ∈ T

Φl(k) + ZWl(k) + WTl (k)ZT < −εI1 (28)

where

Wl(k) =w∑

i=1

w∑j=1

hki hk

j

(Wij,l + Ei,lFl(k)HT

ij,l

).

In addition, from (9), we can deduce, for each l ∈ T

Wl(k)ϕ1(k) = 0.

Applying Lemma 1 gives

ϕT1 (k)Φl(k)ϕ1(k) < −εϕT

1 (k)I1ϕ1(k) (29)

for each l ∈ T. Therefore, it follows from (12)–(14), (25), and(29) that, for each l ∈ T

E [V (ξk+1 , rk+1)|ξk , rk = l] − V (ξk , l) < −εξT (k)ξ(k).(30)

In the case when τ(k) = τ2 or τ(k) = τ1 , we can follow asimilar line as earlier to have (30). Hence, from (30) and theresult in [4], it can be deduced that the uncertain FMJS in (9) isrobustly stochastically stable.

Remark 1: In the proof of Theorem 1, the Lyapunov functional(15) is not only mode-dependent but also fuzzy-basis-dependentowing to the use of the matrices Pl(hk ) and Q1(hk ). More-over, this functional also depends on both the upper and lowerbounds of the interval time-varying delay by introducing someadditional terms into V4(ξk , rk , k). Hence, this Lyapunov func-tional contains information on the Markovian jump parameters,the fuzzy-basis functions, and the interval time-varying delay.This implies that the condition in Theorem 1 is less conservativethan those derived by using mode-independent or fuzzy-basis-independent Lyapunov functionals.

Remark 2: By using the Finsler’s lemma, we derive the anal-ysis conditions (10)–(14) in an enlarged space. This methodguarantees that the Lyapunov matrices are separated from thesystem matrices. Therefore, it is easy to realize the controllersynthesis by utilizing Theorem 1.

We are now in a position to give the main result on the designof the robust fuzzy output feedback controller for the FMJS in(1)–(3).

Theorem 2: Given integers τ2 and τ1 with τ2 > τ1 > 0, thenthere exists a fuzzy output feedback controller in the formof (7) and (8) such that the resulting closed-loop system in(9) is robustly stochastically stable for any τ(k) satisfying0 < τ1 ≤ τ(k) ≤ τ2 , if there exist matrices M1i , M2i , N1i , N2i ,S1i , S2i , K1i,l , K2i,l , K3i,l , Pi,l > 0, Q1i > 0, i ∈ S, l ∈ T,

Q2 > 0, Q3 > 0, R1 > 0, R2 > 0, X =[

X11 X12∗ X22

]≥ 0,

Y =[

Y11 Y12∗ Y22

]≥ 0, Z1 , L1 , U , and scalars εij,l > 0, i, j ∈

S, l ∈ T such that (31) at the bottom of the next page and thefollowing LMIs hold for each l ∈ T

Giitv ,l εii,lΞi,l Υii,l Γ1i,l 0∗ −εii,lI 0 0 εii,lI∗ ∗ −εii,lI 0 0∗ ∗ ∗ −I 0∗ ∗ ∗ ∗ −I

< 0, i, t, v ∈ S

(32)



X11 X12 M1i

∗ X22 M2i

∗ ∗ R1

> 0, i ∈ S (33)

Y11 Y12 S1i

∗ Y22 S2i

∗ ∗ R2

> 0, i ∈ S (34)

X11 + Y11 X12 + Y12 N1i

∗ X22 + Y22 N2i

∗ ∗ R1 + R2

> 0, i ∈ S (35)

where

Gijtv ,l = Φij tv ,l + Wij,l

Wij,l =[

W1 W2ij,l W3ij,l

∗ 0 0

]W1 =

[−L1 − LT

1 −I − UT

∗ −Z1 − ZT1

]W2ij,l =

[Ai,lL1 + Bi,lK2j,l Ai,l

K3j,l ZT1 Ai,l + K1j,lCi,l

]W3ij,l =

[Adi,lL1 Adi,l 0n×4n

0 ZT1 Adi,l + K1j,lCdi,l 0n×4n

]

Φij tv ,l =

Φ11 Φ12 0 0 0∗ Φ22 Φ23 S1i −N1i

∗ ∗ Φ33 S2i −N2i

∗ ∗ ∗ −Q2 0∗ ∗ ∗ ∗ −Q3

Φ11 =

m∑f =1

πlf Pt,f + τ2R1 + σR2

Φ12 = −τ2R1 − σR2

Φ22 = −Pi,l + (σ + 1)Q1i + Q2 + Q3 + M1i + MT1i

+ τ2X11 + σY11 + τ2R1 + σR2

Φ23 = −M1i + MT2i + N1i − S1i + τ2X12 + σY12

Φ33 = −Q1v − M2i − MT2i + N2i + NT

2i − S2i − ST2i

+ τ2X22 + σY22

Ξi,l = [ETi,l 0n×9n ]T

Υij,l = [ 0n×2n H1i,lL1 + H3i,lK2j,l H1i,l

H2i,lL1 H2i,l 0n×4n ]T

Γ1i,l = [ Σ1ii,l Σ2i,l Σ3 ]

Σ1ij,l = [ 0n×n ATdi,lZ1 + CT

di,lKT1j,l 0n×8n ]T

Σ2i,l = [ 0n×n ETi,lZ1 0n×8n ]T

Σ3 = [ 0n×4n L1 0n×5n ]T

Γ2ij,l = [ Σ1ij,l + Σ1j i,l Σ1ij,l Σ2ij,l

Σ2i,l Σ2j,l Σ3 Σ3 Σ4ij,l ]

Σ1ij,l = [ 0n×n (Ci,l − Cj,l)T (K1j,l − K1i,l)

T 0n×8n ]T

Σ2ij,l = [ 0p×n (Bi,l − Bj,l)T Z1 0p×8n ]T

Σ3 = [ 0n×2n L1 0n×7n ]T

Σ4ij,l = [ 0p×2n K2j,l − K2i,l 0p×7n ]T

Γ3ij,l = [ εij,lIn×n 0n×n ]

Γ4ij,l = [ 0n×n εji,lIn×n ]

σ = τ2 − τ1 .

Furthermore, a desired fuzzy output feedback controller is givenin the form of (7) and (8) with the parameters as follows:

Aci,l = Z−T2

(K3i,l − ZT

1 Ai,lL1 − K1i,lCi,lL1

− ZT1 Bi,lK2i,l

)L−1

2 (36)

Bci,l = Z−T2 K1i,l , Cci,l = K2i,lL

−12 , i ∈ S, l ∈ T (37)

where Z2 and L2 are some nonsingular matrices satisfyingZT

2 L2 = U − ZT1 L1 .

Proof: Similar to [23], we choose the relaxed variable Zintroduced in Theorem 1 as

Z =[

ZT

08n×2n

]Z =

[Z1 Z3Z2 Z4

].

Define

Z−1 =[

L1 L3L2 L4

]L =

[L1 IL2 0

]and for each l ∈ T and i ∈ S, denote

U = ZT1 L1 + ZT

2 L2 Θ = diag (L,L,L, L, L)

K1i,l = ZT2 Bci,l K2i,l = Cci,lL2

K3i,l = ZT1 Ai,lL1 + K1i,lCi,lL1 + ZT

1 Bi,lK2i,l

+ ZT2 Aci,lL2

Gijtv ,l + Gjitv ,l εij,lΞi,l εji,lΞj,l Υij,l Υj i,l Γ2ij,l 0∗ −εij,lI 0 0 0 0 Γ3ij,l

∗ ∗ −εji,lI 0 0 0 Γ4ij,l

∗ ∗ ∗ −εij,lI 0 0 0∗ ∗ ∗ ∗ −εji,lI 0 0∗ ∗ ∗ ∗ ∗ −I 0∗ ∗ ∗ ∗ ∗ ∗ −I

< 0, 1 ≤ i < j ≤ w, t, v ∈ S (31)



Pi,l = LT Pi,lL Q1i = LT Q1iL Q2 = LT Q2L

Q3 = LT Q3L M1i = LT M1iL M2i = LT M2iL

N1i = LT N1iL N2i = LT N2iL S1i = LT S1iL

S2i = LT S2iL R1 = LT R1L R2 = LT R2L

X11 = LT X11L X12 = LT X12L X22 = LT X22L

Y11 = LT Y11L Y12 = LT Y12L Y22 = LT Y22L.

Then, for each l ∈ T, pre- and postmultiplying (11) bydiag(ΘT , I2n×2n ) and its transpose, respectively, we obtain Giitv ,l εii,lΞi,l Υii,l

∗ −εii,lI 0

∗ ∗ −εii,lI

+ Λ1ΛT2 + Λ2ΛT

1 < 0,

i, t, v ∈ S (38)

where

Λ1 =[

Σ1ii,l Σ2i,l

02n×n 02n×n

]Λ2 =

[Σ3 010n×n

02n×n Σ4i,l

]Σ4i,l = [ εii,lIn×n 0n×n ]T .

Note that (38) holds for each l ∈ T if the following inequalityholds for each l ∈ T and i, t, v ∈ S Giitv ,l εii,lΞi,l Υii,l

∗ −εii,lI 0

∗ ∗ −εii,lI

+ [ Λ1 Λ2 ][

ΛT1

ΛT2

]< 0.

(39)

Applying the Schur complements to (39), we obtain (32).Next, we observe that the following equation holds for each

l ∈ T and i, j ∈ S,(ZT

1 Ai,lL1 + K1j,lCi,lL1 + ZT1 Bi,lK2j,l + ZT

2 Acj,lL2)

+(ZT

1 Aj,lL1 + K1i,lCj,lL1 + ZT1 Bj,lK2i,l + ZT

2 Aci,lL2)

= K3i,l + K3j,l + ZT1 (Bi,l − Bj,l) (K2j,l − K2i,l)

+ (K1j,l − K1i,l) (Ci,l − Cj,l) L1 . (40)

Now, for each l ∈ T, pre- and postmultiplying (10) bydiag(ΘT , I4n×4n ) and its transpose, respectively, and using(40), we have, for t, v ∈ S

Gijtv ,l + Gjitv ,l εij,lΞi,l εji,lΞj,l Υij,l Υj i,l


+ Λ3ΛT

4 + Λ4ΛT3 < 0, 1 ≤ i < j ≤ w, (41)

where

Λ3 =[

Σ1ij,l + Σ1j i,l Σ1ij,l Σ2ij,l Σ2i,l Σ2j,l

04n×n 04n×n 04n×p 04n×n 04n×n

]Λ4 =

[Σ3 Σ3 Σ4ij,l 010n×n 010n×n

04n×n 04n×n 04n×p Σ5ij,l Σ5ij,l

]Σ5ij,l = [ εij,lIn×n 0n×3n ]T

Σ5ij,l = [ 0n×n εij,lIn×n 0n×2n ]T .

TABLE IMODES OF THE PARAMETERS a1 , a2 , AND b

Then, it can be seen that, for each l ∈ T and t, v ∈ S, (41) holdsif the following inequality holds for each l ∈ T and t, v ∈ S

Gijtv ,l + Gjitv ,l εij,lΞi,l εji,lΞj,l Υij,l Υj i,l


+ [ Λ3 Λ4 ] [ Λ3 Λ4 ]T < 0, 1 ≤ i < j ≤ w (42)

which, by using the Schur complements, implies (31). Pre- andpostmultiplying (12)–(14) by diag(LT , LT , LT ) and its trans-pose, respectively, we obtain (33)–(35). Therefore, we derivethe conditions in (31)–(35).

Remark 3: Theorem 2 provides a delay-dependent conditionfor the existence of robust fuzzy dynamic output feedback con-trollers for uncertain discrete-time FMJSs. It is worth pointingout that (31)–(35) are in the form of LMIs that are easy to besolved. Thus, our main result in Theorem 2 is effective for robustoutput feedback stabilization of uncertain discrete-time FMJSswith time-varying delays.

IV. SIMULATION EXAMPLE

In this section, we provide a simulation example to illustratethe design approach of the fuzzy output feedback controllerdeveloped in the previous section. Consider the following un-certain discrete-time Markovian jump nonlinear system withinterval time-varying delays:

x1(k + 1) = −x21(k) + a1x2(k) + a2x1(k)x1(k − τ(k))

+ 0.02c1(k)x1(k)x2(k) + u1(k)

+ 0.01c2(k)x2(k − τ(k))

− 0.2x1(k)x2(k − τ(k)) (43)

x2(k + 1) = 0.1x1(k) + bx2(k) − 0.1x1(k)x2(k − τ(k))

+ 0.02c1(k)x2(k) + 0.5u2(k) (44)

y(k) = 0.6x1(k) (45)

where the parameters a1 , a2 , and b have two modes as shownin Table I; τ(k) is an interval time-varying delay satisfying0 < τ1 ≤ τ(k) ≤ τ2 ; c1(k) and c2(k) are uncertain parameterssatisfying c1(k) ∈ [−1, 1] and c2(k) ∈ [−1, 1]. The nonlinearsystem switches between the two modes, and the transitionprobability matrix is given by

Π =[

0.75 0.250.30 0.70

].

Similar to [27], we assume that x1(k) ∈ [−0.5, 0.5] andset the membership functions as µ11(x1(k)) = 1

2 (1 − 2x1(k)),



µ21(x1(k)) = 12 (1 + 2x1(k)). Then, we represent the Marko-

vian jump nonlinear time-delay system in (43)–(45) as the fol-lowing T–S model:

Plant rule i: IF x1(k) is µi1(x1(k)), THEN

x(k + 1) = [Ai,l + ∆Ai,l(k)]x(k) + [Adi,l + ∆Adi,l(k)]

× x(k − τ(k)) + [Bi,l + ∆Bi,l(k)]u(k)

y(k) = Cix(k), i = 1, 2, l = 1, 2

where

A1,1 =[

0.5 0.30.1 1.0

]A1,2 =

[0.5 0.40.1 1.06

]A2,1 =

[−0.5 0.30.1 1.0

]A2,2 =

[−0.5 0.40.1 1.06

]Ad1,1 =

[−0.05 0.1

0 0.05

]Ad1,2 =

[−0.07 0.1

0 0.05

]Ad2,1 =

[0.05 −0.10 −0.05

]Ad2,2 =

[0.07 −0.10 −0.05

]B1,1 = B2,1 = B1,2 = B2,2 =

[1.0 00 0.5

]C1,1 = C2,1 = C1,2 = C2,2 = [ 0.6 0 ] .

The uncertain parameters ∆Ai,l(k), ∆Adi,l(k), and ∆Bi,l(k),for i = 1, 2 and l = 1, 2 can be represented in the form of (4)with

E11 = E12 =[−0.05 0.10.1 0

]E21 = E22 =

[0.05 0.10.1 0

]H11 = H12 =

[0 0.20 1

]H21 = H22 =

[0 00 0.1

]H31 = H32 =

[0 00 0

].

The aim is to develop a fuzzy output feedback controllersuch that the resulting closed-loop system is robustly stochas-tically stable. The comparison between the conditions in (31)–(35) and those obtained by replacing the Lyapunov functionalin (15) with a mode-independent and fuzzy-basis-independentLyapunov functional will be given by finding the maximum al-lowable size of τ2 with τ1 = 2 for the aforesaid robust controlproblem. By using the sufficient conditions obtained by mode-independent and fuzzy-basis-independent Lyapunov function-als, we obtain that the maximum allowable size of τ2 withτ1 = 2 is 3. However, using Theorem 2, we obtain that the cor-responding maximum value is 4. This means that the conditionsin Theorem 2 are less conservative than those derived by us-ing mode-independent and fuzzy-basis-independent Lyapunovfunctionals for this example. Then, choosing τ1 = 2, τ2 = 4 andsolving the LMIs in (31)–(35), we obtain the parameters of thefuzzy output feedback controller as follows:

Ac1,1 =[

0.0472 −0.07400.8470 −0.4661

]Bc1,1 =

[0.9509−6.5443

]

Fig. 1. Operation mode, time-varying delay, and state responses of the open-loop system.

Fig. 2. Operation mode, time-varying delay, and control results of the closed-loop system.

Ac1,2 =[−0.0473 −0.01260.8994 −0.4884

]Bc1,2 =

[1.6787−6.7662

]Ac2,1 =

[0.6172 −0.18750.6597 −0.3377

]Bc2,1 =

[−5.5554−5.1108

]



Ac2,2 =[

0.5431 −0.14900.6978 −0.3407

]Bc2,2 =

[−4.9470−5.3518

]Cc1,1 =

[−0.0524 0.08970.0242 0.4374

]Cc1,2 =

[−0.0561 0.11150.0016 0.4498

]Cc2,1 =

[0.0961 0.05160.0451 0.3995

]Cc2,2 =

[0.0972 0.06630.0287 0.4056

].

Now, we set the initial conditions as r(0) = 1 and φ(k) =[0.2,−0.2]T , k = −4,−3, . . . , 0. We further assume thatc1(k) = cos(k) and c2(k) = sin(k).

The simulations of the operation mode, the time-varying de-lay, and the state responses of the open-loop system are given inFig. 1(a), Fig. 1(b), and Fig. 1(c), respectively. Now, we applythe designed fuzzy dynamic output feedback controller in theform of (7) and (8) to the nonlinear system in (43)–(45). Thestate responses of the resulting closed-loop system are shownin Fig. 2(c) under the operation mode in Fig. 2(a) and the time-varying delay in Fig. 2(b). The control law is also given in Fig.2(d). These results show that the designed fuzzy output feed-back controller can effectively stabilize the uncertain Markovianjump nonlinear time-delay system in (43)–(45).

V. CONCLUSION

We have studied the problem of delay-dependent robust out-put feedback stabilization for a class of uncertain discrete-timeFMJSs with interval time-varying delays and parameter uncer-tainties. By using the Finsler’s lemma, delay-dependent suffi-cient conditions for the robust stabilization problem have beengiven in terms of LMIs. A desired fuzzy dynamic output feed-back controller can be constructed by solving these LMIs. Asimulation example has also demonstrated the effectiveness ofthe design method.

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Yashun Zhang received the B.E. degree in automa-tion in 2003 and the M.E. degree in control theoryand application in 2006, both from Hefei Universityof Science and Technology, Hefei, China. He is cur-rently working toward the Ph.D. degree at the Schoolof Automation, Nanjing University of Science andTechnology, Nanjing, China.

His current research interests include fuzzy con-trol, sliding mode control, and nonlinear control.

Shengyuan Xu received the B.Sc. degree fromHangzhou Normal University, Hangzhou, China, in1990, the M.Sc. degree from Qufu Normal Univer-sity, Qufu, China in 1996, and the Ph.D. degreefrom Nanjing University of Science and Technology,Nanjing, China, 1999.

From 1999 to 2000, he was a Research Asso-ciate in the Department of Mechanical Engineering,University of Hong Kong, Hong Kong. From Decem-ber 2000 to November 2001, and December 2001 toSeptember 2002, he was a Postdoctoral Researcher

in CESAME at the Universite Catholique de Louvain, Belgium, and the Depart-ment of Electrical and Computer Engineering, University of Alberta, Canada,respectively. From September 2002 to September 2003, and September 2003to September 2004, he was a William Mong Young Researcher and an Hon-orary Associate Professor, respectively, both in the Department of MechanicalEngineering, University of Hong Kong. Since November 2002, he has been aProfessor with the Department of Automation, Nanjing University of Scienceand Technology. His current research interests include robust filtering and con-trol, singular systems, time-delay systems, nonlinear systems, neural networksand multidimensional systems.

Prof. Xu is a member of the Editorial Boards of the Multidimensional Sys-tems and Signal Processing and an Associate Editor of the Circuits Systemsand Signal Processing. He was a recipient of the National Excellent DoctoralDissertation Award in the year 2002 from the Ministry of Education of China.In the year 2006, he obtained a grant from the National Science Foundationfor Distinguished Young Scholars of China. He was awarded a Cheung KongProfessorship in the year 2008 from the Ministry of Education of China.

Baoyong Zhang received the B.Sc. degree in mathe-matics in 2003 and the M.Sc. degree in control theoryin 2006, both from Qufu Normal University, Qufu,China. He is currently working toward the Ph.D. de-gree at the School of Automation, Nanjing Universityof Science and Technology, Nanjing, China.

His current research interests include robust con-trol and filtering, nonlinear systems, time-delay sys-tems, and stochastic systems.



Representation of Uncertain Multichannel DigitalSignal Spaces and Study of Pattern Recognition

Based on Metrics and Difference Values on Fuzzyn-Cell Number Spaces

Guixiang Wang, Peng Shi, Senior Member, IEEE, and Paul Messenger

Abstract—In this paper, we discuss the problem of character-ization for uncertain multichannel digital signal spaces, proposeusing fuzzy n-cell number space to represent uncertain n-channeldigital signal space, and put forward a method of constructing suchfuzzy n-cell numbers. We introduce two new metrics and conceptsof certain types of difference values on fuzzy n-cell number spaceand study their properties. Further, based on the metrics or differ-ence values appropriately defined, we put forward an algorithmicversion of pattern recognition in an imprecise or uncertain environ-ment, and we also give practical examples to show the applicationand rationality of the proposed techniques.

Index Terms—Difference values, fuzzy n-cell numbers, metrics,n-dimensional fuzzy vectors, pattern recognition, uncertain mul-tichannel digital signals.

I. INTRODUCTION

I T IS known that in a precise or certain environment, mul-tichannel digital signals can be represented by elements of

multidimensional Euclidean space, i.e., crisp multidimensionalvectors. If, however, we wish to study multichannel digital sig-nals in an imprecise or uncertain environment, then the signalsthemselves are imprecise or have no certain bound, and it be-comes unwise to use crisp multidimensional vectors to representthem. In this paper, we recommend using fuzzy n-cell numbersto represent imprecise or uncertain multichannel digital signalsand put forward a method of constructing such fuzzy n-cellnumbers.

The concept of general fuzzy numbers was introduced byChang and Zadeh [2] in 1972 with the consideration of the prop-

Manuscript received July 16, 2008; revised October 15, 2008; acceptedNovember 29, 2008. First published January 9, 2009; current version publishedApril 1, 2009. This work was supported in part by the Natural Science Founda-tions of China under Grant 60772006, by the Natural Science Foundations ofZhejiang Province, China, under Grant Y7080044, and by the Engineering andPhysical Sciences Research Council, U.K., under Grant EP/F0219195.

G. Wang is with the Institute of Operational Research and Cyber-netics, Hangzhou Dianzi University, Hangzhou 310018, China (e-mail:[email protected]).

P. Shi is with the Department of Computing and Mathematical Sciences, Uni-versity of Glamorgan, Pontypridd CF37 1DL, U.K. He is also with the Institutefor Logistics and Supply Chain Management, School of Science and Engineer-ing, Victoria University, Melbourne, Vic. 8001, Australia, and also with theSchool of Mathematics and Statistics, University of South Australia, Adelaide,S.A. 5095, Australia (e-mail: [email protected]).

P. Messenger is with the Department of Computing and MathematicalSciences, University of Glamorgan, Pontypridd CF37 1DL, U.K. (e-mail:[email protected]).


erties of probability functions. Since then, both the numbers andthe problems in relation to them (see, for example, [3]–[6], [11],[16], and [19]–[21]) have been widely studied. With the develop-ment of theories and applications of fuzzy numbers, this conceptbecomes more and more important. Kaleva [7], however, used aspecial type of n-dimensional fuzzy number, whose sets of cutsare all hyperrectangles. In 2002, we carefully studied the specialtype of n-dimensional fuzzy number and called it fuzzy n-cellnumber in [14] and [15]. It has been demonstrated that the fuzzyn-cell number is used much more conveniently than generaln-dimensional fuzzy numbers in theoretical investigations andin some fields of application in [14], [15], and [17]. On the otherhand, n-dimensional fuzzy vector is also an important concept,which is the Cartesian product of n 1-D fuzzy numbers. In 1985,Kaufmann and Gupta [8] had already studied fuzzy vectors.Soon after, Miyakawa et al. [9], Nakamura [10], and Ramik andNakamura [12] also studied the problems of theories and appli-cations in relation to fuzzy vectors. In 1997, Butnariu [1] studiedmethods of solving optimization problems and linear equationsin the space of fuzzy vectors. Recently, Wang et al. [14] showedthat fuzzy n-cell numbers and n-dimensional fuzzy vectors canrepresent each other and obtained the representations of the jointmembership function and the edge membership functions of afuzzy n-cell number.

In a previous paper [15], we defined a metric DL on the fuzzyn-cell number space and studied its properties. We again stud-ied this type of metric in [14] with regard to two fuzzy n-cellnumbers in the form of n-dimensional fuzzy vectors. Althoughmetric DL can be more conveniently used in applications andtheoretical investigations, it has some shortcomings, i.e., it has atendency to be rougher, and cannot really characterize the degreeof difference of two fuzzy n-cell numbers in some applications(see Example 3.1 in Section III of this paper). In this paper, inorder to discuss the problem of pattern recognition in an impre-cise or uncertain environment based on degree of difference, wedefine two new metrics and some concepts of difference valueson fuzzy n-cell number space, which may better characterizethe degree of difference of two fuzzy n-cell numbers in someapplications, and study their properties.

It is well known that pattern recognition is an important fieldof research. In this aspect, research achievements are many (forexample, see [13]). In this paper, as applications of the metricsand difference values (defined by us), we study the problem ofpattern recognition in an imprecise or uncertain environment,

1063-6706/$25.00 © 2009 IEEE



put forward an algorithmic version of pattern recognition basedon the metrics or difference values (defined by us) of fuzzyn-cell numbers, and also give examples to show the applicationand rationality of the method.

This paper is organized as follows. In Section II, we givean example to show how to set up fuzzy n-cell numbers torepresent imprecise or uncertain multichannel digital signals. InSection III, we define two new metrics and study their properties.In Section IV, we introduce concepts of difference values of twofuzzy n-cell numbers and examine their properties. In Section V,an algorithmic version of pattern recognition is given based onthe metrics or the difference values defined by us and examplesare also given to show the application and rationality of themethod. Finally, in Section VI, we give a brief conclusion ofthis paper.

II. REPRESENTATIONS OF UNCERTAIN MULTICHANNEL

DIGITAL SIGNALS

A fuzzy set of the Euclidean space Rn is a function u: Rn →[0, 1]. For fuzzy set u, we denote [u]r = x ∈ Rn : u(x) ≥ rfor r ∈ [0, 1] and [u]0 = x ∈ Rn : u(x) > 0 (the closure ofx ∈ Rn : u(x) > 0). If u is a normal and fuzzy convex fuzzyset of Rn, u(x) is upper semicontinuous, and [u]0 is com-pact, then we call u an n-dimensional fuzzy number and de-note the n-dimensional fuzzy number space by En . If u ∈ E,and for each r ∈ [0, 1], [u]r is a hyperrectangle, i.e., there ex-ist ui(r), ui(r) ∈ R with ui(r) ≤ ui(r), (i = 1, 2, . . . , n) suchthat [u]r =

∏ni=1 [ui(r), ui(r)], then we call u a fuzzy n-cell

number and denote the fuzzy n-cell number space by L(En ). Ann-dimensional fuzzy vector is an ordered class (u1 , u2 , . . . , un ),where ui ∈ E (i.e., E1), i = 1, 2, . . . , n. We have shown in [14]that fuzzy n-cell numbers and n-dimensional fuzzy vectors canrepresent each other, and as the representation is unique, L(En )and the n-dimensional fuzzy vector space (i.e., the Cartesian

product

n︷︸︸︷E × E × · · · × E) may be regarded as identical.

When exploring and discussing some quantity, properties, orlaws of movement of phenomena/objects in the physical world,it is essential for us to establish the description space of them.For instance, when the quantity in question is only the one witha single factor, we can take it as a dot in real number field R,i.e., the space of quantities corresponding to single factor can bedescribed by 1-D Euclidean space R. Similarly, we can describethe quantities with n factors, using n-dimensional Euclideanspace Rn . However, in the physical world, many phenomenaare imprecise or uncertain (such as having no certain bound).When the quantity discussed by us possesses some impreciseor uncertain attributes, it is unsuitable that we still use Rn torepresent the space of the quantities (see Remark 2.1). It isour opinion that using the fuzzy n-cell number space discussedin [14] and [15] to describe the quantities with some uncertainfactors and discuss these quantities in this n-dimensional fuzzyvector space is a more suitable method to reveal the objectivelaws of things in physical world (see Remark 2.1).

In the following example, we demonstrate how we constructa fuzzy n-cell number to represent a quantity that possesses

some uncertain attributes based on statistical data. About thealgorithmic version of such fuzzy n-cell numbers, we can seethe first or second step of the algorithmic version in Section V.

Example 2.1: It is well known that different kinds of terrainor landcover possess different reflections of the electromagneticspectrum. Based on this principle, one can set up a method torecognize the category of landcover, a challenging remote sens-ing classification problem, using spectral and terrain features forvegetation classification in some zone. In remote sensing classi-fication, the colligation of all species covering a zone of 4500 m2

can be boiled down to an element of remote sensing space. Weuse “Korean pine accounts for the main part” to denote forestthat mainly contains Korean pines. Because in different “Koreanpine accounts for the main part” areas, there are many differentfactors, such as the difference of the density of Korean pines,of the species and quantity of other plants, of the physiognomy,etc., the values of reflections of the electromagnetic spectrumare also different. Therefore, “Korean pine accounts for the mainpart” should not be a certain crisp value but a fuzzy set withoutcertain bound. So, using a fuzzy number to represent the spec-tral sensitivity level of the “Korean pine accounts for the mainpart” is more suitable than using a crisp number. Suppose thatwe use four wave bands: MSS-4, MSS-5, MSS-6, and MSS-7.We take ten samples and acquire the following data for somezone of “Korean pine accounts for the main part”:

MSS-4 MSS-5 MSS-6 MSS-7sample 1 15.01 13.30 40.50 19.37sample 2 15.60 12.56 38.81 16.35sample 3 15.82 12.79 37.70 18.16sample 4 14.90 11.70 35.50 14.75sample 5 16.10 13.80 42.10 20.75sample 6 13.80 11.94 32.10 15.54sample 7 15.90 10.98 30.87 14.29sample 8 16.82 13.67 37.64 18.62sample 9 15.50 12.58 36.10 18.02sample 10 15.38 12.48 34.08 17.45.

We can directly work out the following means µi (i =1, 2, 3, 4) and standard deviations σi (i = 1, 2, 3, 4) from thedata:

MSS-4 MSS-5 MSS-6 MSS-7µi : µ1 = 15.46 µ2 = 12.58 µ3 = 36.54 µ4 = 17.33σi : σ1 = 1.22 σ2 = 0.88 σ3 = 3.55 σ4 = 2.08.

From the means and the standard deviations, with

ui(xi)

=

xi − (µi − 2σi)

2σi, if xi ∈ [µi − 2σi, µi ] ∩ (0,+∞)

(µi + 2σi) − xi

2σi, if xi ∈ (µi, µi + 2σi ] ∩ (0,+∞)

0, if xi /∈ [µi−2σi, µi+2σi ] ∩ (0,+∞)

i = 1, 2, 3, 4

we can define four triangular model 1-D fuzzy numbersu1 , u2 , u3 , and u4 , which correspond to MSS-4, MSS-5, MSS-6,


WANG et al.: REPRESENTATION OF UNCERTAIN MULTICHANNEL DIGITAL SIGNAL SPACES AND STUDY 423

and MSS-7, respectively

u1(x1) =

x1 − 13.022.44

, if x1 ∈ [13.02, 15.46]

17.9 − x1

2.44, if x1 ∈ (15.46, 17.9]

0, if x1 /∈ [13.02, 17.9]

u2(x2) =

x2 − 10.821.76

, if x2 ∈ [10.82, 12.58]

14.34 − x2

1.76, if x2 ∈ (12.58, 14.34]

0, if x2 /∈ [10.82, 14.34]

u3(x3) =

x3 − 29.447.1

, if x3 ∈ [29.44, 36.54]

43.64 − x3

7.1, if x3 ∈ (36.54, 43.64]

0, if x3 /∈ [29.44, 43.64]

u4(x4) =

x4 − 13.174.16

, if x4 ∈ [13.17, 17.33]

21.49 − x4

4.16, if x4 ∈ (17.33, 21.49]

0, if x4 /∈ [13.17, 21.49].

By [14, Th. 3.1 and Th. 3.2], we know that u1 , u2 , u3 , andu4 determine a fuzzy four-cell number u = (u1 , u2,u3 , u4) andthe membership function of u is

u(x1 , x2 , x3 , x4) = minu1(x1), u2(x2), u3(x3), u4(x4)(x1 , x2 , x3 , x4) ∈ R4 .

Then, u can be used to represent “Korean pine accounts for themain part.”

Likewise, from the means and the standard deviations, ac-cording to

vi(xi)= exp(− (xi−µi)2

2σ2i

), if xi ∈ (0,+∞)

0, if xi /∈ (0,+∞)i= 1, 2, 3, 4

we can also define four Gaussian model 1-D fuzzy num-bers v1 , v2 , v3 , and v4 , which correspond to MSS-4, MSS-5,MSS-6, and MSS-7, respectively

v1(x1) = exp(− (x1 − 15.46)2

2.98

), if x1 ∈ (0,+∞)

0, if x1 /∈ (0,+∞) v2(x2) = exp(− (x2 − 12.58)2

1.56

), if x2 ∈ (0,+∞)

0, if x2 /∈ (0,+∞)

v3(x3) = exp(− (x3 − 36.54)2

25.21

), if x3 ∈ (0,+∞)

0, if x3 /∈ (0,+∞) v4(x4) = exp(− (x4 − 17.33)2

8.65

), if x4 ∈ (0,+∞)

0, if x4 /∈ (0,+∞)

and obtain the membership function of the fuzzy four-cellnumber v = (v1 , v2,v3 , v4) determined by v1 , v2 , v3 , and v4as v(x1 , x2 , x3 , x4) = minv1(x1), v2(x2), v3(x3), v4(x4),(x1 , x2 , x3 , x4) ∈ R4 . Then, the fuzzy four-cell number v canalso be used to represent the “Korean pine accounts for the mainpart.”

Remark 2.1: Of course, if the quantity to describe is pre-cise and certain, we should use a crisp multidimensional vec-tor to represent it. However, if the quantity to describe isimprecise and uncertain, such as “Korean pine accounts forthe main part,” then using a fuzzy n-cell number to rep-resent it is better than using a crisp n-dimensional vector.If we narrowly use a crisp multidimensional vector, such as(15.46, 12.58, 36.54, 17.33) (i.e., the mean vector), to repre-sent “Korean pine accounts for the main part,” then it can-not clearly tell us the relationship of “Korean pine accountsfor the main part” and the zone whose value of reflection ofelectromagnetic spectrum is (15.16, 12.80, 37.50, 16.79) since(15.16, 12.80, 37.50, 16.79) = (15.46, 12.58, 36.54, 17.33). Ifwe use fuzzy n-cell number v = (v1 , v2,v3 , v4) to representit, then we can almost affirm that the zone whose valueis (15.16, 12.80, 37.50, 16.79) is a part of “Korean pine ac-counts for the main part” since v(15.16, 12.80, 37.50, 16.79) =min(0.94, 0.94, 0.93, 0.93) = 0.93, i.e., the degree of the zonethat is “Korean pine accounts for the main part” is 0.93.

III. METRICS ON FUZZY n-CELL NUMBER SPACE

Diamond and Kloeden [3] studied the metric dp( · , · ) [notethat in this paper, we rewrite dp( · , · ) as Dp( · , · )] on generaln-dimensional fuzzy number space En , which is defined byDp(u, v) = (

∫ 10 [d([u]r , [v]r )]pdr)1/p for any u, v ∈ En , and

point out that the metric Dp is complete.We studied the metrics D and DL on L(En ) in [15], but

the two metrics seem to be “rough” in certain applications (seeExample 3.1). In the following, other metrics are defined onL(En ), which better reveal the difference between two differentuncertain quantities (see Example 3.1). Their properties are alsodiscussed such that they may be used appropriately.

We denote LC(Rn ) = A : there exist ai ≤ bi, i = 1,2, . . . , n such that A =

∏ni=1 [ai, bi ], where

∏ni=1 [ai, bi ] is

the Cartesian product [a1 , b1 ] × [a2 , b2 ] × · · · × [an , bn ].Theorem 3.1: We define mappings

dα : LC(Rn ) × LC(Rn ) → [0,+∞)

and

dα : LC(Rn ) × LC(Rn ) → [0,+∞)



r almost everywhere on [0, 1], so we know that u = v holdsby [18, Lemma 2.1]. Thus, Dα,p(u, v) = 0 ⇔ u = v holds.Likewise, we can prove that Dα,p(u, v) = 0 ⇔ u = v, so 3) ofthe theorem holds. From Theorem 3.1, we have

Dα,p(u, v) =(∫ 1

0

[r

n∑i=1

αi max|ui(r)

− vi(r)|, |ui(r) − vi(r)|]p

dr

)1/p

≤[∫ 1

0(r[dα ([u]r , [w]r ) + dα ([w]r , [v]r )])pdr

]1/p

≤(∫ 1

0[r · dα ([u]r , [w]r )]pdr

)1/p

+

(∫ 1

0[r · dα ([w]r , [v]r )]pdr

)1/p

= Dα,p(u,w) + Dα,p(w, v).

The proofs of Dα,p(u, v) ≤ Dα,p(u,w) + Dα,p(w, v), 5)and 6) can be similarly proved.

Remark 3.1: From Theorems 3.1 and 3.2, we know that dα , dα

and Dα,p , Dα,p are metrics on LC(Rn ) and L(En ), respec-tively, and satisfy translation invariance and absolute homo-geneity. Also, from the factor r of the integrands in the defini-tions of Dα,p(u, v) and Dα,p(u, v), we can see that the biggerthe degrees of the points are, which belong to the fuzzy n-cellnumbers u and v, the greater the effects on the metric of u andv. This is true in reality.

Example 3.1: Let u, v, and w be the two-cell numbers definedby u = (u1 , u2), v = (v1 , v2), and w = (w1 , w2), where

ui(x) =

x, if x ∈ [0, 1]

2 − x, if x ∈ (1, 2] i = 1 and 2.

0, if x /∈ [0, 2]

vi(x) =

x − 2, if x ∈ [2, 3]

4 − x, if x ∈ (3, 4]

0, if x /∈ [2, 4]

i = 1, 2

w1(x) =

x, if x ∈ [0, 1]

2 − x, if x ∈ (1, 2]

0, if x /∈ [0, 2]

w2(x) =

x − 2, if x ∈ [2, 3]

4 − x, if x ∈ (3, 4]

0, if x /∈ [2, 4].

Then, we know that ui(r) = r, ui(r) = 2 − r, vi(r) =2 + r, vi(r) = 4 − r (i = 1, 2), w1(r) = r, w1(r) = 2 − r,w2(r) = 2 + r, and w2(r) = 4 − r for r ∈ [0, 1]. From the def-initions of DL , we have DL (u, v) = 2 = DL (u,w), i.e., DL

cannot tell us the difference of DL (u, v) and DL (u,w), sowe say that DL seems to be “rough” (similar proof for D).However, as a matter of fact, DL (u, v) and DL (u,w) shouldhave some difference. Taking α = (1/2, 1/2), from the defi-nitions of Dα,p , we can obtain Dα,p(u, v) = 2/(1 + p)1/p >

1/(1 + p)1/p = Dα,p(u,w), which accords with the fact.If we restrain the metric Dp (i.e., dp( · , · ) defined by

Diamond in [3], see paragraph 1 of this section) on generaln-dimensional fuzzy number space En into L(En ), then it alsobecomes a metric on L(En ). In the following, we give the rela-tionships of the metrics Dα,p , Dα,p , and Dp .

Theorem 3.3: Metrics Dα,p , Dα,p , and Dp satisfy1) (1/2)Dα,p ≤ Dα,p ≤ Dα,p ≤ Dp ≤ D, i.e., (1/2)Dα,p

(u, v) ≤ Dα,p(u, v)≤ Dα,p(u, v) ≤Dp(u, v) ≤ D(u, v)for any u, v ∈ L(En ) (D is discussed in [15]);

2) Dα,p ≤(1/(p + 1)1/p

)DL ≤

(1/(p + 1)1/p

)D, i.e.,

Dα,p(u, v)≤ (1/(p + 1)1/p)DL (u, v) ≤(1/(p + 1)1/p

)D(u, v) for any u, v ∈ L(En ).

Proof: For any u, v ∈ L(En ) and r ∈ [0, 1], by the definitionsof dα and dα , we have the following equation shown at thebottom of this page.

12dα ([u]r , [v]r ) =

12

n∑i=1

αi max|ui(r) − vi(r)|, |ui(r) − vi(r)|

=12

n∑i=1

αi

|ui(r) − vi(r)| + |ui(r) − vi(r)| + ||ui(r) − vi(r)| − |ui(r) − vi(r)||2

≤n∑

i=1

αi

|ui(r) − vi(r)| + |ui(r) − vi(r)| + |ui(r) − vi(r)| + |ui(r) − vi(r)|4

≤n∑

i=1

αi

|ui(r) − vi(r)| + |ui(r) − vi(r)|2

(= dα ([u]r , [v]r ))

≤n∑

i=1

αi

2max|ui(r) − vi(r)|, |ui(r) − vi(r)|2

= dα ([u]r , [v]r ).



From this, we can directly obtain (1/2)Dα,p(u, v) ≤Dα,p(u, v) ≤ Dα,p(u, v).

By [15, Th. 4.4], we know dL ≤ d ≤ √ndL , where

dL (A,B) = max1≤i≤n|ai − bi |, |ai − bi | for any A =∏ni=1 [ai, ai ] ∈ LC(Rn ) and B =

∏ni=1 [bi, bi ] ∈ LC(Rn ).

Therefore, for any u, v ∈ L(En ), we have

Dα,p(u, v) =(∫ 1

0

[r

n∑i=1

αi max|ui(r) − vi(r)|,

|ui(r) − vi(r)|]p

dr

)1/p

≤(∫ 1

0

[r

n∑i=1

αi max1≤j≤n

|uj (r) − vj (r)|,

|uj (r) − vj (r)|]p

dr

)1/p

=(∫ 1

0

[rdL ([u]r , [v]r )

n∑i=1

αi

]pdr

)1/p

≤(∫ 1

0[d([u]r , [v]r )]pdr

)1/p

(=Dp(u, v))

≤(∫ 1

0

[sup

r∈[0,1]d([u]r , [v]r )

]pdr

)1/p

=

(∫ 1

0[D(u, v)]pdr

)1/p

≤ D(u, v).

Thus, we also obtain Dα,p(u, v) ≤ Dp(u, v) ≤ D(u, v) and theproof of 1) of the theorem is complete.

From

Dα,p(u, v) =(∫ 1

0

[r

n∑i=1

αi max|ui(r) − vi(r)|,

|ui(r) − vi(r)|]p

dr

)1/p

≤(∫ 1

0[rdL ([u]r , [v]r )]pdr

)1/p

≤(∫ 1

0

[r sup

r∈[0,1]dL ([u]r , [v]r )

]pdr

)1/p

=

(∫ 1

0[rDL (u, v)]pdr

)1/p

≤ D(u, v)

(∫ 1

0rpdr

)1/p

=1

(p + 1)1/pD(u, v)

and [15, Th. 4.5], we can see that 2) of the theoremholds.

Remark 3.2: From 1) of Theorem 3.3, we see that(1/2)Dα,p ≤ Dα,p ≤ Dα,p , i.e., Dα,p and Dα,p are equivalent,so we know that Dα,p and Dα,p induce equivalent topologieson L(En ) by the knowledge of topological space.

IV. DIFFERENCE VALUES ON FUZZY n-CELL NUMBER SPACE

In Section III, we discussed metrics on L(En ). But some-times, these have some shortcomings demonstrating the differ-ence of two objects. For example, we consider that the degreeof difference of 1 and 2 is bigger than the degree of differenceof 1010 and 1010 + 1 though their metrics (Euclidean metric)both measure 1. A mapping from the Cartesian product X × Xof a set X into R needs to satisfy stronger conditions so that itcan become a metric, and this brings limitations in some appli-cations. The measure used to characterize the differences doesnot need to satisfy all metric conditions, for example, when weset up a method of pattern recognition based on the principle ofminimal difference (i.e., the principle of maximal likelihood),the measure used to characterize the differences does not needto satisfy all metric conditions. To conveniently set up methodsof pattern recognition using fuzzy n-cell numbers, we intro-duce the concepts of difference values on L(En ) and study theproperties.

Let u ∈ L(En ) and α = (α1 , α2 , . . . , αn ) ∈ Rn satisfy∑ni=1 αi = 1 and αi ≥ 0 (i = 1, 2, . . . , n). We denote

Mα (u) =∑n

i=1 αi

∫ 10 r[ui(r) + ui(r)]dr, M(u) = Mα (u) as

α = ((1/n), (1/n), . . . , (1/n)), and M(u) = M1(u) as u ∈ E.Definition 4.1: Let u, v ∈ L(En ) with Mα (u), Mα (v) ≥ 0

and Mα (u) + Mα (v) = 0. We denote

Lα,a(u, v) =∑n

i=1 αi

∫ 10 2r|ui(r) − vi(r)|dr

[Mα (u) + Mα (v)]a

and

Rα,a(u, v) =∑n

i=1 αi

∫ 10 2r|ui(r) − vi(r)|dr

[Mα (u) + Mα (v)]a

and call Lα,a(u, v) and Rα,a(u, v) a left difference value and aright difference value of u and v (with respect to the weight αand parameter a), respectively. And we denote

∆α,a(u, v) =12[Lα,a(u, v) + Rα,a(u, v)]

i.e.

∆α,a(u, v)=∑n

i=1 αi

∫ 10 r|(ui(r)−vi(r)|+|ui(r)−vi(r)|]dr(∑n

i=1 αi

∫ 10 r[ui(r)+vi(r)+ui(r)+vi(r)]dr

)a

and call ∆α,a(u, v) a difference value of u and v (with respect tothe weight α and parameter a), where α = (α1 , α2 , . . . , αn ) ∈Rn with

∑ni=1 αi = 1 and αi ≥ 0 (i = 1, 2, . . . , n), and

a ∈ (0,+∞). Specially, we denote ∆a(u, v) = ∆α,a(u, v) asα = ((1/n), (1/n), . . . , (1/n)), and ∆a(u, v) = ∆1,a(u, v) asu, v ∈ E.



Remark 4.1:1) Generally speaking, we consider that the degree of the

difference of two numbers is related not only to the metricof them but also to the sizes of them. As the metrics are thesame, the bigger the sizes of the two numbers, the smallerthe degree of their difference. The denominator [Mα (u) +Mα (v)]a in the definition of ∆α,a just plays the action (seeExample 4.1), and the exponent a in [Mα (u) + Mα (v)]a

can be properly chosen accordingly to the case in question.2) Taking the note that

∑ni = 1

αi [(M(ui) + M(vi))/ (M(ui) + M(vi))]a = 1 holds as α = ((1/n), (1/n), . . . , (1/n)) and a = 1, and

∑ni = 1

αi [(M(ui) + M(vi))/(Mα (u) + Mα (v))]a = 1 does not necessarilyhold as α = ((1/n), (1/n), . . . , (1/n)) or a = 1, from

∆α,a(u, v)

=∑n

i=1 αi

∫ 10 r[|ui(r) − vi(r)| + |ui(r) − vi(r)|]dr(∑n

i=1 αi

∫ 10 r[ui(r) + vi(r) + ui(r) + vi(r)]dr

)a

=n∑

i=1

αi

( ∫ 10 r[ui(r)+vi(r)+ui(r)+vi(r)]dr∑n

i=1 αi

∫ 10 r[ui(r)+vi(r)+ui(r)+vi(r)]dr

)a

×∫ 1

0 r[|ui(r) − vi(r)| + |ui(r) − vi(r)|]dr(∫ 10 r[ui(r) + vi(r) + ui(r) + vi(r)]dr

)a

=n∑

i=1

αi

(M(ui) + M(vi)Mα (u) + Mα (v)

)a

∆a(ui, vi)

we can directly see that ∆1(u, v) is a convex combinationof ∆1(ui, vi), i = 1, 2, . . . , n, but ∆α,a(u, v) is not neces-sarily a convex combination of ∆a(ui, vi), i = 1, 2, . . . , nas α = ((1/n), (1/n), . . . , (1/n)) or a = 1.

Example 4.1: Let u, v, u′, and v′ be the two-cell numbersdefined by u = (u1 , u2), v = (v1 , v2), u′ = (u′

1 , u′2), and v′ =

(v′1 , v

′2), where

ui(xi) =

xi, if xi ∈ [0, 1]

2 − xi, if xi ∈ (1, 2]

0, if xi /∈ [0, 2]

vi(xi) =

−10 + xi, if xi ∈ [10, 11]

12 − xi, if xi ∈ (11, 12]

0, if xi /∈ [10, 12]

u′i(xi) =

−100 + xi, if xi ∈ [100, 101]

102 − xi, if xi ∈ (101, 102]

0, if xi /∈ [100, 102]

v′i(xi) =

−110 + xi, if xi ∈ [10, 11]

112 − xi, if xi ∈ (11, 12]0, if xi /∈ [10, 12]

and i = 1, 2. Then, we know that ui(r) = r, ui(r) = 2 − r,

vi(r) = 10 + r, vi(r) = 12 − r, u′i(r) = 100 + r, u′

i(r) =102 − r, v′i(r) = 110 + r, and v′

i(r) = 112 − r (r ∈ [0, 1], i =

1, 2). From the definitions of Dα,p and ∆α,a , we can ob-tain Dα,p(u, v), Dα,p(u′, v′),∆α,a(u, v), and ∆α,a(u′, v′),as shown at the bottom of the next page, so we haveDα,p(u, v) = Dα,p(u′, v′) and ∆α,a(u, v) > ∆α,a(u′, v′).

Property 4.1: Let u, v ∈ L(En ) with Mα (u), Mα (v) ≥0, and Mα (u) + Mα (v) = 0, α = (α1 , α2 , . . . , αn ) ∈ Rn with∑n

i=1 αi = 1 and αi > 0 (i = 1, 2, . . . , n), and a ∈ (0,+∞).Then

1) ∆α,a(u, v) ≥ 0;2) ∆α,a(u, v) = 0 if and only if u = v;3) ∆α,a(u, v) = ∆α,a(v, u);4) ∆α,a(u,w) ≥ ∆α,a(v, w) for any w ∈ L(En ) and u ≤

v ≤ w;5) ∆α,a(u + w, v + w) ≤ ∆α,a(u, v) for any w ∈ L(En )

and w ≥ (0, 0, . . . , 0);6) ∆α,a(ku, kv) = k1−a∆α,a(u, v) for any k > 0.Proof: It is obvious that the conclusions 1) and 3) hold. The

proof of conclusion 2) can also be completed by imitating theproof of 3) of Theorem 3.2 by using [18, Lemma 2.1].

From u ≤ v ≤ w, we know that ui(r) ≤ vi(r) ≤ wi(r) andui(r) ≤ vi(r) ≤ wi(r) (i = 1, 2, . . . , n), so |ui(r) − wi(r)| ≥|vi(r) − wi(r)| and |ui(r) − wi(r)| ≥ |vi(r) − wi(r)|. There-fore, we have

n∑i=1

αi

∫ 1

0r[|ui(r) − wi(r)| + |ui(r) − wi(r)|]dr

≥n∑

i=1

αi

∫ 1

0r[|vi(r) − wi(r)| + |vi(r) − wi(r)|]dr.

On the other hand, from ui(r) ≤ vi(r) ≤ wi(r) andui(r) ≤ vi(r) ≤ wi(r) (i = 1, 2, . . . , n), we can alsosee that Mα (u) =

∑ni=1 αi

∫ 10 r[ui(r) + ui(r)]dr ≤∑n

i=1 αi

∫ 10 r[vi(r) + vi(r)]dr = Mα (v), so we can ob-

tain 0 ≤ Mα (u) ≤ Mα (v) ≤ Mα (w). Thus, we have

∆α,a(u,w)

=∑n

i=1 αi

∫ 10 r[|ui(r) − wi(r)| + |ui(r) − wi(r)|]dr

(Mα (u) + Mα (w))a

≥∑n

i=1 αi

∫ 10 r[|vi(r) − wi(r)| + |vi(r) − wi(r)|]dr

(Mα (v) + Mα (w))a

= ∆α,a(v, w)

so conclusion 4) holds.From w ≥ 0, we know that Mα (w) ≥ 0, so we have

∆α,a(u + w, v + w), as shown at the bottom of the next page,i.e., conclusion 5) holds.

For any k > 0, we have

∆α,a(ku, kv)

=

∑ni=1 αi

∫ 10 r[|(ku)i(r)−(kv)i(r)|+|(ku)i(r)−(kv)i(r)|]dr(∑n

i=1 αi

∫ 10 r[(ku)i(r)+(kv)i(r)+(ku)i(r)+(kv)i(r)]dr

)a



=k∑n

i=1 αi

∫ 10 r[|ui(r) − vi(r)| + |ui(r) − vi(r)|]dr

ka(∑n

i=1 αi

∫ 10 r[ui(r) + vi(r) + ui(r) + vi(r)]dr)a

=k1−a∆α,a(u, v)

so conclusion 6) holds. Therefore, the proof of the theorem iscomplete.

Remark 4.2:1) Although the conclusion 4) of Theorem 4.1 holds, u ≤

v ≤ w does not imply ∆α,a(u,w) ≥ ∆α,a(u, v) (seeExample 4.2). Comparing

∆α,a(u,w)

=∑n

i=1 αi

∫ 10 r[|ui(r) − wi(r)| + |ui(r) − wi(r)|]dr

(Mα (u) + Mα (w))a

Dα,p(u, v)

=12

[∫ 1

0[r(α1(|r − 10 − r| + |2 − r − 12 + r|)+α2(|r − 10 − r| + |2 − r − 12 + r|))]pdr

]1/p

=12

(20p

(1 + p)

)1/p

Dα,p(u′, v′)

=12

[∫ 1

0[r(α1(|100 + r − 110 − r| + |102 − r − 112 + r|)+α2(|100+r − 110 − r|+|102−r−112 + r|))]pdr

]1/p

=12

(20p

(1 + p)

)1/p

∆α,a(u, v)

=α1∫ 1

0 r(|r − 10 − r| + |2 − r − 12 + r|)dr + α2∫ 1

0 r(|r − 10 − r| + |2 − r − 12 + r|)dr(α1∫ 1

0 r(r + 10 + r + 2 − r + 12 − r)dr + α2∫ 1

0 r(r + 10 + r + 2 − r + 12 − r)dr)a

=1012a

∆α,a(u′, v′)

=α1∫ 1

0 r(|100 + r − 110 − r| + |102 − r − 112 + r|)dr + α2∫ 1

0 r(|100 + r − 110 − r| + |102 − r − 112 + r|)dr(α1∫ 1

0 r(100 + r + 110 + r + 102 − r + 112 − r)dr + α2∫ 1

0 r(100 + r + 110 + r + 102 − r + 112 − r)dr)a

=10

212a

∆α,a(u + w, v + w) =

∑ni=1 αi

∫ 10 r[|(u + w)i(r) − (v + w)i(r)| + |(u + w)i(r) − (v + w)i(r)|]dr(∑n

i=1 αi

∫ 10 r[(u + w)i(r) + (v + w)i(r) + (u + w)i(r) + (v + w)i(r)]dr

)a

=∑n

i=1 αi

∫ 10 r[|ui(r) + wi(r) − vi(r) − wi(r)| + |ui(r) + wi(r) − vi(r) − wi(r)|]dr(∑n

i=1 αi

∫ 10 r[ui(r) + wi(r) + vi(r) + wi(r) + ui(r) + wi(r) + vi(r) + wi(r)]dr

)a

=∑n

i=1 αi


(Mα (u) + Mα (v) + 2Mα (w))a

≤∑n

i=1 αi


(Mα (u) + Mα (v))a

= ∆α,a(u, v)



to

∆α,a(u, v)

=∑n

i=1 αi


(Mα (u) + Mα (v))a

we can see that the untruth of ∆α,a(u,w) ≥∆α,a(u, v) is caused only by (Mα (u) + Mα (w))a ≥(Mα (u) + Mα (v))a , and when (Mα (u) + Mα (w))a and(Mα (u) + Mα (v))a are properly smaller, u ≤ v ≤ w canimply ∆α,a(u,w) ≥ ∆α,a(u, v). So, in general, we maychoose a in (0, 1] such that ∆α,a can reasonably char-acterize the degree of the difference of two fuzzy n-cellnumbers.

2) Generally speaking, the difference value ∆α,a does notsatisfy the property of the triangular inequality, i.e., theinequality ∆α,a(u, v) ≤ ∆α,a(u,w) + ∆α,a(w, v) doesnot necessarily hold for u, v, w ∈ L(En ). Example 4.3can show it.

Example 4.2: Let u, v, and w be the two-cell numbers definedby: u = (0, 0), v = (v1 , v2), and w = (w1 , w2), where

vi(xi) =

−1 + xi, if xi ∈ [1, 2]

3 − xi, if xi ∈ (2, 3]

0, if xi /∈ [1, 3]

wi(xi) =

−2 + xi, if xi ∈ [2, 3]

4 − xi, if xi ∈ (3, 4]

0, if xi /∈ [2, 4]

and i = 1, 2. Then, we know that ui(r) = 0, ui(r) = 0, vi(r) =1 + r, vi(r) = 3 − r, wi(r) = 2 + r, and wi(r) = 4 − r (r ∈[0, 1], i = 1, 2), so we have u ≤ v ≤ w, but we can see from∆α,a(u,w) ≤ ∆α,a(u, v) from ∆α,2(u,w), and ∆α,2(u, v), asshown at the bottom of this page.

Example 4.3: Let u, v, and w be the two-cell numbers definedby: u = (u1 , u2), v = (1, 1), and w = (2, 2), where

ui(xi) =

1910

+ xi, if xi ∈[−19

10,− 9

10

]110

− xi, if xi ∈(− 9

10,

110

]0, if xi /∈

[−19

10,

110

]and i = 1, 2. Then, we know that ui(r) = −(19/10) +r, ui(r) = (1/10) − r, vi(r) = 1, vi(r) = 1, wi(r) = 2, andwi(r) = 2 (r ∈ [0, 1], i = 1, 2), so we have ∆α,1(u, v) =19,∆α,1(u,w) = 29/11, and ∆α,1(v, w) = 1/3, whichimplies that ∆α,1(u, v) > ∆α,1(u,w) + ∆α,1(v, w).

Example 4.4: Let u, v, and w be the two-cell numbers (seeFig. 1) defined by: u = (u1 , u2), v = (2, 2), and w = (w1 , w2),where

ui(xi) =

xi, if xi ∈ [0, 1]2 − xi, if xi ∈ (1, 2]0, if xi /∈ [0, 2]

wi(xi) =

−1 + xi, if xi ∈ [1, 2]3 − xi, if xi ∈ (2, 3]0, if xi /∈ [1, 3]

and i = 1, 2. Then, we know that ui(r) = r, ui(r) = 2 −r, vi(r) = 2, vi(r) = 2, wi(r) = 1 + r, and wi(r) = 3 − r (r ∈[0, 1], i = 1, 2), so we can obtain that Dα,p(u, v) = 1/2 =Dα,p(u,w) and ∆α,a(u, v) = 2/3a = ∆α,a(u,w). However,it is obvious (see Fig. 1) that the degree of the difference ofu and v is different from the degree of the difference of uand w.

In fact, sometimes, the degree of the difference of two fuzzynumbers is not only related with the metric and the sizes ofthem, but also related with the degree of fuzzy (we call it fuzzydegree) of them. Example 4.4 shows that for the u, v, and wgiven, the metric Dα,p and the difference value ∆α,a cannottell us the difference of the degree of the difference u and vwith the degree of the difference of u and w since Dα,p(u, v)= Dα,p(u,w) and ∆α,a (u, v) = ∆α,a(u,w), but we see that thetwo degrees of the differences indeed have some differences.In order to overcome the defect, we introduce the followingconcept.

∆α,2(u,w) =α1∫ 1

0 r[|0 − 2 − r| + |0 − 4 + r|]dr + α2∫ 1

0 r[|0 − 2 − r| + |0 − 4 + r|]dr(α1∫ 1

0 r[0 + 2 + r + 0 + 4 − r]dr + α2∫ 1

0 r[0 + 2 + r + 0 + 4 − r]dr)2

=13

∆α,2(u, v) =α1∫ 1

0 r(|0 − 1 − r| + |0 − 3 + r|)dr + α2∫ 1

0 r(|0 − 1 − r| + |0 − 3 + r|)dr(α1∫ 1

0 r(0 + 1 + r + 0 + 3 − r)dr + α2∫ 1

0 r(0 + 1 + r + 0 + 3 − r)dr)2

=12.



Fig. 1. Fuzzy two-cell numbers u, v, and w in Example 4.4.

Definition 4.2: Let u, v ∈ L(En ). We denote

Λα,a(u, v)

=

(n∑

i=1

αi

∫ 1

0r[|(ui(r) − vi(r)| + |ui(r) − vi(r)|]dr

)

× exp

(a

n∑i=1

αi

(∣∣∣∣∣∫ u∗

i (1)

ui (0)ui(t)dt −

∫ v ∗i (1)

vi (0)vi(t)dt

∣∣∣∣∣+

∣∣∣∣∣∫ ui (0)

u∗i(1)

ui(t)dt −∫ v i (0)

v ∗i(1)

vi(t)dt

∣∣∣∣∣))

and call Λα,a(u, v) a difference value of u and v(with respect to the weight α and parameter a), whereu∗

i (1) =(ui(1) + ui(1)

)/2, v∗i (1) =

(vi(1) + vi(1)

)/2, and

α = (α1 , α2 , . . . , αn ) ∈ Rn with∑n

i=1 αi = 1 and αi ≥ 0(i = 1, 2, . . . , n) and a ∈ (0,+∞). We denote Λa(u, v) =Λα,a(u, v) as α = ((1/n), (1/n), . . . , (1/n)) and Λa(u, v) =Λ1,a(u, v) as u, v ∈ E.

Example 4.5: Let u, v, and w be the two-cell numbers definedin Example 4.4. Then

Λα,1(u, v) =

(2∑

i=1

αi

∫ 1

0r(|r − 2| + |2 − r − 2|)dr

)

× exp

(2∑

i=1

αi

(∣∣∣∣∣∫ 1

0t dt −

∫ 2

21 dt

∣∣∣∣∣

+

∣∣∣∣∣∫ 2

1(2 − t)dt −

∫ 2

21 dt

∣∣∣∣∣))

=∫ 1

02r dr exp(1)

= e

and

Λα,1(u,w) =

(2∑

i=1

αi

∫ 1

0r(|r−1−r|+|2−r−3+r|)dr

)

× exp

(2∑

i=1

αi

(∣∣∣∣∣∫ 1

0t dt −

∫ 2

1(t − 1)dt

∣∣∣∣∣+

∣∣∣∣∣∫ 2

1(2 − t)dt −

∫ 3

2(3 − t)dt

∣∣∣∣∣))

=∫ 1

02r dr exp(0)

= 1

so we see that Λα,1(u, v) > Λα,1(u,w). Therefore, in this case,the difference value Λα,a is more suitable than metrics anddifference value ∆α,a to characterize the degree of differenceof two fuzzy n-cell numbers.

Property 4.2: Letu, v ∈ L(En ), α = (α1 , α2 , . . . , αn ) ∈ Rn

with∑n

i=1 αi = 1 and αi > 0 (i = 1, 2, . . . , n), and a ∈(0,+∞). Then

1) Λα,a(u, v) ≥ 0;2) Λα,a(u, v) = 0 if and only if u = v;3) Λα,a(u, v) = Λα,a(v, u);4) Λα,a(u + b, v + b) = Λα,a(u, v) for any b ∈ R;5) Λα,a(ku, kv) = |k|Λα,|k |a(u, v) for any k ∈ R.Proof: It is obvious that the conclusions 1) and 3) hold. The

proof of conclusion 2) can also be completed by imitating theproof of 3) of Theorem 3.2 by using [18, Lemma 2.1].

For any b ∈ R and i = 1, 2, . . . , n, we have

(ui + b)(t) = supr ∈ [0, 1] : t ∈ [(ui + b)(r), (ui + b)(r)]= supr ∈ [0, 1] : t − b ∈ [ui(r), ui(r)]= ui(t − b).

Hence

Λα,a(u + b, v + b)

=

(n∑

i=1

αi

∫ 1

0r|(ui(r) + b − vi(r) − b|

+ |ui(r) + b − vi(r) − b|]dr

)

× exp

(a

n∑i=1

αi

(∣∣∣∣∣∫ u∗

i (1)+b

ui (0)+b

(ui + b)(t)dt



−∫ v ∗

i (1)+b

vi (0)+b

(vi + b)(t)dt

∣∣∣∣∣+

∣∣∣∣∣∫ ui (0)+b

u∗i(1)+b

(ui+b)(t)dt−∫ v i (0)+b

v ∗i(1)+b

(vi+b)(t)dt

∣∣∣∣∣))

=

(n∑

i=1

αi

∫ 1

0r|(ui(r) − vi(r)| + |ui(r) − vi(r)|]dr

)

× exp

(a

n∑i=1

αi

(∣∣∣∣∣∫ u∗

i (1)

ui (0)(ui + b)(s + b)ds

−∫ v ∗

i (1)

vi (0)(vi+b)(s+b)ds

∣∣∣∣∣+∣∣∣∣∣∫ ui (0)

u∗i(1)

(ui+b)(s+b)ds

−∫ v i (0)

v ∗i(1)

(vi+b)(s+b)ds

∣∣∣∣∣))

=

(n∑

i=1

αi

∫ 1

0r|(ui(r) − vi(r)| + |ui(r) − vi(r)|]dr

)

× exp

(a

n∑i=1

αi

(∣∣∣∣∣∫ u∗

i (1)

ui (0)(ui)(s)ds−

∫ v ∗i (1)

vi (0)(vi)(s)ds

∣∣∣∣∣+

∣∣∣∣∣∫ ui (0)

u∗i(1)

(ui)(s)ds −∫ v i (0)

v ∗i(1)

(vi)(s)ds

∣∣∣∣∣))

= Λα,a(u, v)

i.e., conclusion 4) holds.For any k > 0, we have

Λα,a(ku, kv)

=

(n∑

i=1

αi

∫ 1

0r[|(ku)i(r)−(kv)i(r)|+|(ku)i(r)−(kv)i(r)|]dr

)

exp

(a

n∑i=1

αi

(∣∣∣∣∣∫ (ku)∗i (1)

(ku)i (0)(ku)i(t)dt

−∫ (kv )∗i (1)

(kv )i (0)(kv)i(t)dt

∣∣∣∣∣+∣∣∣∣∣∫ (ku)i (0)

(ku)∗i(1)

(ku)i(t)dt

−∫ (kv )i (0)

(kv )∗i(1)

(kv)i(t)dt

∣∣∣∣∣))

=

(n∑

i=1

αi

∫ 1

0r[|kui(r)−kvi(r)| + |kui(r) − kvi(r)|]dr

)

× exp

(a

n∑i=1

αi

(∣∣∣∣∣∫ ku∗

i (1)

kui (0)ui

(t

k

)dt

−∫ kv ∗

i (1)

kvi (0)vi

(t

k

)dt

∣∣∣∣∣

+

∣∣∣∣∣∫ kui (0)

ku∗i(1)

ui

(t

k

)dt −∫ kv i (0)

kv ∗i(1)

vi

(t

k

)dt

∣∣∣∣∣))

=

(k

n∑i=1

αi

∫ 1

0r[|ui(r) − vi(r)| + |ui(r) − vi(r)|]dr

)

× exp

(ka

n∑i=1

αi

(∣∣∣∣∣∫ u∗

i (1)

ui (0)ui(s)ds−

∫ v ∗i (1)

vi (0)vi(s)ds

∣∣∣∣∣+

∣∣∣∣∣∫ ui (0)

u∗i(1)

ui(s)ds −∫ v i (0)

v ∗i(1)

vi(s)ds

∣∣∣∣∣))

= |k|[Λα,|k |a(u, v)].

For any k < 0, we have

Λα,a(ku, kv)

=

(n∑

i=1

αi

∫ 1

0r[|(ku)i(r) − (kv)i(r)|

+ |(ku)i(r) − (kv)i(r)|]dr

)

× exp

(a

n∑i=1

αi

(∣∣∣∣∣∫ (ku)∗i (1)

(ku)i (0)(ku)i(t)dt

−∫ (kv )∗i (1)

(kv )i (0)(kv)i(t)dt

∣∣∣∣∣+∣∣∣∣∣∫ (ku)i (0)

(ku)∗i(1)

(ku)i(t)dt

−∫ (kv )i (0)

(kv )∗i(1)

(kv)i(t)dt

∣∣∣∣∣))

=

(n∑

i=1

αi

∫ 1

0r[|kui(r)−kvi(r)|+|kui(r)−kvi(r)|]dr

)

× exp

(a

n∑i=1

αi

(∣∣∣∣∣∫ ku∗

i (1)

kui (0)ui

(t

k

)dt

−∫ kv ∗

i (1)

kvi (0)vi

(t

k

)dt

∣∣∣∣∣+∣∣∣∣∣∫ kui (0)

ku∗i(1)

ui

(t

k

)dt

−∫ kvi (0)

kv ∗i(1)

vi

(t

k

)dt

∣∣∣∣∣))

=

(|k|

n∑i=1

αi

∫ 1


)

× exp

(a

n∑i=1

αi

(∣∣∣∣∣∫ u∗

i (1)

ui (0)kui(s)ds−

∫ v ∗i (1)

vi (0)kvi(s)ds

∣∣∣∣∣+

∣∣∣∣∣∫ ui (0)

u∗i(1)

kui(s)ds −∫ v i (0)

v ∗i(1)

kvi(s)ds

∣∣∣∣∣))



=

(|k|

n∑i=1

αi

∫ 1


)

× exp

(|k|a

n∑i=1

αi

(∣∣∣∣∣∫ u∗

i (1)

ui (0)ui(s)ds−

∫ v ∗i (1)

vi (0)vi(s)ds

∣∣∣∣∣+

∣∣∣∣∣∫ ui (0)

u∗i(1)

ui(s)ds −∫ v i (0)

v ∗i(1)

vi(s)ds

∣∣∣∣∣))

= |k|[Λα,|k |a(u, v)].

If k = 0, it is obvious that Λα,a(ku, kv) = |k|Λα,|k |a(u, v)holds, so conclusion 5) holds. Therefore, the proof of the theo-rem is complete.

At the end of the section, combining the definitions of ∆α,a

and Λα,a , we give the following definition of difference valueΓα,a .

Definition 4.3: Let u, v ∈ L(En ) with Mα (u), Mα (v) ≥ 0and Mα (u) + Mα (v) = 0. We denote Γα,a(u, v), as shown atthe bottom of this page, and call Γα,a(u, v) a difference valueof u and v [with respect to α and a = (a1 , a2)], where α =(α1 , α2 , . . . , αn ) ∈ Rn with

∑ni=1 αi = 1 and αi ≥ 0 (i =

1, 2, . . . , n), and a = (a1 , a2) ∈ (0,+∞) × (0,+∞). We de-note Γa(u, v) = Γα,a(u, v) as α = ((1/n), (1/n), . . . , (1/n))and Γa(u, v) = Γ1,a(u, v) as u, v ∈ E.

Likewise, we have the following properties about the differ-ence value Γα,a .

Property 4.3: Let u, v ∈ L(En ) with Mα (u), Mα (v) ≥0 and Mα (u) + Mα (v) = 0, α = (α1 , α2 , . . . , αn ) ∈ Rn with∑n

i=1 αi = 1 and αi > 0 (i = 1, 2, . . . , n), and a = (a1 , a2)∈ (0,+∞) × (0,+∞). Then

1) Γα,a(u, v) ≥ 0;2) Γα,a(u, v) = 0 if and only if u = v;3) Γα,a(u, v) = Γα,a(v, u);4) Γα,a(u + b, v + b) ≤ Γα,a(u, v) for any b ∈ [0,+∞);5) Γα,a(ku, kv) = k2−a1 Γα,b(u, v) for any k ∈ [0,+∞),

where b = (a1 , ka2).Proof: The proofs of the properties can be completed similarly

with the proofs of Properties 4.1 and 4.2, respectively, so weomit it.

V. PATTERN RECOGNITION BASED ON METRICS AND

DIFFERENCE VALUES

In Sections III and IV, we discussed metrics and differencevalues on L(En ). In this section, we put forward an algo-rithmic version of pattern recognition in an imprecise or un-certain environment based on the metrics and difference val-ues defined by us and give examples to show the application(see Example 5.1) and rationality (see Example 5.2) of themethod.

Consider a problem to identify an object (denoted by O) be-longing to some one of l classes (denoted by C1 , C2 , . . . , Cl) inan imprecise or uncertain environment. Let the objects have ncharacteristics. Since the problem discussed by us takes on someimprecise or uncertain attributes, it is unsuitable (see Remark2.1) that we use a crisp n-dimensional vector (i.e., a standardn-dimensional real number vector) to express the n charactervalues of Ci (i = 1, 2, . . . , l) or O. Therefore, using the methodof statistics, we construct l + 1 fuzzy n-cell numbers to expressthe n character values of C1 , C2 , . . . , Cl and O, respectively,and then put forward an algorithmic version of pattern recog-nition based on the metrics or the difference values definedby us.

A. Algorithmic Version of Pattern Recognition Based onMetrics

1) First Step: Depending on the practicality, we first findout one domain of the jth character value of Ci for each i (i =1, 2, . . . , l) and j (j = 1, 2, . . . , n) and denote the said domainby Di

j .We arbitrarily take mi samples in Ci (i = 1, 2, . . . , l) and gain

mi measure values (denoted by ci1 , c

i2 . . . , ci

m i) of the n charac-

ters of the mi samples and denote cim = (ci

m1 , cim2 , . . . , c

imn )

(m = 1, 2, . . . ,mi), i.e., we gain the following tables as shownby C1 , C2 , and Cl at the bottom of this page.

For Ci (i = 1, 2, . . . , l), we directly work out the follow-ing means µi

j = (1/mi)∑mi

k=1 cikj and standard deviations

σij =√

(1/(mi − 1))∑mi

k=1 (cikj − µi

j )2 (j = 1, 2, . . . , n) of

the n character values of Ci (i = 1, 2, . . . , l), respectively.We construct triangular model 1-D fuzzy numbers ui

j

Γα,a(u, v) =exp(a2∑n

i=1 αi

(∣∣ ∫ u∗i (1)

ui (0) ui(t)dt −∫ v ∗

i (1)vi (0) vi(t)dt| + |

∫ ui (0)u∗

i(1) ui(t)dt −

∫ v i (0)v ∗

i(1) vi(t)dt

∣∣))(∑ni=1 αi


)a1

×(

n∑i=1

αi

∫ 1


)

C1 :

c111 c1

12 · · · c11n

c121 c1

22 · · · c12n

......

...c1m 1 1 c1

m 1 2 · · · c1m 1 n

C2 :

c211 c2

12 · · · c21n

c221 c2

22 · · · c22n

......

...c2m 2 1 c2

m 2 2 · · · c2m 2 n

· · · Cl :

cl11 cl

12 · · · cl1n

cl21 cl

22 · · · cl2n

......

...clm l 1 cl

m l 2 · · · clm l n.



(i = 1, 2, . . . , l, j = 1, 2, . . . , n) as follows:

uij (x) =

x − (µij − 2σi

j )2σi

j

, if x ∈ [µij − 2σi

j , µij ] ∩ Di

j

(µij + 2σi

j ) − x

2σij

, if x ∈ (µij , µ

ij + 2σi

j ] ∩ Dij

0, if x/∈[µij−2σi

j , µij+2σi

j ]∩Dij

i = 1, 2, . . . , l, j = 1, 2, . . . , n

or construct Gaussian model 1-D fuzzy numbers vij (i =

1, 2, . . . , l, j = 1, 2, . . . , n) as follows:

vij (x) =

exp

(−

(x − µij )

2

2σi2

j

), if x ∈ Di

j

0, if x /∈ Dij

i = 1, 2, . . . , l j = 1, 2, . . . , n.

We construct fuzzy n-cell numbers ui = (ui1 , u

i2 , . . . , u

in )(

ui(x1 , x2 , . . . , xn ) = minui

1(x1), ui2(x2), . . . , ui

n (xn ))

and vi = (vi1 , vi

2 , . . . , vin ) (vi(x1 , x2 , . . . , xn ) = minvi

1(x1),vi

2(x2), . . . , vin (xn )), i = 1, 2, . . . , l, and use ui or vi to

express the ith class Ci (i = 1, 2, . . . , l).2) Second Step: For the object O to be recognized, taking t

samples in O, we can gain t classes of data about the n charactersof O as follows:

O :

o11 o12 · · · o1n

o21 o22 · · · o2n...

......

ot1 ot2 · · · otn.

We work out the following means (denoted by o1 , o2 , . . . , on )and standard deviations (denoted by s1 , s2 , . . . , sn ) of the ncharacter values of O:

oi =1t

t∑k=1

oki, i = 1, 2, . . . , n

and

si =

√√√√ 1t − 1

t∑k=1

(oki − oi)2 , i = 1, 2, . . . , n.

We construct triangular model 1-D fuzzy numbers wi (i =1, 2, . . . , n) as follows:

wi(x)=

x−(oi−2si)2si

, if x ∈ [oi−2si, oi ]∩(∪lj=1D

ji )

(oi+2si)−x

2si, if x ∈ (oi , oi+2si ]∩(∪l

j=1Dji )

0, if x /∈ [oi−2si, oi+2si ]∩(∪lj=1D

ji )

i = 1, 2, . . . , n

or construct Gaussian model 1-D fuzzy numbers w′i (i =

1, 2, . . . , n) as follows: v′i(x) = exp

(− (x − oi)2

2s2i

), if x ∈ ∪l

j=1Dji

0, if x /∈ ∪lj=1D

ji

i = 1, 2, . . . , n.

We construct fuzzy n-cell numbers w = (w1 , w2 , . . . , wn )(w(x1 , x2 , . . . , xn ) = min w1(x1), w2(x2), . . . , wn (xn ))and w′ = (w′

1 , w′2 , . . . , w′

n ) (w′(x1 , x2 , . . . , xn ) =min w′

1(x1), w′2(x2), . . . , w′

n (xn )), and use w or w′ toexpress the object O.

3) Third Step: Taking proper α = (α1 , α2 , . . . , αn ) with∑ni=1 αi = 1 and αi ≥ 0, i = 1, 2, . . . , n, and p ≥ 1, we com-

pute the metrics

Dα,p(w, uj )

=12

(∫0

1[r

n∑i=1

αi(|wi(r) − uji (r)|

+ |wi(r) − uji (r)|)

]p

dr

)1/p

, j = 1, 2, . . . , l

or

Dα,p(w′, vj )

=12

(∫ 1

0

[r

n∑i=1

αi(|w′i(r) − vj

i (r)|

+ |w′i(r) − vj

i (r)|)]p

dr

)1/p

, j = 1, 2, . . . , l.

4) Fourth Step: We choose uj0 in u1 , u2 , . . . , ul , or vj ′0 in

v1 , v2 , . . . , vl such that

Dα,p(w, uj0 )

= minDα,p(w, u1), Dα,p(w, u2), . . . , Dα,p(w, ul)

or

Dα,p(w′, vj ′0 )

= minDα,p(w′, v1), Dα,p(w′, v2), . . . , Dα,p(w′, vl).

Then, we can consider that object O belongs to the j0th classCj0 , or belongs to the j′0 th class Cj ′

0.

Remark 5.1: In the third and fourth steps of the aforemen-tioned method, we can use the metric Dα,p to replace the metricDα,p , as a result of which, we can also set up a method basedon the metric Dα,p .

B. Algorithmic Version of Pattern Recognition Based onDifference Values

1) First Step and the Second Step: They are same, respec-tively, with the first step and the second step of the method ofpattern recognition based on the metric, as mentioned earlier.

2) Third Step: Taking proper α = (α1 , α2 , . . . , αn ) with∑ni=1 αi = 1 and αi ≥ 0, i = 1, 2, . . . , n, and a = (a1 , a2) ∈

(0,+∞) × (0,+∞), we compute the difference valuesΓα,a(w, uj ) or Γα,a(w′, vj ), as shown at the bottom of the nextpage.



3) Fourth Step: We choose uj0 in u1 , u2 , . . . , ul or vj ′0 in

v1 , v2 , . . . , vl such thatΓα,a(w, uj0 )

= minΓα,a(w, u1),Γα,a(w, u2), . . . ,Γα,a(w, ul)or

Γα,a(w′, vj ′0 )

= minΓα,a(w′, v1),Γα,a(w′, v2), . . . ,Γα,a(w′, vl).Then, we can consider that object O belongs to the j0 th class

Cj0 , or belongs to the j′0 th class Cj ′0.

Remark 5.2: In the third and fourth steps of the aforemen-tioned method, we can have the difference value ∆α,a or Λα,a

to replace the difference value Γα,a , as a result of which, wecan also set up a method based on the difference value ∆α,a orΛα,a .

In order to be more obvious, we may use the following dia-gram to illustrate the methods set up by us.

Example 5.1: Suppose that some terrain consists of five dif-ferent types of land-based cover: C1 : road; C2 : farm or crop;

C3 : Korean pine accounts for the main part; C4 : boreal andbroad-leaf mixture forest; and C5 : birch forest. For the fivetypes of land cover (C1 , C2 , C3 , C4 , C5) and by using the fourwave bands: MSS-4, MSS-5, MSS-6, and MSS-7, we take tensamples and acquire the following data:

MSS-4 MSS-5 MSS-6 MSS-7sample 1 18.62 20.71 58.20 26.72sample 2 19.76 17.01 51.02 24.32sample 3 18.24 19.46 48.12 26.33sample 4 18.76 15.95 56.35 22.89

C1 : sample 5 18.96 18.78 45.32 28.32sample 6 19.90 20.13 50.82 25.05sample 7 19.16 18.58 52.30 22.21sample 8 19.36 17.32 55.02 25.28sample 9 19.36 17.98 48.26 23.86sample 10 18.48 16.46 46.63 27.46





Γα,a(w, uj ) =exp(

a2∑n

i=1 αi

(∣∣∣∣ ∫ w ∗i (1)

wi (0) wi(t)dt −∫ uj

i∗(1)

uji(0)

uji (t)dt

∣∣∣∣+ ∣∣∣∣ ∫ wi (0)w ∗

i(1) wi(t)dt −

∫ uji(0)

uji∗(1)

uji (t)dt

∣∣∣∣))(∑ni=1 αi

∫ 10 r[wi(r) + uj

i (r) + wi(r) + uji (r)]dr

)a1

×(

n∑i=1

αi

∫ 1

0r[|(wi(r) − uj

i (r)| + |wi(r) − uji (r)|]dr

)j = 1, 2, . . . , l

Γα,a(w′, vj ) =exp(

a2∑n

i=1 αi

(∣∣∣∣ ∫ w ′i∗(1)

w ′i(0) w′

i(t)dt −∫ v j

i∗(1)

v ji(0)

vji (t)dt

∣∣∣∣+ ∣∣∣∣ ∫ w ′i (0)

w ′i∗(1) w′

i(t)dt −∫ v j

i(0)

v ∗i(1) vj

i (t)dt

∣∣∣∣))(∑ni=1 αi

∫ 10 r[w′

i(r) + vji (r) + w′

i(r) + vji (r)]dr

)a1

×(

n∑i=1

αi

∫ 1

0r[|(w′

i(r) − vji (r)| + |w′

i(r) − vji (r)|]dr

)j = 1, 2, . . . , l.






C5 : sample 5 18.21 13.90 46.38 23.41sample 6 17.23 13.73 45.03 23.76sample 7 15.89 12.98 44.32 23.15sample 8 17.02 13.25 45.38 22.61sample 9 15.97 12.70 43.96 23.11sample 10 17.67 13.34 46.09 23.62.

By

µij =

110

10∑k=1

cikj , i = 1, 2, 3, 4, 5, j = 1, 2, 3, 4

and

σij =

√√√√ 110 − 1

10∑k=1

(cikj − µi

j )2 ,

i = 1, 2, 3, 4, 5, j = 1, 2, 3, 4

we can work out the following means and standard deviations:

µ11 = 19.06, σ1

1 = 0.55, µ12 = 18.24, σ1

2 = 1.58µ1

3 = 51.20, σ13 = 4.28, µ1

4 = 25.24, σ14 = 1.98

µ21 = 21.89, σ2

1 = 2.88, µ22 = 24.68, σ2

2 = 4.82µ2

3 = 47.37, σ23 = 4.09, µ2

4 = 21.63, σ24 = 2.39

µ31 = 15.46, σ3

1 = 1.22, µ32 = 12.58, σ3

2 = 0.88µ3

3 = 36.54, σ33 = 3.55, µ3

4 = 17.33, σ34 = 2.08

µ41 = 16.22, σ4

1 = 0.64, µ42 = 12.78, σ4

2 = 0.58µ4

3 = 42.41, σ43 = 2.87, µ4

4 = 21.22, σ44 = 1.50

µ51 = 17.01, σ5

1 = 0.82, µ52 = 13.20, σ5

2 = 0.42µ5

3 = 45.00, σ53 = 0.94, µ5

4 = 23.20, σ54 = 0.42.

Taking Dji = (0,+∞)(i = 1, 2, 3, 4 and j = 1, 2, 3, 4, 5),

then according to

uij (x) =

x − (µij − 2σi

j )2σi

j

, if x ∈ [µij − 2σi

j , µij ] ∩ Di

j

(µij + 2σi

j ) − x

2σij

, if x ∈ (µij , µ

ij + 2σi

j ] ∩ Dij

0, if x /∈ [µij−2σi

j , µij+2σi

j ]∩Dij

i = 1, 2, 3, 4, 5, j = 1, 2, 3, 4

we have

u11(x) =

x − 17.961.10

, if x ∈ [17.96, 19.06]

20.16 − x

1.10, if x ∈ (19.06, 20.16]

0, if x /∈ [17.96, 20.16]

u12(x) =

x − 15.083.16

, if x ∈ [15.08, 18.24]

21.40 − x

3.16, if x ∈ (18.24, 21.40]

0, if x /∈ [15.08, 21.40]

u13(x) =

x − 42.648.56

, if x ∈ [42.64, 51.20]

59.76 − x

8.56, if x ∈ (51.20, 59.76]

0, if x /∈ [42.64, 59.76]

u14(x) =

x − 21.283.96

, if x ∈ [21.28, 25.24]

29.20 − x

3.96, if x ∈ (25.24, 29.20]

0, if x /∈ [21.28, 29.20]

u21(x) =

x − 16.135.76

, if x ∈ [16.13, 21.89]

27.65 − x

5.76, if x ∈ (21.89, 27.65]

0, if x /∈ [16.13, 27.65]

u22(x) =

x − 15.049.64

, if x ∈ [15.04, 24.68]

34.32 − x

9.64, if x ∈ (24.68, 34.32]

0, if x /∈ [15.04, 34.32]

u23(x) =

x − 39.198.18

, if x ∈ [39.19, 47.31]

55.55 − x

8.18, if x ∈ (47.31, 55.55]

0, if x /∈ [39.19, 55.55]

u24(x) =

x − 16.854.78

, if x ∈ [16.85, 21.63]

26.41 − x

4.78, if x ∈ (21.63, 26.41]

0, if x /∈ [16.85, 26.41]

u31(x) =

x − 13.022.44

, if x ∈ [13.02, 15.46]

17.90 − x

2.44, if x ∈ (15.46, 17.90]

0, if x /∈ [13.02, 17.90]



u32(x) =

x − 10.82

1.76, if x ∈ [10.82, 12.58]

14.34 − x

1.76, if x ∈ (12.58, 14.34]

0, if x /∈ [10.82, 14.34]

u33(x) =

x − 29.44

7.10, if x ∈ [29.44, 36.54]

43.64 − x

7.10, if x ∈ (36.54, 43.64]

0, if x /∈ [29.44, 43.64]

u34(x) =

x − 13.17

4.16, if x ∈ [13.17, 17.33]

21.49 − x

4.16, if x ∈ (17.33, 21.49]

0, if x /∈ [13.17, 21.49]

u41(x) =

x − 14.94

1.28, if x ∈ [14.94, 16.22]

17.50 − x

1.28, if x ∈ (16.22, 17.50]

0, if x /∈ [14.94, 17.50]

u42(x) =

x − 11.62

1.16, if x ∈ [11.62, 12.78]

13.94 − x

1.16, if x ∈ (12.78, 13.94]

0, if x /∈ [11.62, 13.94]

u43(x) =

x − 36.67

5.74, if x ∈ [36.67, 42.41]

48.15 − x

5.74, if x ∈ (42.41, 48.15]

0, if x /∈ [36.67, 48.15]

u44(x) =

x − 18.22

3.00, if x ∈ [18.22, 21.22]

24.22 − x

3.00, if x ∈ (21.22, 24.22]

0, if x /∈ [18.22, 24.22]

u51(x) =

x − 15.37

1.64, if x ∈ [15.37, 17.01]

18.65 − x

1.64, if x ∈ (17.01, 18.65]

0, if x /∈ [15.37, 18.65]

u52(x) =

x − 12.36

0.84, if x ∈ [12.36, 13.20]

14.04 − x

0.84, if x ∈ (13.20, 14.04]

0, if x /∈ [12.36, 14.04]

u53(x) =

x − 43.02

1.98, if x ∈ [43.02, 45.00]

46.98 − x

1.98, if x ∈ (45.00, 46.98]

0, if x /∈ [43.02, 46.98]

u54(x) =

x − 22.36

0.84, if x ∈ [22.36, 23.20]

24.04 − x

0.84, if x ∈ (23.20, 24.04]

0, if x /∈ [22.36, 24.04]

and for r ∈ [0, 1]

u11(r) = 1.10r + 17.96 u1

1(r) = 20.16 − 1.10r

u12(r) = 3.16r + 15.08 u1

2(r) = 21.40 − 3.16r

u13(r) = 8.56r + 42.64 u1

3(r) = 59.76 − 8.56r

u14(r) = 3.96r + 21.28 u1

4(r) = 29.20 − 3.96r

u21(r) = 5.76r + 16.13 u2

1(r) = 27.65 − 5.76r

u22(r) = 9.64r + 15.04 u2

2(r) = 34.32 − 9.64r

u23(r) = 8.18r + 39.19 u2

3(r) = 55.55 − 8.18r

u24(r) = 4.78r + 16.85 u2

4(r) = 26.41 − 4.78r

u31(r) = 2.44r + 13.02 u3

1(r) = 17.90 − 2.44r

u32(r) = 1.76r + 10.82, u3

2(r) = 14.34 − 1.76r

u33(r) = 7.10r + 29.44 u3

3(r) = 43.64 − 7.10r

u34(r) = 4.16r + 13.17 u3

4(r) = 21.49 − 4.16r

u41(r) = 1.28r + 14.94 u4

1(r) = 17.50 − 1.28r

u42(r) = 1.16r + 11.62 u4

2(r) = 13.94 − 1.16r

u43(r) = 5.74r + 36.67 u4

3(r) = 48.15 − 5.74r

u44(r) = 3.00r + 18.22 u4

4(r) = 24.22 − 3.00r

u51(r) = 1.64r + 15.37 u5

1(r) = 18.65 − 1.64r

u52(r) = 0.84r + 12.36 u5

2(r) = 14.04 − 0.84r

u53(r) = 1.98r + 43.02 u5

3(r) = 46.98 − 1.98r

u54(r) = 0.84r + 22.36 u5

4(r) = 24.04 − 0.84r.

Thus, the fuzzy four-cell numbers ui = (ui1 , u

i2 , u

i3 , u

i4), i =

1, 2, 3, 4, 5, i.e.

ui(x1 , x2 , x3 , x4) = minui1(x1), ui

2(x2), ui3(x3), ui

4(x4)(x1 , x2 , x3 , x4) ∈ R4 , i = 1, 2, 3, 4, 5

can be used to represent Ci, i = 1, 2, 3, 4, 5, respectively.Using the same four wave bands: MSS-4, MSS-5, MSS-6,

and MSS-7, we now proceed to examine some zone (i.e., ob-ject, denoted by O) 12 times, stochastically, at various timesor positions, or using various viewers, and obtain the following



data:

MSS-4 MSS-5 MSS-6 MSS-7sample 1 18.31 13.13 45.41 23.52sample 2 17.21 12.71 45.67 23.49sample 3 17.00 12.57 44.82 22.56sample 4 16.32 13.06 43.53 22.77sample 5 18.01 13.91 45.31 23.31

O sample 6 16.11 13.59 46.97 23.61sample 7 16.21 12.56 43.96 23.14sample 8 17.13 13.32 46.01 22.84sample 9 16.20 12.34 43.81 23.13sample 10 17.51 13.34 45.89 23.52sample 11 17.02 13.78 45.21 23.18sample 12 18.02 13.25 44.56 23.23.

We can work out the following means and standarddeviations:

o1 = 17.09, s1 = 0.77 o2 = 13.17, s2 = 0.50o3 = 45.10, s3 = 1.01 o4 = 23.19, s4 = 0.33.

So, we can obtain the fuzzy four-cell number o =(o1 , o2 , o3 , o4), i.e.

o(x1 , x2 , x3 , x4) = mino1(x1), o2(x2), o3(x3), o4(x4)(x1 , x2 , x3 , x4) ∈ R4

which can be used to represent O, where

o1(x) =

x − 15.55

1.54, if x ∈ [15.55, 17.09]

18.63 − x

1.54, if x ∈ (17.09, 18.63]

0, if x /∈ [15.55, 18.63]

o2(x) =

x − 12.17

1.00, if x ∈ [12.17, 13.17]

14.17 − x

1.00, if x ∈ (13.17, 14.17]

0, if x /∈ [12.17, 14.17]

o3(x) =

x − 43.08

2.02, if x ∈ [43.08, 45.10]

47.12 − x

2.02, if x ∈ (45.10, 47.12]

0, if x /∈ [43.08, 47.12]

o4(x) =

x − 22.53

0.66, if x ∈ [22.53, 23.19]

23.79 − x

0.66, if x ∈ (23.19, 23.79]

0, if x /∈ [22.53, 23.79]

and for r ∈ [0, 1]

o1(r) = 1.54r + 15.55 o1(r) = 18.63 − 1.54r

o2(r) = 1.00r + 12.17 o2(r) = 14.17 − 1.00r

o3(r) = 2.02r + 43.08 o3(r) = 47.12 − 2.02r

o4(r) = 0.66r + 22.53 o4(r) = 23.79 − 0.66r.

Taking α = ((1/4), (1/4), (1/4), (1/4)) and a =((1/5), (1/5)), by Γα,a(u, v), as shown at the bot-tom of this page, we can obtain Γα,a(o, u1) =3.36,Γα,a(o, u2) = 10.34,Γα,a(o, u3) = 4.27,Γα,a(o, u4) =1.11, and Γα,a(o, u5) = 0.03. From Γα,a(o, u5) =minΓα,a(o, u1),Γα,a(o, u2),Γα,a(o, u3),Γα,a(o, u4),Γα,a

(o, u5), we know that O belongs to C5 , i.e., the zone measuredby us is covered by birch forest.

Remark 5.3: Although, only for example, using mean vec-tors (µi

1 , µi2 , µ

i3 , µ

i4), i = 1, 2, 3, 4, 5 and (o1 , o2 , o4 , o4) to rep-

resent, respectively, C1 , C2 , C3 , C4 and O, we perhaps alsojudge that O belongs to C5 by the usual Euclidean metrics,and we still emphasize that using fuzzy n-cell numbers to dealwith imprecise or uncertain quantities is better than using crispn-dimensional vectors. The following example (to simplify andshorten the problem, we consider only a 1-D case) will showthis.

Example 5.2: The following two classes of ferrous quantities(in kilograms per hundred kilogram) of minerals come from twodifferent mine areas (denoted by A and B)

A : 10.20, 11.76, 8.31, 9.02, 9.63, 8.33, 11.36, 12.30,

12.03, 7.98

B : 52.33, 79.34, 34.51, 62.34, 82.26, 28.36, 17.37, 25.32,

10.11, 8.34.

Suppose that one group (denoted by C) of minerals comesfrom the one of A and B. The problem to be solved is to identifyif C comes from A or B. We take samples and acquire thefollowing data for C:

C : 18.20, 20.31, 76.02, 9.36, 28.56, 23.32, 15.36, 20.51,

13.27, 20.32.

We can work out: µA = 10.09, σA = 1.67, µB =40.03, σB = 27.43, µC = 24.52, and σC = 18.86.

If we use crisp means to represent A,B, and C, then wehave A = 10.09, B = 40.03, C = 24.52, and d(C,A) = 14.43< 15.51 = d(C,B). If we regard d(C,A) < d(C,B) as ev-idence, we can draw a conclusion that C comes from A.

Γα,a(u, v) =exp(

a2∑4

i=1 αi

(∣∣∣∣ ∫ u∗i (1)

ui (0) ui(t)dt −∫ v ∗

i (1)vi (0) vi(t)dt

∣∣∣∣+ ∣∣∣∣ ∫ ui (0)u∗

i(1) ui(t)dt −

∫ v i (0)v ∗

i(1) vi(t)dt

∣∣∣∣))(∑4i=1 αi


)a1

×4∑

i=1

αi

∫ 1




However, the conclusion does not accord with fact. We shouldnote that although d(C,A) < d(C,B), the difference of d(C,A)and d(C,B) is small. Furthermore, from the statistical data, wecan see that the ferrous quantities of minerals coming from Aare more coincident, but B and C are not. It is almost impos-sible that some minerals in C come from A, such as mineralswith ferrous quantities 76.02 and 28.56. So we may judge thatC comes from B.

If we use fuzzy one-cell numbers to represent A,B, and C,then we have

A(x) =

x − 6.75

3.34, if x ∈ [6.75, 10.09]

13.43 − x

3.34, if x ∈ (10.09, 13.43]

0, if x /∈ [6.75, 13.43]

B(x) =

x + 14.83

54.86, if x ∈ [0, 40.03]

94.89 − x

54.86, if x ∈ (40.03, 94.89]

0, if x /∈ [0, 94.89]

C(x) =

x + 13.20

37.72, if x ∈ [0, 24.52]

62.24 − x

37.72, if x ∈ (24.52, 62.24]

0, if x /∈ [0, 62.24]

A(r) = 3.34r + 6.75

B(r) =

54.86r − 14.83, if r ∈ (0.27, 1]

0, if r ∈ [0, 0.27]

C(r) =

37.72r − 13.20, if r ∈ (0.35, 1]0, if r ∈ [0, 0.35]

A(r) = 13.43 − 3.34r

B(r) = 94.89 − 54.86r

C(r) = 62.24 − 37.72r

so

Γ(0.1,0.1)(C,A) = e3.27(14.01/3.487) = 106.39 > 21.77

= e1.178(15.38/4.019) = Γ(0.1,0.1)(C,B).

Thus, we can affirm that C comes from B.

VI. CONCLUSION

In this paper, we have suggested using fuzzy n-cell numbersto represent imprecise or uncertain multichannel digital signalsand have put forward a method (see the first or second stepof the algorithmic version in Section V, or see Example 2.1)of constructing such fuzzy n-cell numbers. Although the met-rics D and DL have been studied formerly in [14] and [15],in view of the roughness of D and DL , we have defined twonew metrics on fuzzy n-cell number space in order that theycan better characterize the degree of the difference of twoobjects in some imprecise or uncertain environment, and wehave studied their properties (Section III). In some applications,

metrics are unsuitable for use in finding the difference of twofuzzy n-cell numbers, so we introduced the concepts of dif-ference values ∆α,a ,Λα,a , and Γα,a , studied their properties,and showed the rationality for their use in characterizing thedegree of the difference of two fuzzy n-cell numbers by re-marks and examples (see Section IV). Finally, in Section V,we put forward an algorithmic version of pattern recognition inan imprecise or uncertain environment based on the metrics anddifference values defined by us and gave examples to show theapplication (see Example 5.1) and rationality (see Example 5.2)of the methods.

ACKNOWLEDGMENT

The authors would like to thank the Associate Editor andreviewers for their valuable suggestions and comments that havegreatly improved the presentation of the paper.

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[17] G. Wang and C. Zhao, “The measurability and (K)-integrability aboutfuzzy n-cell number value mappings,” J. Fuzzy Math., vol. 13, pp. 117–128, 2005.

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[19] T. Wang, Y. Chen, and S. Tong, “Fuzzy reasoning models and algorithmson type-2 fuzzy sets,” Int. J. Innovative Comput. Inf. Control, vol. 4,no. 10, pp. 2451–2460, 2008.

[20] M. Shimakawa, “A proposal of extension fuzzy reasoning method,” Int.J. Innovative Comput. Inf. Control., vol. 4, no. 10, pp. 2603–2615, 2008.

[21] K. Shinkai, “Decision analysis of fuzzy partition tree applying fuzzytheory,” Int. J. Innovative Comput. Inf. Control., vol. 4, no. 10, pp. 2581–2594, 2008.



Guixiang Wang received the B.Sc. degree in mathe-matics from Hebei Normal University, Shijiazhuang,China, in 1982, the M.Sc. degree in mathematics fromHebei University, Baoding, China, in 1989, and thePh.D. degree in mathematics from Harbin Institute ofTechnology, Harbin, China, in 2002.

From 1992 to 1999, he was an Associate Profes-sor at the Agricultural University of Hebei, Baoding,China. In 2002, he joined the Hangzhou Dianzi Uni-versity, Hangzhou, China, as a Professor. From 2003to 2005, he was a Postdoctoral Fellow at Harbin Engi-

neering University, Harbin, China. From 2007 to 2008, he was a Visiting SeniorFellow at the University of Glamorgan, Pontypridd, U.K. His current researchinterests include fuzzy set theory and application, nonlinear systems, signal andinformation processing, data fusion, and pattern recognition.

Peng Shi (SM’97) received the B.Sc. degree in math-ematics from Harbin Institute of Technology, Harbin,China, in 1982, the M.E. degree in control theory fromHarbin Engineering University, Harbin, in 1985, thePh.D. degree in electrical engineering from the Uni-versity of Newcastle, Newcastle, Australia, in 1994,the Ph.D. degree in mathematics from the Univer-sity of South Australia, Adelaide, Australia, in 1998,and the Doctor of Science (Higher Doctorate) degreefrom the University of Glamorgan, Pontypridd, U.K.,in 2006.

He was a Lecturer at Heilongjiang University (1985–1989), the Universityof South Australia (1997–1999), and a Senior Scientist at the Defence Scienceand Technology Organisation, Department of Defence, Australia (1999–2005).In 2004, he joined the University of Glamorgan as a Professor. His current re-search interests include robust control and filtering, fault detection techniques,Markov decision processes, and optimization techniques. He has authored orcoauthored a number of papers in these areas. In addition, he is a coauthor of thetwo research monographs: Fuzzy Control and Filtering Design for UncertainFuzzy Systems (Berlin, Jermany, Springer-Verlag, 2006) and Methodologies forControl of Jump Time-Delay Systems (Boston, MA, Kluwer, 2003). He is cur-rently the Editor-in-Chief of the International Journal of Innovative Computingand Information and Control.

Prof. Shi is an Associate Editor for a number of other journals, such as theIEEE TRANSACTIONS ON AUTOMATIC CONTROL, the IEEE TRANSACTIONS ON

SYSTEMS, MAN AND CYBERNETICS, PART B, and the IEEE TRANSACTIONS ON

FUZZY SYSTEMS. He is a Fellow of the Institute of Mathematics and its Appli-cations (U.K.).

Paul Messenger received the MMath Master’s degree in mathematics and thePh.D. degree in relativity theory from the University of Glamorgan, Wales, U.K.,in 2000 and 2005, respectively.

He is currently a Full-Time Lecturer in applied mathematics at the Universityof Glamorgan. His research interests are relativity theory, signal processing andrelated topics, and severe weather mathematics. He also does some outreachwork to local schools, explaining topics like relativity, etc., to local schoolchildren.



H∞ Fuzzy Control for Systems With Repeated ScalarNonlinearities and Random Packet Losses

Hongli Dong, Zidong Wang, Senior Member, IEEE, and Huijun Gao, Member, IEEE

Abstract—This paper is concerned with the H∞ fuzzy controlproblem for a class of systems with repeated scalar nonlinearitiesand random packet losses. A modified Takagi–Sugeno (T–S) fuzzymodel is proposed in which the consequent parts are composed ofa set of discrete-time state equations containing a repeated scalarnonlinearity. Such a model can describe some well-known nonlin-ear systems such as recurrent neural networks. The measurementtransmission between the plant and controller is assumed to be im-perfect and a stochastic variable satisfying the Bernoulli randombinary distribution is utilized to represent the phenomenon of ran-dom packet losses. Attention is focused on the analysis and design ofH∞ fuzzy controllers with the same repeated scalar nonlinearitiessuch that the closed-loop T–S fuzzy control system is stochasticallystable and preserves a guaranteed H∞ performance. Sufficientconditions are obtained for the existence of admissible controllers,and the cone complementarity linearization procedure is employedto cast the controller design problem into a sequential minimiza-tion one subject to linear matrix inequalities, which can be readilysolved by using standard numerical software. Two examples aregiven to illustrate the effectiveness of the proposed design method.

Index Terms—Diagonally dominant matrix, fuzzy systems, H∞control, linear matrix inequality (LMI), random packet losses, re-peated scalar nonlinearity.

I. INTRODUCTION

S INCE the concept of fuzzy sets was introduced by Zadehin 1965, fuzzy logic control has developed into one of

the most important and successful branch of automation andcontrol theory. In the past few decades, the fuzzy logic theoryhas been demonstrated to be effective in dealing with a vari-ety of complex nonlinear systems, which has therefore receiveda rapidly growing interest in the literature. In particular, thecontrol technique based on the so-called Takagi–Sugeno (T–S)fuzzy model has attracted much attention. The common prac-

Manuscript received September 18, 2008; revised November 23, 2008;accepted January 13, 2009. First published February 2, 2009; current ver-sion published April 1, 2009. This study was supported in part by the Engi-neering and Physical Sciences Research Council (EPSRC), U.K. under GrantGR/S27658/01, in part by the Royal Society, U.K., in part by the National Out-standing Youth Science Fund under Grant 60825303, in part by the National973 Program of China under Grant 2009CB320600, in part by the ResearchFound for the Doctoral Programme of Higher Education of China under Grant20070213084, in part by the Heilongjiang Outstanding Youth Science Fund un-der Grant JC200809, and in part by the Alexander von Humboldt Foundation,Germany.

H. Dong is with the Space Control and Inertial Technology Research Center,Harbin Institute of Technology, Harbin 150001, China, and also with the Collegeof Electrical and Information Engineering, Daqing Petroleum Institute, Daqing163318, China (e-mail: [email protected]).

Z. Wang is with the Department of Information Systems and Comput-ing, Brunel University, Uxbridge, Middlesex UB8 3PH, U.K. (e-mail: [email protected]).

H. Gao is with the Space Control and Inertial Technology Research Cen-ter, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]).


tice based on this technique is as follows. Fuzzy models haveprovided an approach to represent complex nonlinear systemsto a set of linear local models by using fuzzy sets and fuzzyreasoning. The overall fuzzy model of the system is achieved bysmoothly blending these local models together through member-ship functions. It has a convenient and simple dynamic structuresuch that the existing results for linear systems theory can bereadily extended for this class of nonlinear systems and, as aresult, a great number of important results has been reportedin the literature. For example, the problem of stability analysishas been investigated in [2], [3], [9], [10], and [25], and thestabilizing as well as H∞ control designs have been reportedin [1], [4], [8], [12], [19], [21], [22], [24] and [29].

The T–S fuzzy model has been widely employed to rep-resent or approximate a nonlinear system, which is describedby a family of fuzzy IF–THEN rules that represent local lin-ear input–output relations of the system. Nevertheless, the localmodel is not necessarily a linear one but sometimes a “simple” or“slightly” nonlinear system whose dynamics can be thoroughlyinvestigated. A good example of such a simple nonlinear sys-tem is the recurrent neural network that involves a nonlinear butknown activation function. Therefore, there has appeared initialresearch interest focusing on the extended T–S model whose sys-tem dynamics is captured by a set of fuzzy implications whichcharacterize local relations in the state space [15], [16], [30]. Inthis case, the local dynamics of each fuzzy rule is expressed by awell-studied nonlinear system, and the overall fuzzy model canbe achieved by fuzzy “blending” of these simple nonlinear localsystems. For example, a modified T–S fuzzy model has beenproposed in [16] in which the consequent parts are composed ofa set of stochastic Hopfield neural networks with time-varyingdelays, and a stability criterion has been derived in terms oflinear matrix inequalities (LMIs). The results of [16] have thenbeen extended in [15] to deal with the stability analysis prob-lem for T–S fuzzy cellular neural networks with time-varyingdelays. Motivated by the works in [15], [16] and [30], in thispaper, we will consider a more general yet well-known nonlin-earity, namely repeated scalar nonlinearity [5], [7], [11] whichcovers some typical classes of nonlinearities such as the semi-linear function, the hyperbolic tangent function that has beenextensively used for activation function in neural networks, thesine function, etc.

On the other hand, we notice that, in almost all aforemen-tioned literature concerning T–S fuzzy control systems, it hasbeen implicitly assumed that the communication between thephysical plant and controller is perfect, i.e., the signals trans-mitted from the plant always arrive at the controller withoutany information losses, and vice versa. Such an assumption,however, is not always true in practice. For example, due tothe unreliability of the network links, a networked control

1063-6706/$25.00 © 2009 IEEE


DONG et al.: H∞ FUZZY CONTROL FOR SYSTEMS WITH REPEATED SCALAR NONLINEARITIES AND RANDOM PACKET LOSSES 441

system (NCS) typically exhibits significant communication de-lays and data loss across the network, which gives rise to con-siderable research attention on how to design control systemswith the simultaneous consideration of the data loss issue (alsocalled package dropout or missing measurements). For exam-ple, in [27] and [28], the control and filtering problems foruncertain discrete-time stochastic systems with missing mea-surements have been investigated when data travel along un-reliable communication channels in a large, wireless, multihopsensor network. The problem of optimally controlling a lineardiscrete-time plant has been studied in [17] when some of themeasurement and control packets were missing. However, to thebest of the authors’ knowledge, the T–S fuzzy control problemfor nonlinear systems under unreliable communication linkshas not been fully investigated, which motivates our presentresearch.

In this paper, the H∞ fuzzy control problem is addressedfor a class of nonlinear systems under unreliable communi-cation links. The nonlinear system is described by a discrete-time state equation involving a repeated scalar nonlinearity thattypically appears in recurrent neural networks. The commu-nication links, existing between the plant and controller, areassumed to be imperfect, and the packet loss phenomena aremodeled by a Bernoulli random binary distributed white se-quence with a known conditional probability. The objective isto analyze and design a fuzzy controller such that the closed-loopfuzzy control system is stochastically stable while preserving aguaranteed H∞ performance. Sufficient conditions that involvematrix equalities are obtained for the existence of admissiblecontrollers, and the CCL procedure is employed to cast the non-convex feasibility problem into a sequential minimization prob-lem subject to LMIs, which can then be readily solved by usingstandard numerical software. Two numerical examples are givento illustrate the effectiveness of the proposed design method.

The rest of this paper is organized as follows. Section II for-mulates the problem under consideration. The stability condi-tion and H∞ performance of the closed-loop T–S fuzzy controlsystem are given in Section III. The H∞ fuzzy controller designproblem is solved in Section IV. The validity of this approachis demonstrated by illustrative examples in Section V. Finally,in Section VI, the conclusion is given.

Notation. The notation used in the paper is fairly standard.The superscript “T ” stands for matrix transposition, R

n de-notes the n-dimensional Euclidean space, R

m×n is the set of allreal matrices of dimension m × n; I and 0 represent the identitymatrix and zero matrix, respectively. The notation P > 0 meansthat P is real symmetric and positive definite; tr(M) refers tothe trace of the matrix M ; the notation ‖A‖ refers to the normof a matrix A defined by ‖A‖ =

√tr(AT A) and ‖ · ‖2 stands

for the usual l2 norm. In symmetric block matrices or com-plex matrix expressions, we use an asterisk (∗) to represent aterm that is induced by symmetry, and diag . . . stands fora block-diagonal matrix. In addition, Ex and Ex| y will,respectively, mean expectation of x and expectation of x con-ditional on y. The set of all nonnegative integers is denoted byI+ and the set of all nonnegative real numbers is represented by

R+ . Ξ denotes the class of all continuous nondecreasing con-

vex functions φ : R+ → R

+ such that φ(0) = 0 and φ(x) > 0for x > 0. Matrices, if their dimensions are not explicitly stated,are assumed to be compatible for algebraic operations.


A. Physical Plant

In this paper, we consider the following discrete-time fuzzysystems with repeated scalar nonlinearities: Plant rule i: IF θ1(k) is Mi1 and θ2(k) is Mi2 and · · · and

θp(k) is Mip THEN

xk+1 = Aif(xk ) + B2iuk + B1iwk

zk = Cif(xk ) + D2iuk + D1iwk

i = 1, . . . , r

(1)

where Mij is the fuzzy set; xk ∈ Rn represents the state vector;

uk ∈ Rm is the input vector; wk ∈ R

p is the exogenous distur-bance input which belongs to l2 [0,∞); zk ∈ R

q is the controlledoutput; Ai, B2i , B1i , Ci, D2i , and D1i are all constant matriceswith compatible dimensions; r is the number of IF-THEN rules;θk = [θ1(k), θ2(k), . . . , θp(k)] is the premise variable vector. Itis assumed that the premise variables do not depend on the inputvariable uk , which is needed to avoid a complicated defuzzifi-cation process of fuzzy controllers. f is a nonlinear functionsatisfying the following assumption as in [6].

Assumption 1: The nonlinear function f : R → R in system(1) satisfies

∀a, b ∈ R |f(a) + f(b)| ≤ |a + b| . (2)

In the sequel, for the vector x = [x1 x2 · · · xn ]T , wedenote

f(x)= [f(x1) f(x2) · · · f(xn )]T .

Remark 1: The model (1) is called a system with repeatedscalar nonlinearity [5], [7], [11]. Note that f is odd (by puttingb = −a) and 1-Lipschitz (by putting b = −b). Therefore, f en-capsulates some typical classes of nonlinearities, such as

1) the semilinear function (i.e., the standard saturationsat(s) := s if |s| ≤ 1 and sat(s) := sgn(s) if |s| > 1);

2) the hyperbolic tangent function that has been extensivelyused for activation function in neural networks;

3) the sine function, etc.Given a pair of (xk , uk ), the final outputs of the fuzzy system

are inferred as follows:

xk+1 =r∑

i=1

hi(θk )[Aif(xk ) + B2iuk + B1iwk ]

zk =r∑

i=1

hi(θk )[Cif(xk ) + D2iuk + D1iwk ]

(3)

where the fuzzy basis functions are given by

hi(θk ) =ϑi(θk )∑ri=1 ϑi(θk )

with ϑi(θk ) = Πpj=1Mij (θj (k)). Mij (θj (k)) represents the

grade of membership of θj (k) in Mij . Here, ϑi(θk ) has the



following basic property:

ϑi(θk ) ≥ 0, i = 1, 2, . . . , r,

r∑i=1

ϑi(θk ) > 0, ∀k

and therefore

hi(θk ) ≥ 0, i = 1, 2, . . . , r,

r∑i=1

hi(θk ) = 1 ∀k.

B. Controller

In this paper, we consider the following fuzzy control law forthe fuzzy system (3): Controller rule i: IF θ1(k) is Mi1 and θ2(k) is Mi2 and · · ·

and θp(k) is MipTHEN

uck = Kif(xk ), i = 1, 2, . . . , r.

Here, f(xk ) ∈ Rn is the input to the controller; uck ∈ Rm is theoutput of the controller; Ki are the gain matrices of the controllerto be designed. Hence, the controller can be represented by thefollowing input–output form:

uck =r∑

i=1

hi(θk )Kif(xk ). (4)


Due to the existence of the packet losses between the physicalplant and controller, the measurement of the plant is probably notequivalent to the input to the controller [i.e., f(xk ) = f(xk )],and the output of the controller is probably not equivalent to theinput of the plant (i.e., uk = uck ). As a result, we model thepacket losses phenomena via a stochastic approach as follows:

f(xk ) = αkf(xk ) uk = βkuck

where αk and βk are two independent Bernoulli processes.αk models the unreliable nature of the link from the sensorto the controller, and βk models that from the controller tothe actuator. Obviously, αk = 0 holds when the communicationlink fails (i.e., data are lost), and αk = 1 means successful trans-mission. The same happens for βk . A natural assumption on thesequence αk and βk can be made as follows:

Prob αk = 1 = E αk = α Prob αk = 0 = 1 − α

Prob βk = 1 = E βk = β Prob βk = 0 = 1 − β.


uk =r∑

i=1

hi(θk )βkαkKif(xk ). (5)

Note that such a stochastic Bernoulli approach has been ex-tensively used for dealing with data missing problems; see,e.g., [13] and [26] and the references therein.

D. Closed-Loop System

In this paper, we introduce another Bernoulli process ρkwith ρk

∆= αkβk . It is easy to know that ρk = 1 when both αk = 1

and βk = 1 are true, and ρk = 0 otherwise. Then, we have

Probρk = 1 = E ρk := ρ = αβ

Probρk = 0 = 1 − ρ = 1 − αβ.

uk =r∑

i=1

hi(θk )ρkKif(xk ). (6)

The closed-loop T–S fuzzy control system can now be obtainedfrom (3) and (6) that

xk+1 =r∑

i=1

r∑j=1

hi(θk )hj (θk )[Aij f(xk ) + B1iwk ]

zk =r∑

i=1

r∑j=1

hi(θk )hj (θk )[Cij f(xk ) + D1iwk ]

(7)

where Aij = Ai + ρB2iKj + ρkB2iKj , Cij =Ci+ ρD2iKj +ρkD2iKj , and ρk = ρk − ρ. It is clear that Eρk = 0 andEρ2

k = ρ(1 − ρ).Before formulating the problem to be investigated, we first

introduce the following definitions and lemmas.Definition 1 [23]: The solution xk = 0 of the closed-loop T–S

fuzzy control system in (7) with wk ≡ 0 is said to be stochasti-cally stable if, for any ε > 0, there exists a δ > 0 such that

E ‖xk‖ < ε (8)

whenever k ∈ I+ and ‖x0‖ < δ.

Definition 2: A square matrix P= [pij ] ∈ R

n×n is calleddiagonally dominant if for all i = 1, . . . , n

pii ≥∑j =i

|pij | . (9)

Lemma 1 [6]: If P > 0 is diagonally dominant, then for allnonlinear functions f satisfying (2), the following inequalityholds for all xk ∈ R

n :

fT (xk )Pf(xk ) ≤ xTk Pxk . (10)

Remark 2: It will be seen later that the purpose of requiringthe matrix P to satisfy (10) is to admit the quadratic Lyapunovfunction V (xk ) = xT

k Pxk .Lemma 2 [23]: If there exist a Lyapunov function V (xk ) and

a function φ(x) ∈ Ξ satisfying the following conditions

V (0) = 0 (11)

φ(‖xk‖) ≤ V (xk ) (12)

E V (xk+1) − E V (xk ) < 0, k ∈ I+ (13)

then the solution xk = 0 of the closed-loop T–S fuzzy controlsystem in (7) with wk ≡ 0 is stochastically stable.

Consider the fuzzy control problem in the presence of packetlosses phenomena and suppose the parameter ρ describing



intermittent transmission is known. We are now in a position tostate the problem of H∞ fuzzy control for systems with repeatedscalar nonlinearities and random packet losses as follows.

Problem H∞ fuzzy control with data loss (HFCDL): Given ascalar γ > 0, design a controller in the form of (4) such that

1) (stochastic stability) the closed-loop T–S fuzzy controlsystem in (7) is stochastically stable in the sense ofDefinition 1;

2) (H∞ performance) under zero initial condition, the con-trolled output zk satisfies

‖z‖E ≤ γ‖w‖2 (14)

where

‖z‖E

= E

√√√√ ∞∑

k=0

zTk zk

and ‖ · ‖2 stands for the usual l2 norm.

If the earlier two conditions are satisfied, the closed-loopT–S fuzzy control system is said to be stochastically stable witha guaranteed H∞ performance γ, and the problem HFCDL issolved.

III. H∞ FUZZY CONTROL PERFORMANCE ANALYSIS

In this section, the problem HFCDL formulated in the previ-ous section will be tackled via a quadratic approach describedin the following theorem.

Theorem 1: Consider the fuzzy system in (3) and supposethe gain matrices Ki (i = 1, . . . , r) of the controllers in (4)are given. The closed-loop fuzzy system in (7) is stochasticallystable with a guaranteed H∞ performance γ if there exists apositive diagonally dominant matrix P satisfying

ΨTii PΨii + ΛT

iiΛii − L < 0, i = 1, 2, . . . , r (15)

(Ψij + Ψj i)T P (Ψij + Ψj i) + (Λij + Λj i)T (Λij + Λj i)

− 4L < 0, 1 ≤ i < j ≤ r (16)

where

Ψij =[

Ai + ρB2iKj B1i

qB2iKj 0

]L =

[P 0

0 γ2I

]

Λij =[

Ci + ρD2iKj D1i

qD2iKj 0

]P =

[P 0

0 P

]q =

√ρ(1 − ρ).

Proof: In order to show that the fuzzy system in (7) is stochas-tically stable with a guaranteed H∞ performance γ under condi-tions (15) and (16), we define the following Lyapunov functioncandidate

V (xk ) = xTk Pxk . (17)

When wk ≡ 0, the difference of the Lyapunov function is cal-culated as

E∆V (xk ) = E V (xk+1)|xk − V (xk )

= E

r∑

i=1

r∑j=1

hi(θk )hj (θk )Aij f(xk )

T

P

×

r∑i=1

r∑j=1

hi(θk )hj (θk )Aij f(xk )

∣∣∣∣∣∣xk

− xT

k Pxk

= E

r∑

i=1

r∑j=1

r∑s=1

r∑t=1

hi(θk )hj (θk )hs(θk )ht(θk )

× fT (xk )ATijPAstf(xk )

∣∣xk

− xT

k Pxk

= E

r∑

i=1

r∑j=1

r∑s=1

r∑t=1

hi(θk )hj (θk )hs(θk )ht(θk )

× fT (xk )(

Aij + Aji

2

)T

P

(Ast + Ats

2

)

× f(xk )|xk

− xT

k Pxk

≤ E

r∑

i=1

r∑j=1

hi(θk )hj (θk )fT (xk )

×(

Aij + Aji

2

)T

P

(Aij + Aji

2

)

× f(xk )|xk

− xT

k Pxk . (18)

Note that in the earlier inequality, the elementary inequality of2aT b ≤ aT a + bT b for a, b ∈ R

n has been used.According to Lemma 1, we have

E∆V (xk ) ≤ E

r∑

i=1

r∑j=1

hi(θk )hj (θk )fT (xk )

×[(

Aij + Aji

2

)T

P

(Aij + Aji

2

)− P

]

× f(xk )|xk

= fT (xk )r∑

i=1

h2i (θk )(ΠT

ii PΠii − P )f(xk )



+12fT (xk )

r∑i,j=1,i<j

hi(θk )hj (θk )

×[(Πij + Πj i)

T P (Πij + Πj i) − 4P]f(xk )

where

Πij =[

Ai + ρB2iKj

qB2iKj

].

By the Schur complement lemma, we know from inequalities(15) and (16) that

ΠTii PΠii − P < 0, i = 1, 2, . . . , r

(Πij + Πj i)T P (Πij + Πj i) − 4P < 0, 1 ≤ i < j ≤ r.

Thus, we have

E V (xk+1) − E V (xk ) < 0

which satisfies (13). Taking φ(‖xk‖) = λmin(P )x2k such that

φ(·) ∈ Ξ, we obtain

φ(‖xk‖) = λmin(P )‖xk‖2 = λmin(P )xTk xk

≤ xTk Pxk = V (xk )

which satisfies (12). Considering V (0) = 0, it follows readilyfrom Lemma 2 that the closed-loop system in (7) with wk ≡ 0is stochastically stable.

Next, the H∞ performance criteria for the closed-loop systemin (7) will be established. Assuming zero initial conditions, anindex is introduced as follows:

J = E V (xk+1)| ξk + E

zTk zk

∣∣ ξk

− γ2wT

k wk − fT (xk )Pf(xk ).

Defining

ξk = [xTk wT

k ]T ηk = [ fT (xk ) wTk ]T

Gij = [Aij B1i ] Hij = [Cij D1i ]

similar to the derivation of (18), we have

J = E

r∑

i=1

r∑j=1

hi(θk )hj (θk )Gij ηk

T

P

r∑i=1

r∑j=1

hi(θk )

× hj (θk )Gij ηk

∣∣∣∣∣∣ ξk

+ E

r∑

i=1

r∑j=1

hi(θk )hj (θk )Hij ηk

T

×

r∑i=1

r∑j=1

hi(θk )hj (θk )Hij ηk

∣∣∣∣∣∣ ξk

− ηTk Lηk

≤ E

r∑

i=1

r∑j=1

hi(θk )hj (θk )ηTk

(Gij + Gji

2

)T

P

×(

Gij + Gji

2

)ηk

∣∣∣∣ ξk

+ E

r∑

i=1

r∑j=1


(Hij + Hji

2

)T

×(

Hij + Hji

2

)ηk

∣∣∣∣ ξk

− ηT

k Lηk

=r∑

i=1

r∑j=1


(Ψij + Ψj i

2

)T

P

×(

Ψij + Ψj i

2

)ηk +

r∑i=1

r∑j=1


×(

Λij + Λj i

2

)T (Λij + Λj i

2

)ηk − ηT

k Lηk

= ηTk

r∑i=1

h2i (θk )(ΨT

ii PΨii + ΛTiiΛii − L)ηk +

12ηT

k

×r∑

i,j=1,i<j

hi(θk )hj (θk )

×[(Ψij + Ψj i)T P (Ψij + Ψj i)

+ (Λij + Λj i)T (Λij + Λj i) − 4L]ηk

From inequalities (15) and (16), we know that J ≤ 0 and, ac-cording to Lemma 1, we have

ExT

k+1Pxk+1∣∣ ξk

+ E

zTk zk

∣∣ ξk

− γ2wT

k wk−xTk Pxk ≤ 0.

Taking mathematical expectation on both sides, we obtain

E

xTk+1Pxk+1

∣∣ ξk

− E

xT

k Pxk

∣∣ ξk

+ EzT

k zk

− γ2wTk wk ≤ 0.

For k = 0, 1, 2, . . . , summing up both sides under zero initialcondition and considering ExT

∞Px∞ ≥ 0, we arrive at

E

∞∑k=0

zTk zk

−

∞∑k=0

γ2wTk wk ≤ 0

which is equivalent to (14). The proof is now completed. Remark 3: In Theorem 1, with given controller gain and dis-

turbance attenuation level γ, we obtain the stochastic stabilityconditions of the nominal fuzzy system (7), which are repre-sented via a set of matrix inequalities in (15) and (16). We willshow later in the next section that such inequalities can be con-verted into LMIs when designing the actual controllers. Notethat the feasibility of LMIs can be easily checked by using theMATLAB LMI toolbox.

Remark 4: Let us now consider the standard H∞ performancecriterion for a discrete-time fuzzy closed-loop system with per-fect communication links between the plant and controller. Inthis case, we have ρ = 1 in (7) and then the inequalities (15)



and (16) reduce to

XTii PXii + Y T

ii Yii − L < 0, i = 1, 2, . . . , r (19)

(Xij + Xji)T P (Xij + Xji) + (Yij + Yji)T (Yij + Yji)

− 4L < 0, 1 ≤ i < j ≤ r (20)

where

Xij = [ Ai + B2iKj B1i ] Yij = [ Ci + D2iKj D1i ] .

Later, we will show via simulation that, in the case there indeedexist random packet losses, the main results given by Theorem 1will provide much improved performance over the standardH∞ approach that does not take into account the data missingproblem.

IV. H∞ FUZZY CONTROLLER DESIGN

In this section, we aim at designing a controller in the form of(4) based on Theorem 1, i.e., we are interested in determining thecontroller parameters such that the closed-loop fuzzy system in(7) is stochastically stable with a guaranteed H∞ performance.The following theorem provides sufficient conditions for theexistence of such H∞ fuzzy controller for system (7).

Theorem 2: Consider the fuzzy system in (3). There exists astate-feedback controller in the form of (4) such that the closed-loop system in (7) is stochastically stable with a guaranteed H∞

performance γ, if there exist matrices 0 < P= [pij ], L > 0,

Mi, i = 1, . . . , r, R = RT = [rij ] satisfying

−L ∗ ∗ ∗ ∗ ∗0 −γ2I ∗ ∗ ∗ ∗

AiL + ρB2iMi B1i −L ∗ ∗ ∗qB2iMi 0 0 −L ∗ ∗

CiL + ρD2iMi D1i 0 0 −I ∗qD2iMi 0 0 0 0 −I

< 0

i = 1, 2, . . . , r (21)[Υ11 ∗Υ21 Υ22

]< 0 (22)

pii −∑j =i

(pij + 2rij ) ≥ 0 (23)

rij ≥ 0 ∀i = j (24)

pij + rij ≥ 0 ∀i = j (25)

PL = I (26)

where

Υ11 =[−4L ∗

0 −4γ2I

]

Υ21 =

(Ai + Aj )L + ρB2 B1i + B1j

qB2 0

(Ci + Cj )L + ρD2 D1i + D1j

qD2 0

Υ22 = diag −L,−L,−I,−IB2 = B2iMj + B2jMi D2 = D2iMj + D2jMi

1 ≤ i < j ≤ r.

Furthermore, if the earlier conditions have feasible solutions,the gain Ki of the subsystem controller in (4) is given by

Ki = MiL−1 . (27)

Proof: From Theorem 1, we know that the closed-loop systemin (7) is stochastically stable with a guaranteed H∞ performanceγ if there exists a diagonally dominant matrix P > 0 satisfying(15) and (16). By the Schur complement, the following inequal-ities are obtained:

−P ∗ ∗ ∗ ∗ ∗0 −γ2I ∗ ∗ ∗ ∗

Ai + ρB2iKi B1i −P−1 ∗ ∗ ∗qB2iKi 0 0 −P−1 ∗ ∗

Ci + ρD2iKi D1i 0 0 −I ∗qD2iKi 0 0 0 0 −I

< 0

(28)[Υ11 ∗Υ21 Υ22

]< 0 (29)

where

Υ11 =[−4P ∗

0 −4γ2I

]

Υ21 =

(Ai + Aj ) + ρB2 B1i + B1j

qB2 0

(Ci + Cj ) + ρD2 D1i + D1j

qD2 0

Υ22 = diag

−P−1 ,−P−1 ,−I,−I

B2 = B2iKj + B2jKi D2 = D2iKj + D2jKi.

Performing congruence transformations to inequalities (28) and(29) by diag

P−1 , I, I, I, I, I

, we have

−P−1 ∗ ∗ ∗ ∗ ∗0 −γ2I ∗ ∗ ∗ ∗

(Ai + ρB2iKi)P−1 B1i −P−1 ∗ ∗ ∗qB2iKiP

−1 0 0 −P−1 ∗ ∗(Ci + ρD2iKi)P−1 D1i 0 0 −I ∗

qD2iKiP−1 0 0 0 0 −I

< 0[

Υ11 ∗ˇΥ21 Υ22

]< 0

where

Υ11 =[−4P−1 ∗

0 −4γ2I

]



Υ21 =

(Ai + Aj )P−1 + ρB2 B1i + B1j

qB2 0

(Ci + Cj )P−1 + ρD2 D1i + D1j

qD2 0

Υ22 = diag

−P−1 ,−P−1 ,−I,−I

B2 = (B2iKj + B2jKi)P−1

D2 = (D2iKj + D2jKi)P−1 .

Defining L = P−1 and Mi = KiP−1 , we can obtain (21) and

(22) readily. Furthermore, from (23) to (25), we have

pii ≥∑j =i

(pij + 2rij ) =∑j =i

(|pij + rij | + |−rij |) ≥∑j =i

|pij |

which guarantees the positive definite matrix P to be diagonallydominant, and the proof is then complete.

It is worth noting that, by far, we are unable to apply theLMI approach in the design of controller because of the matrixequality in Theorem 2. Fortunately, this problem can be ad-dressed with help from the cone complementarity linearization(CCL) algorithm proposed in [14]. The basic idea behind theCCL algorithm is that if the LMI[

P I

I L

]≥ 0

is feasible in the n × n matrix variables L > 0 and P > 0, thentr(PL) ≥ n; and tr(PL) = n if and only if PL = I . Basedon this, it is likely to solve the equalities in (26) by using theCCL algorithm. In view of this observation, we put forwardthe following nonlinear minimization problem involving LMIconditions instead of the original nonconvex feasibility problemformulated in Theorem 2.

The nonlinear minimization problem: min tr(PL) subject to(21)–(25) and [

P I

I L

]≥ 0. (30)

If the solution of min tr(PL) subject to (21)–(25) exists andmin tr(PL) = n, then the conditions in Theorem 2 are solvable.

Finally, the following algorithm is suggested to solve theearlier problem.

Algorithm HinfFC (HinfFC: H∞ Fuzzy Control)Step 1: Find a feasible set (P(0) , L(0) ,Mi (0) , R(0)) satisfy-

ing (21)–(25) and (30). Set q = 0.Step 2: According to (21)–(25) and (30), solve the LMI

problem: min tr(PL(q) + P(q)L).Step 3: Substitute the obtained matrix variables (P,L,Mi,

R) into (15) and (16). If conditions (15) and (16)are satisfied with |tr(PL) − n| < τ for some suffi-ciently small scalar τ > 0, then output the feasiblesolutions. Exit.

Step 4: If K > N , where N is the maximum number ofiterations allowed. Exit. Else, set q = q + 1 and goto Step 2.

Remark 5: As is well known, the packet dropout problem mayoccurs in the area of networked fault detection and isolation(FDI); see, e.g., [18] and [20] and the references therein. Oneof the future research topics would be the extension of our mainresults to network-based FDI and filtering problems.

Remark 6: Our main results are based on the LMI conditions.The LMI control toolbox implements state-of-the-art interior-point LMI solvers. While these solvers are significantly fasterthan classical convex optimization algorithms, it should be keptin mind that the complexity of LMI computations remains higherthan that of solving, say, a Riccati equation. For instance, prob-lems with a thousand design variables typically take over an houron today’s workstations. However, research on LMI optimiza-tion is a very active area in the applied math, optimization andthe operations research community, and substantial speedupscan be expected in the future.

V. ILLUSTRATIVE EXAMPLES

In this section, two simulation examples are presented toillustrate the fuzzy controller design method developed in thispaper.

Example 1: Consider a T–S fuzzy model (1) with repeatedscalar nonlinearities and random packet losses. The rules aregiven as follows:Plant rule 1: IF f1(xk ) is h1(f1(xk )) THEN

xk+1 = A1f(xk ) + B21uk + B11wk

zk = C1f(xk ) + D21uk + D11wk .(31)

Plant rule 2: IF f1(xk ) is h2(f1(xk )) THEN

xk+1 = A2f(xk ) + B22uk + B12wk

zk = C2f(xk ) + D22uk + D12wk .(32)

Controller rule 1: IF f1(xk ) is h1(f1(xk )) THEN uck =K1 f(xk ).Controller rule 2: IF f1(xk ) is h2(f1(xk )) THEN uck =

K2 f(xk ).The final outputs of the fuzzy system are inferred as follows:

xk+1 =2∑

i=1

hi(f1(xk ))[Aif(xk ) + B2iuk + B1iwk ]

zk =2∑

i=1

hi(f1(xk ))[Cif(xk ) + D2iuk + D1iwk ].

(33)

The model parameters are given as follows:

A1 =

1.0 0.31 0

0 0.33 0.21

0 0 −0.52

C1 =

0.2 0 0

0 0 0

0 0 0.1

A2 =

0.8 −0.38 0

−0.2 0 0.21

0.1 0 −0.55

B21 =

1 1

0 1

0 1



B11 =

0.1

0

0

C2 =

−0.12 0 0.1

0 0 0

0 0 0.1

B12 =

0

0.12

0

D21 =

1 1

0 1

0 1

B22 =

1 0

0 1

0 1

D12 =

0

0

0.22

D11 =

0.15

0

0

D22 =

1 1

0 1

0 1

and γ = 0.8. The membership function is assumed to be

h1(f1(xk )) =

1, f1(x0) = 0

| sin(f1(x0))|/f1(x0), else(34)

h2(f1(xk )) = 1 − h1(f1(xk )). (35)

Earlier, the nonlinear function f(xk ) = sin(xk ) satisfies (2).Our aim is to design a state-feedback paralleled controller inthe form of (4) such that the system (33) is stochastically stablewith a guaranteed H∞ norm bound γ.

Let ρ = 0.8. By applying Theorem 2 with help from Algo-rithm HinfFC, we can obtain admissible solutions as follows:

L =

3.0596 −0.4887 0.0629

−0.4887 0.7434 0.3136

0.0629 0.3136 0.8091

P =

0.3874 0.2799 −0.1065

0.2799 2.0222 −0.9528

−0.1065 −0.9528 1.8081

K1 =

[−0.4004 −0.0364 −0.1353

−0.0157 −0.1077 0.0209

]

K2 =[−0.1291 0.1054 −0.2146

0.0513 −0.0160 0.0166

].

First, we assume the initial condition to be

x0 = [ 0 0.01 0 ]T (36)

and the external disturbance wk ≡ 0. Fig. 1 gives the stateresponses for the uncontrolled fuzzy systems, which are ap-parently unstable. Fig. 2 gives the state simulation results ofthe closed-loop fuzzy system, from which we can see that theclosed-loop system is stochastically stable.

Next, to illustrate the disturbance attenuation performance,we choose the initial condition x0 ≡ 0 and the external distur-bance wk as follows:

wk =

0.2, 20 ≤ k ≤ 30−0.2, 40 ≤ k ≤ 50

0, else(37)

Fig. 3 shows the controller output, and Fig. 4 shows the evolu-tion of the state variables. The disturbance input wk and con-trolled output zk are depicted in Fig. 5. It can be calculated that

Fig. 1. State evolution xk of uncontrolled systems.

Fig. 2. State evolution xk of controlled systems.

‖z‖2 = 0.1669 and ‖w‖2 = 0.9381, and therefore, γ = 0.4218,which stays below the prescribed upper bound γ∗ = 0.8.

Example 2: In this example, we aim to show the advantageof considering the probabilistic packet losses. We demonstratethrough numerical simulation that, when the measurement trans-mission between the plant and controller is indeed imperfect, thedesign method proposed in this paper gives better performancethan the standard H∞ approach without taking into account thepacket loss problem. For this purpose, we let

A1 =

1.0 0.31 00 0.33 0.210 0 −1.5

and the other system data of (7) be the same as those inExample 1 with ρ = 0.6. First, we assume the initial condi-tion to be x0 = [ 0 0.01 0.2 ]T , and the external disturbancewk ≡ 0.



Fig. 3. Controllers uk .

Fig. 4. State evolution xk of controlled systems when w(k) = 0.

Fig. 5. Controlled output zk and disturbance input wk .

Fig. 6. Closed-loop state variables by taking the standard H∞ performancecriterion.

Fig. 7. Closed-loop state variables by taking the Theorem 2.

In such a case, if we use the standard H∞ performance crite-rion described in (19) and (20), the controller gain matrices canbe obtained as follows:

K1 =[−0.3231 0.0460 −0.7181

−0.0076 −0.1092 0.6953

]

K2 =[−0.3231 0.0597 −0.2481

0.0129 −0.0018 0.2066

]and the evolution of the corresponding state variables is givenin Fig. 6. If we use Theorem 2, which accounts for the packetlosses, the controller gain matrices are obtained as

K1 =[−1.7265 0.0129 −1.7443

0.3543 −0.1478 1.5592

]

K2 =[−0.8813 0.4437 −0.1694

0.1682 −0.1474 0.9820

]with the evolution of the state variables depicted in Fig. 7.



Fig. 8. Closed-loop state variables by taking the standard H∞ performancecriterion when w(k) = 0.

Fig. 9. Closed-loop state variables by taking the Theorem 2 when w(k) = 0.

Furthermore, to illustrate the disturbance attenuation perfor-mance, we set the initial condition x0 ≡ 0 and the externaldisturbance wk be the same as that in (37). Again, we apply thestandard H∞ performance criterion as described in (19) and(20) and Theorem 2, and display the evolutions of the statevariables under these two situations in Figs. 8 and 9, respec-tively. The disturbance attenuation performance indexes canalso be calculated as γ = 0.4670 and γ = 0.4445, respectively.By comparing Fig. 6 with Fig. 7 and noticing the H∞ perfor-mance indexes, we can conclude that the controller design byTheorem 2 has given better dynamical behavior as well as abetter disturbance rejection attenuation level, which confirmsour theoretical analysis for the problem of H∞ fuzzy control forsystems with repeated scalar nonlinearities and random packetlosses.

VI. CONCLUSION

In this paper, we have investigated the H∞ fuzzy control prob-lem for systems with repeated scalar nonlinearities and randompacket losses. The nonlinear system is described by a discrete-time state equation containing a repeated scalar nonlinearityand the control strategy takes the form of parallel distributedcompensation. The missing measurements are modeled by astochastic variable satisfying the Bernoulli random binary dis-tribution. The quadratic Lyapunov function has been used todesign H∞ fuzzy controllers such that, for the admissible ran-dom measurement missing and repeated scalar nonlinearities,the closed-loop T–S fuzzy control system is stochastically sta-ble and preserves a guaranteed H∞ performance. By using theCCL algorithm, sufficient conditions have been established thatensure the stochastic stability of the closed-loop system, and thecontroller gains have been obtained by the solution of a set ofLMIs. Two illustrative simulation examples have been given toillustrate the effectiveness of the proposed design method.

We would like to point out that our main results can beextended to more general/practical systems such as Ito-typestochastic systems, Markovian jumping systems and time-delaysystems, and the corresponding results will appear in the nearfuture.

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Hongli Dong received the B.Sc. degree in computerscience and technology from Heilongjiang Instituteof Science and Technology, Harbin, China, in 2000,and the M.Sc. degree in control theory and engineer-ing from Daqing Petroleum Institute, Daqing, China,in 2003. She is currently working toward the Ph.D.degree in control science and engineering from theHarbin Institute of Technology, Harbin.

She is currently a Lecturer at Daqing PetroleumInstitute. Her current research interests include robustcontrol and networked control systems. She is an ac-

tive reviewer for many international journals.

Zidong Wang (M’03–SM’03) was born in Jiangsu,China, in 1966. He received the B.Sc. degree in math-ematics from Suzhou University, Suzhou, China, in1986, and the M.Sc. degree in applied mathematicsand the Ph.D. degree in electrical and computer en-gineering from Nanjing University of Science andTechnology, Nanjing, China, in 1990 and 1994, re-spectively.

He is currently a Professor of dynamical systemsand computing, Brunel University, Uxbridge, Mid-dlesex, U.K. His current research interests include

dynamical systems, signal processing, bioinformatics, and control theory andapplications. He has published more than 120 papers in refereed internationaljournals.

Dr. Wang is currently serving as an Associate Editor for 12 international jour-nals including IEEE TRANSACTIONS ON AUTOMATIC CONTROL, IEEE TRANSAC-TIONS ON NEURAL NETWORKS, IEEE TRANSACTIONS ON SIGNAL PROCESSING,IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C, andIEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY.

Huijun Gao (M’06) was born in HeilongjiangProvince, China, in 1976. He received the M.S. de-gree in electrical engineering from Shenyang Univer-sity of Technology, Shengyang, China, in 2001, andthe Ph.D. degree in control science and engineeringfrom Harbin Institute of Technology, Harbin, China,in 2005.

He is currently a Professor at Harbin Instituteof Technology. His current research interests includenetwork-based control, robust control/filter theory,model reduction, time-delay systems, and multidi-

mensional systems, and their applications. He has published more than 80papers in refereed international journals.

Dr. Gao is an Associate Editor or Member of the editorial board forseveral journals, such as IEEE TRANSACTIONS ON SYSTEMS, MAN, AND

CYBERNETICS—PART B, International Journal of Systems Science, Journal ofIntelligent and Robotics Systems, and Journal of the Franklin Institute.



The Model of Fuzzy Variable Precision Rough SetsSuyun Zhao, Student Member, IEEE, Eric C. C. Tsang, Member, IEEE, and Degang Chen

Abstract—The fuzzy rough set (FRS) model has been introducedto handle databases with real values. However, FRS was sensitiveto misclassification and perturbation (here misclassification meanserror or missing values in classification, and perturbation meanssmall changes of numerical data). The variable precision rough sets(VPRSs) model was introduced to handle databases with misclas-sification. However, it could not effectively handle the real-valueddatasets. Now, it is valuable from theoretical and practical aspectsto combine FRS and VPRS so that a powerful tool, which not onlycan handle numerical data but also is less sensitive to misclassifi-cation and perturbation, can be developed. In this paper, we setup a model named fuzzy VPRSs (FVPRSs) by combining FRS andVPRS with the goal of making FRS a special case. First, we studythe knowledge representation ways of FRS and VPRS, and then,propose the set approximation operators of FVPRS. Second, weemploy the discernibility matrix approach to investigate the struc-ture of attribute reductions in FVPRS and develop an algorithm tofind all reductions. Third, in order to overcome the NP-completeproblem of finding all reductions, we develop some fast heuristicalgorithms to obtain one near-optimal attribute reduction. Finally,we compare FVPRS with RS, FRS, and several flexible RS-basedapproaches with respect to misclassification and perturbation. Theexperimental comparisons show the feasibility and effectiveness ofFVPRS.

Index Terms—Attribute reduction, fuzzy sets, knowledge repre-sentation, rough sets (RSs).

I. INTRODUCTION

ROUGH set (RS) theory, as one kind of generalization ofset theory, was proposed to deal with uncertainty and in-

discernibility [1], [2]. It assumed that objects characterized bythe same information were indiscernible (similar) in the view ofthe available information about them [3]. The indiscernibilityrelation generated in this way was the mathematical basis ofRS. It made RS work well on some problems, but it also limitedthe further applications of RS. For example, RS could not workeffectively on the datasets with real values. Furthermore, RSwas sensitive to misclassification and perturbation. Here, mis-classification represents erroneous or missing classification and

Manuscript received October 4, 2008; accepted December 30, 2008. Cur-rent version published April 1, 2009. This work was supported by the HongKong Research Grants Council (RGC) Competitive Earmarked Research Grant(CERG) under Grant PolyU 5273/05E (B-Q943) and Grant PolyU 5281/07E(B-Q06C). The work of D. Chen was supported by the National Science Financeof China (NSFC) under Grant 70871036.

S. Zhao is with the Department of Computing, Hong Kong PolytechnicUniversity, Kowloon, Hong Kong, and also with the School of Mathematicsand Computer Sciences, Hebei University, Baoding 071002, China (e-mail:[email protected]).

E. C. C. Tsang is with the Department of Computing, Hong Kong PolytechnicUniversity, Kowloon, Hong Kong (e-mail: [email protected]).

D. Chen is with the Department of Mathematics and Physics,North China Electric Power University, Beijing 102206, China (e-mail:[email protected]).



perturbation represents small change of real values. In recentyears, a great number of researchers have paid attention to thedevelopment of RS. Many extensions of RS, such as general-ization of approximation space, concepts approximation, roughfuzzy sets, fuzzy rough sets (FRSs) and variable precision roughsets (VPRSs), covering RSs, have been proposed and studied.Interested readers may consult [3] for summary of some of theseextensions. The focus of this paper is on two types of general-izations: FRSs and VPRSs.

Since theories of fuzzy sets and RSs were distinct and com-plementary on dealing with uncertainty, many researchers wereinspired to construct some new models, in which both indiscerni-bility and fuzziness were considered. The results of these stud-ies led to the construction of FRSs. Recently, most researchersfocused on the study of knowledge representation of FRS [5]–[15]. These studies mainly have been done from two branches:the constructive and axiomatic. The axiomatic branch took thelower and upper approximation operators as the primary notionand focused on studying the mathematical structure of FRS,such as algebraic and topologic structures [10]–[15]. On thecontrary, the constructive branch focused on studying the con-struction of the lower and upper approximation operators andwas often used to develop the applications of FRS [5]–[9].

Attribute reduction, as one important application of FRS, hasbeen studied in [16]–[23]. Shen and Jensen first proposed theattribute reduction method with FRS by keeping dependencyfunction invariant [18]–[21]. Their work performed well onsome practical applications, but their algorithms lacked math-ematical foundation and theoretical analysis. Unlike [18]–[21],an attribute reduction method by adopting information entropyto measure attribute significance was proposed in [22]. How-ever, the notions of attribute reduction in [22] were not basedon knowledge representation of FRS. That is to say, the knowl-edge representation part and the attribute reduction part wereindependent. By considering the limitations of the aforemen-tioned attribute reduction methods, the authors in [16] proposeda unified framework of attribute reduction with FRS that notonly proposed a formal notion of attribute reduction based onthe lower approximation operators, but also analyzed the math-ematical structure of attribute reduction. Although this methodproposed in [16] worked well on some datasets with fuzzifiedvalues, by experiments we found that their attribute reductionresults on the real-valued datasets were not compact becausethe lower approximation operators in FRS could not effectivelydeal with the problem of misclassification and were sensitive toperturbation of the original data [17], [50]. Now, it is necessaryand important to improve FRS to a robust model with respect tomisclassification and perturbation.

As a robust model, VPRS was proposed to handle the prob-lems with misclassification by incorporating a controlled de-gree of misclassification into knowledge representation of RS

1063-6706/$25.00 © 2009 IEEE



formalism [24]. Recently, many researchers have generalizedVPRS to some more general application areas [25]–[27]. Someof them have been focused on the parameter, i.e., the vari-able precision. For example, one generalization of VPRS withasymmetric bounds was proposed that allowed the lower andupper approximation operators with different variable preci-sions [25]. More researches have been put forward on theinvestigation of attribute reduction with VPRS. For example,by considering the unreasonable definition of attribute reduc-tion, some new concepts named beta lower and upper dis-tribution attribute reductions were then proposed in [26]. Al-though VPRS and its existing generalizations worked well onthe problems with misclassification, they could not effectivelydeal with the numerical problems with misclassification and/orperturbation.

By considering the characteristics of FRS and VPRS, someresearchers were inspired to construct a model, which can handlethe numerical problems with misclassification and/or perturba-tion, by combining FRS and VPRS. Thus, several flexible RS-based generalizations were proposed. For example, VPRS wasintroduced into a specific FRS framework (VPRS-FRS) in [28].It replaced the normal aggregation by β-precision aggregationto weaken the significant influence of some misclassificationsamples on the values of lower and upper approximation oper-ators. Furthermore, VPRS was generalized into a general FRSframework (VPFRS) [37]–[39]. In this generalization, misclas-sification was controlled by a crisp cut set that was introducedto the lower and upper approximation operators of a generalFRS framework. VPRS-FRS and VPFRS were effective to han-dle misclassification. However, they could not effectively dealwith the problems with perturbation. Recently, a robust modelwith respect to both misclassification and perturbation, vaguelyquantified rough sets (VQRSs), has been proposed [40]. It in-troduced a fuzzy quantification measure, i.e., quantifier, into theset approximation operators. This quantifier made VQRS workwell on deducing the influence of misclassification and pertur-bation, but it also limited the further applications of VQRS.For example, it was difficult for VQRS to design an attributereduction method since some important properties such as theapproximation quality being a monotonic measure did not holdin VQRS.

Now, it is valuable from both theoretical and practical aspectsto set up a framework by combining FRS and VPRS. From thetheoretical aspect, we can improve the knowledge representationpower of FRS and set up a robust framework so that a furthergeneralization of RSs is gained. From the practical aspect, wecan propose some new attribute reduction methods that are lesssensitive to both misclassification and perturbation in a fuzzyinformation system.

The rest of this paper is organized as follows. Section II re-views RS, FRS, and several RS-based flexible methods with re-spect to misclassification and/or perturbation. Section III buildsa framework named fuzzy VPRSs (FVPRSs) and develops somealgorithms to find the reductions by employing the discernibilitymatrix approach. In Section IV, we by experiments demonstratethe feasibility and effectiveness of FVPRS. The last sectionconcludes this paper.

II. PRELIMINARIES

In this section, we review RS, FRS, and several flexible RS-based methods with respect to misclassification and/or pertur-bation, which are necessary preliminaries for constructing themodel of FVPRSs.

A. Rough Sets

RS theory proposed by Pawlak [1] was an extension of settheory for the study of intelligent systems characterized byinsufficient and incomplete information. It is formally basedon an information system that is a pair IS = (U,A), whereU is a nonempty finite set of objects called the universe andA is a nonempty finite set of attributes. With every subsetof attributes B ⊆ A, we associate a binary relation definedas IND(B) = (x, y) ∈ U : a(x) = a(y),∀a ∈ B. The setsx ∈ U : [x]B ⊆ X and x ∈ U : [x]B ∩ X = φ are calledB-lower and B-upper approximations of X ⊆ U , and are de-noted by BX and BX , respectively. By using these approxima-tions, knowledge hidden in data can be represented as possibleand certain rules.

If the set of attributes in the information system consists oftwo parts: a set of condition attributes R and a set of decisionattributes D, the information system is called a decision system,or decision table, denoted by DS = (U,R ∪ D). One impor-tant application of RS was attribute reduction that distinguishedRS itself from other approaches since no additional informationsuch as expert knowledge and/or data distribution was neededin the process of attribute reduction. However, RS could noteffectively handle nonsymbolic valued databases with misclas-sification and perturbation. Thus, several generalizations of tra-ditional RSs were considered. Among these generalizations, weemphasize FRS and VPRS in this paper.

B. Fuzzy Rough Sets

1) Fuzzy Logical Operator: To provide a clear introductionof FRS in the constructive branch, we present and exemplifysome fuzzy logical operators used in the construction of the setapproximation operators in FRS [6], [10], [11], [15]: triangularnorm, triangular conorm, negator, dual, T -residuated implica-tion, and its dual operation.

A triangular norm, or shortly T -norm, is a function T :[0, 1] × [0, 1] → [0, 1] that satisfies the following conditions:monotonicity [if x < α, y < β, then T (x, y) ≤ T (α, β)], com-mutativity [T (x, y) = T (y, x)], associativity [T (T (x, y), z) =T (x, T (y, z))], and boundary condition [T (x, 1) = x]. Themost popular continuous T -norms include the standard minoperator TM (x, y) = minx, y and the bounded intersection(also called the Lukasiewicz T -norm) TL (x, y) = max0, x +y − 1.

A triangular conorm, or shortly T -conorm, is an in-creasing, commutative, and associative function S : [0, 1] ×[0, 1] → [0, 1] that satisfies the boundary condition: ∀x ∈[0, 1], S(x, 0) = x. The most well-known continuous. T -conorms include the standard max operator SM (x, y) =maxx, y and the bounded sum SL (x, y) = min1, x + y.


ZHAO et al.: MODEL OF FUZZY VARIABLE PRECISION ROUGH SETS 453

A negator N is a decreasing function N : [0, 1] → [0, 1]that satisfies N(0) = 1 and N(1) = 0. A negator N is calledinvolutive iff N(N(x)) = x for all x ∈ [0, 1]. The standardnegator is defined as NS (x) = 1 − x. Given a negator N ,T -norm T and T -conorm S are called dual with respect to(shortly denoted by w.r.t.) N iff De Morgan laws are satis-fied, i.e., S(N(x), N(y)) = N(T (x, y)) and T (N(x), N(y)) =N(S(x, y)).

Let X : U → [0, 1] be a fuzzy set and F (U) be the fuzzypower set on U , i.e., the collection of all fuzzy sets on U . Forevery X ∈ F (U), the symbol coN X is used to denote the fuzzycomplement of X determined by a negator N , i.e., for everyx ∈ U , (coN X)(x) = N(X(x)).

Given a lower semicontinuous triangular norm T , the resid-uation implication, or called the T -residuated implication isa function ϑ : [0, 1] × [0, 1] → [0, 1] that satisfies ϑ(x, y) =supz |z ∈ [0, 1], T (x, z) ≤ y for every x, y ∈ [0, 1]. T -residuated implications include the Godel implication ϑM basedon

TM : ϑM =

1, x ≤ yy, x > y

and the Lukasiewicz implication ϑL based on TL : ϑL =min1 − x + y, 1.

Given an upper semicontinuous triangular conorm S,the dual of T -residuated implication w.r.t. N is afunction σ : [0, 1] × [0, 1] → [0, 1] that satisfies σ(x, y) =infz |z ∈ [0, 1], S(x, z) ≥ y for every x, y ∈ [0, 1]. Notethat σ(N(x), N(y)) = N(ϑ(x, y)) and ϑ(N(x), N(y)) =N(σ(x, y)) hold for an involutive negator N [15].

2) Fuzzy Rough Set: FRS was first proposed by Dubois andPrade [5], [6] and then studied in [8], [10]–[12], and [15]. Inter-ested readers can consult [15] for more detailed summary of thedevelopment of FRS. Here, we present the general definition ofthe fuzzy approximation operators.

Suppose U is a nonempty universe (may not be finite) and Ris a binary fuzzy relation on U , then the fuzzy approximationoperators can be summarized as follows. For every fuzzy setA ∈ F (U)

RT A(x) = supu∈U

T (R(x, u), A(u))

RS A(x) = infu∈U

S(N(R(x, u)), A(u))

RσA(x) = supu∈U

σ(N(R(x, u)), A(u))

RϑA(x) = infu∈U

ϑ(R(x, u), A(u)).

Note that in the applications of FRS [5], [10], [16]–[22],[28], a binary fuzzy relation R is often specified as a fuzzyT -similarity relation, which is a fuzzy binary relation R on Usatisfying the following conditions: reflexivity (R(x, x) = 1),symmetry [R(x, y) = R(y, x)], and T -transitivity [R(x, y) ≥T (R(x, z), R(z, y))], for every x, y, z ∈ U [10]. In this paper,we focus on studying the fuzzy approximation operators basedon a fuzzy T -similarity relation.

Fig. 1. Fuzzy sets. (a) f1 . (b) f2 . (c) f3 .

From the viewpoint of duality, RT A and RS A are one pair ofdual approximation operators, and RσA and RϑA are anotherpair of dual approximation operators [15].

From the viewpoint of granular computing, these four approx-imation operators can be represented by fuzzy granules RT xλand Rσxλ [14]. They are

RϑA = ∪RT xλ : RT xλ ⊆ A; RT A = ∪RT xλ : xλ ⊆ A

RS A = ∪Rσxλ : Rσxλ ⊆ A; RσA= ∪ Rσxλ : xλ ⊆ A.

There are two conditions RT xλ ⊆ A and Rσxλ ⊆ A in theaforementioned granular representation of fuzzy approximationoperators. These conditions are too harsh for the construction ofthe fuzzy lower approximation operators RϑA and RS A. Thegraphs to illustrate this idea are shown in Figs. 1 and 2.

In Figs. 1 and 2, we use some fuzzy sets to illustrate thefuzzy granules RT xλ and Rσxλ, a box to illustrate a crisp setA. Fig. 1 is the demonstration graph of three fuzzy sets f1 ,f2 , and f3 that satisfy f1 ≥ f2 ≥ f3 . Fig. 2 shows the inclusionrelations between f1 , f2 , f3 , and A, respectively. We find that f3is included in A, whereas only part of f1 and f2 are included inA. From Fig. 2(b), it is easy to see that if we ignore these smallmembership degrees, which may be caused by misclassificationand perturbation, f2 should be included in A. All these show



Fig. 2. Inclusion relations between f1 , f2 , f3 , and A. (a) f1 and A. (b) f2and A. (c) f3 and A.

that the conditions of RT xλ ⊆ A and Rσxλ ⊆ A are too harshfor the construction of the lower approximation operators andlimit the boundary of the positive region. Thus, the knowledgerepresentation power of FRS is relatively weak.

By the aforementioned analysis, we find that the approxima-tion operator of FRS is sensitive to misclassification and per-turbation. Some small misclassification and perturbation mayaffect the values of the positive region and further affect theperformance of attribute reduction with FRS.

C. Several Flexible RS-Based Methods With Respectto Misclassification and Perturbation

In this section, we review two types of RS generalizationswith threshold. One type focused on generalizing the knowl-edge representation power of RS. They are VPRS, VPRS-FRS,VPFRS, and VQRS [24], [28], [37], [40]. The other type fo-cused on studying the criteria of attribute reduction, such as themethods of fuzzy reductions (Fuzzy-RED) [49] and approxi-mate reductions (Approximate-RED) [46]–[48].

1) VPRS: VPRS has been proposed to handle the problemswith misclassification by incorporating a controlled de-gree of misclassification into knowledge representation ofRS formalism. It was a robust model and has been widelyused in practical applications. However, VPRS could noteffectively deal with real-valued databases with misclas-sification and perturbation.

2) VPRS-FRS: VPRS-FRS introduced VPRS into Duboisand Prade’s FRS framework. By considering that one mis-classification sample may significantly influence the valueof fuzzy approximation operators, the authors in [28] in-troduced a concept named β-precision aggregation to thelower and upper approximation operators. Their approxi-mation operators were proposed as follows:

R∗Aβ (x) = max(IR∗Aβ, A(x))

R∗Aβ (x) = min(IR∗Aβ, A(x))

with

IR∗Aβ(x) = maxβ

y (y =x)min(R(x, y), A(y))

and

IR∗Aβ(x) = minβ

y (y =x)max(1 − R(x, y), A(y)).

By employing β-precision aggregation, we find that somemaximum or minimum values of min(R(x, y), A(y)) ormax(1 − R(x, y), A(y)), which may be caused by mis-classification, were omitted by controlling the threshold β.As a result, VPRS-FRS was robust to handle real-valueddatasets with misclassification. However, it was sensitiveto perturbation since perturbation may exist in each valueof min(R(x, y), A(y)) or max(1 − R(x, y), A(y)).

3) VPFRS: The authors in [37] introduced VPRS in a generalFRS framework. Here, we focus on reviewing the lowerapproximation of VPFRS since it was important for at-tribute reduction. The lower approximation was proposedas follows:

µRu F (Xi)

=

inf

x∈Si u

ϑ(µXi(x), µF (x)),

if ∃αu =supα ∈ (0, 1] : eα (Xi, F ) ≤ 1−u0, otherwise

where Siu= sup p(Xi ∩ XF

iα u).

This definition can be equivalently represented asshown as the bottom of the page.



The proof of the equivalence of two statements is presentedas follows.Proof:

Siu= sup p(Xi ∩ XF

iα u) = sup p(Xi) ∩ sup p(XF

iα u)

= sup p(Xi) ∩ XFiα u

(since Aα = x ∈ X,µA (x) ≥ α)= sup p(Xi) ∩ x ∈ X : ϑ(µXi

(x), µF (x))≥α= x ∈ X : ϑ(µXi

(x), µF (x))≥α and µXi(x)>0.

We have

infx∈Si u

ϑ(µXi(x), µF (x))

= infϑ(µX i

(x),µF (x))≥αu

ϑ(µXi(x), µF (x)).

From the equivalent definition of VPFRS, we find that thethreshold in VPFRS was put in the position of aggrega-tion operator. As a result, some misclassification could beomitted by controlling this threshold. However, VPFRSwas still sensitive to perturbation since the introduction ofthe threshold did not reduce the effect of perturbation thatmay exist in each value of ϑ(µXi i

(x), µF (x)).VPRS-FRS and VPFRS shared some strengths: 1) they

took FRS as a special case and 2) they were effective tohandle misclassification. Also, they shared some limita-tions: 1) they were sensitive to perturbation and 2) attributereduction was not mentioned in these models.

4) VQRS: Another flexible generalization is VQRS. In thismodel, a fuzzy quantification measure, named quantifier,was introduced to measure whether one fuzzy set Ry wasmost included in another fuzzy set A: Qu (|Ry ∩ A|/|A|).Then, the lower approximation was proposed asfollows:

R ↓QuA(y) = Qu

(|Ry ∩ A|

|A|

).

VQRS had the following strengths: 1) it was a flexiblemodel since it was less sensitive to misclassification andperturbation; 2) it was suitable for both symbolic problemsand real-valued problems; and 3) it took RS and VPRS asthe special cases. Also, VQRS had some limitations. Onemain limitation was that some important properties aboutattribute reduction did not hold. For example, the approx-imation quality measure did not monotonically increasewith the number of selected attributes. As a result, the al-gorithm of attribute reduction by keeping approximationquality invariant could not be designed in VQRS.

In the following, we review another type of RS generaliza-tions that introduced the threshold in the concept of attribute

reduction. Here, we focus on reviewing the key idea of attributereduction.

1) Approximate-RED: The concept of approximate reductionwas proposed by keeping the quality measure of certainsubset almost unchanged after attribute reduction. Theapproximate reduction proposed in this way was moreflexible than VPRS and FRS because it was not only freefrom the unreasonable behavior of reduction criteria inVPRS, but also free from the harsh criteria of attributereduction in FRS.

2) Fuzzy-RED: In Fuzzy-RED, after describing a monotonicmeasure, the authors in [49] proposed the fuzzy decisionreduction by keeping this measure almost unchanged.Fuzzy-RED was more general than Approximate-REDsince the quality measure was a special case of the mono-tonic measure.

Up until now, we have reviewed two types of flexibleRS-based generalizations. The former focused on studying thelower approximation, while the latter focused on relaxing thecriteria of attribute reduction. The main differences betweenthem are listed as follows: 1) in the former type, the thresholdwas put in set approximation operators, whereas in the latter,it was put in attribute reduction and 2) in the former type, mis-classification and/or perturbation were controlled in knowledgerepresentation, whereas in the latter, they were controlled inattribute reduction.

Note that in this paper, we employ the idea of keeping thequality measure invariant in attribute reduction since misclas-sification and perturbation have been omitted in knowledgerepresentation.

III. FUZZY VARIABLE PRECISION ROUGH SETS MODEL

In this section, we improve the knowledge representationpower of FRS by incorporating one controlled degree into thefuzzy approximation operators so that the undesirable effectof misclassification and perturbation can be weakened. First,we propose four fuzzy approximation operators with variableprecision, and then discuss the properties of them. Second,we propose the concepts of attribute reduction based on theproposed approximation operators. Also, we investigate thestructure of these reductions by employing the discernibilitymatrix approach. Finally, we design some algorithms to findthe reductions. Thus, we build a unified framework namedFVPRSs.

A. Set Approximation in FVPRS and Their Properties

In this section, we first describe some new concepts: fuzzylower and upper approximation operators with variable preci-sion and fuzzy positive region with variable precision. Then,

µRu F (Xi) =

inf

ϑ(µX i(x),µF (x))≥αu

µX i(x)>0

ϑ(µXi(x), µF (x)), if ∃αu = supα ∈ (0, 1] : eα (Xi, F ) ≤ 1 − u

0, otherwise.



we present and prove some properties and theorems that arevaluable for the following construction of discernibility matrix.

Definition 3.1: Given FD = (U,R ∪ D) and A ∈ F (U),the fuzzy lower and upper approximation operators with vari-able precision α ∈ [0, 1), also called the lower and upperapproximation operators of FVPRS, are, respectively, definedas follows. ∀x ∈ U :D3.1.1) Rϑ α

A(x) = infA(u)≤α

ϑ(R(x, u), α)

∧ infA(u)>α

ϑ(R(x, u), A(u));

D3.1.2) RT αA(x) = sup

A(u)≥N (α)T (R(x, u), N(α))

∨ supA(u)<N (α)

T (R(x, u), A(u));

D3.1.3) RS αA(x) = inf

A(u)≤αS(N(R(x, u)), α)

∧ infA(u)>α

S(N(R(x, u)), A(u));

D3.1.4) Rσ αA(x) = sup

A(u)≥N (α)σ(N(R(x, u)), N(α))

∨ supA(u)<N (α)

σ(N(R(x, u)), A(u)).

Since T , S, ϑ, and σ are dual, respectively, we have thefollowing properties.

Proposition 3.1: For A ∈ F (U), the following statementshold.

P3.1.1) RS αA = coN (RT α (coN A)),

RT αA = coN (RS α(coN A));

P3.1.2) Rϑ αA = coN (Rσ α (coN A)),

Rϑ αA = coN (Rσ α (coN A)).

Proof:P3.1.1) ∀x ∈ U

coN (RT α(coN A))(x)

= N(RT α(coN A))(x)

= N

(sup

N (A(u))≥N (α)T (R(x, u), N(α))

∨ supN (A(u))<N (α)

T (R(x, u), coN A(u)))

= infA(u)≤α

N(T (R(x, u), N(α)))

∧ infA(u)>α

N(T (R(x, u), N(A(u))))

= infA(u)≤α

S(N(R(x, u)), α)

∧ infA(u)>α

S(N(R(x, u)), A(u)) = RS αA

coN (RS α(coN A))(x) = N(RS α (coN A))(x)

= N

(inf

coN A(u)≤αS(N(R(x, u)), α)

∧ infcoN A(u)>α

S(N(R(x, u)), coN A(u)))

= supN (A(u))≤α

N(S(N(R(x, u)), α))

∨ supN (A(u))>α

N(S(N(R(x, u)), N(A(u))))

= supA(u)≥N (α)

T (R(x, u), N(α))

∨ supA(u)<N (α)

T (R(x, u), A(u)) = RT αA.

The proof of P3.1.2) is similar to P3.1.1) since ϑ and σ aredual w.r.t. the involutive negator N .

The previous proposition shows that Rϑ αA and Rσ α

A,RS α

A and RT αA are dual w.r.t. the involutive negator N ,

respectively. In the following, we focus on the pair of fuzzyapproximation operators Rϑ α

and RTα. Some properties of an-

other pair of fuzzy approximation operators ofRS αA andRT α

Acan be obtained in a similar way by using the duality.

If A is an arbitrary crisp subset of U , the lower and upper ap-proximation operators of FVPRS degenerate into the followingformulae. ∀x ∈ U :D3.1.5) Rϑ α

A(x) = infA(u)=0

ϑ(R(x, u), α);

D3.1.6) RTαA(x) = sup

A(u)=1T (R(x, u), N(α));

D3.1.7) RS αA(x) = inf

A(u)=0S(N(R(x, u)), α);

D3.1.8) Rσ αA(x) = sup

A(u)=1σ(N(R(x, u)), N(α)).

A fuzzy positive region with variable precision α ∈ [0, 1),also called α-positive region of D relative to R, can be definedas a union of fuzzy lower approximations, i.e., POSRα

(D) =∪

x∈URϑ α

[x]D . If no confusion arises, we still call it a posi-

tive region. If k = γRα(D) = |POSRα

(D)|/|U |, we say thatD is α-dependent on R in a degree k, where k is called α-dependency degree of D on R. Attribute a ∈ P ⊆ R is α-dispensable in P w.r.t. D if PosPα

D = Pos(P −a)αD, oth-

erwise it is α-indispensable in P w.r.t. D. If every attribute inP is α-indispensable, then we say that P is α-independent in Rw.r.t. D.

When α = 0, FRS becomes the special case of FVPRS. Weshow this fact in the following properties.

Proposition 3.2: For α ≤ β and α, β ∈ [0, 1) :P3.2.1) RϑA = Rϑ 0A;P3.2.2) RϑA ⊆ Rϑ α

A;P3.2.3) Rϑ α

A ⊆ Rϑ βA;

P3.2.4) RT A = RT 0 A;P3.2.5) RT α

A ⊆ RT A;P3.2.6) RT β

A ⊆ RT αA;

P3.2.7) POSRD = POSR0 D;P3.2.8) POSRD ⊆ POSRα

D.Proof:

P3.2.1) ∀x ∈ U :

Rϑ 0A(x) = infA(u)≤0

ϑ(R(x, u), 0) ∧ infA(u)>0

ϑ(R(x, u), A(u))

= RϑA(x).

P3.2.2) ∀x ∈ U :

RϑA(x) = infA(u)≤α

ϑ(R(x, u), A(u))∧ infA(u)>α

ϑ(R(x, u), A(u))



≤ infA(u)≤α

ϑ(R(x, u), α) ∧ infA(u)>α

ϑ(R(x, u), A(u))

(since ϑ(·, ·) is monotonically increasingin the right argument)

= RϑA(x) ≤ Rϑ αA(x).

Hence, RϑA ⊆ Rϑ αA.

P3.2.3) ∀x ∈ U :

RϑαA(x) = inf

A(u)≤αϑ(R(x, u), α)∧ inf

α<A(u)≤βϑ(R(x, u), A(u))

∧ infA(u)>β

ϑ(R(x, u), A(u))

≤ infA(u)≤α

ϑ(R(x, u), β) ∧ infα<A(u)≤β

ϑ(R(x, u), β)

∧ infA(u)>β

ϑ(R(x, u), A(u)) = Rϑ βA.

Hence, Rϑ αA ⊆ Rϑ β

A.

P3.2.4) It is straightforward to get RT A = RT0 A by defini-tion of upper approximation of FVPRS.P3.2.5) ∀x ∈ U :

RT A(x) = supA(u)≥N (0)

T (R(x, u), A(u))

∨ supA(u)<N (0)

T (R(x, u), A(u))

≥ supA(u)≥N (α)

T (R(x, u), N(α))

∨ supA(u)<N (α)

T (R(x, u), A(u)) = RT αA(x).

P3.2.6) ∀x ∈ U :

RT αA(x)

= supA(u)≥N (α)

T (R(x, u), N(α))

∨ supN (β )≤A(u)<N (α)

T (R(x, u), N(α))

∨ supA(u)<N (β )

T (R(x, u), A(u))

≥ supA(u)≥N (α)

T (R(x, u), N(β))

∨ supN (β )≤A(u)<N (α)

T (R(x, u), N(β))

∨ supA(u)<N (β )

T (R(x, u), A(u)) = RTβA(x).

Hence, RTβA ⊆ RTα

A.By Properties P3.2.1) and P3.2.2), it is straightforward to get

POSRD = POSR0 D and POSRD ⊆ POSRαD .

It has been proposed and proven in [14] that the elements inMT = RT xλ : x ∈ U, λ ∈ (0, 1] can be applied as the basicgranules for the FRS framework FR

T (U) = A : RϑA = A =RT A. The fuzzy granule RT xλ is important and meaningfulin a fuzzy rough framework. From the theoretical viewpoint, thegranule RT xλ is not only used to construct the fuzzy approxi-mation operators, but also used to study the structure of attribute

Fig. 3. Practical meaning of fuzzy granule RT xλ.

reduction [16]. From the practical viewpoint, the fuzzy granuleRT xλ functions as the crisp granule, i.e., the equivalence class.

In the following, we explain the practical meaning of RT xλby using a demonstration graph (see Fig. 3). The gray squarein Fig. 3 represents the available information R by employingwhich we obtain the values of λ, R(x, y), and RT xλ. First, λ =RϑA(x) is the degree of the object x ∈ U certainly belongingto the decision class A. Second, R(x, y) is the similarity degreebetween x and y. Finally, RT xλ(y) = T (R(x, y), λ) representsthe degree of the object y belonging to the decision class A.The value RT xλ(y) is obtained by aggregating the similarityrelation R(x, y) and the degree of the object x belonging to thedecision class A, i.e., y. The aforementioned analysis shows thatRT xλ measures the supporting degree of condition attributes Rto the decision class A. For example, if R(x, y) = 1 and λ = 1,then the set of condition attributes R supports the decision classA 100% on the objects x and y.

In the following, we focus on the granular representation ofthe fuzzy lower and upper approximation operators by using thebasic granules in FRS. As preliminaries, we need to review aconcept called fuzzy cut set. In fuzzy sets, let f be a fuzzy setdefined on U , then the fuzzy α-cut set of fuzzy set f is defined as

fα (x) =

f(x), f(x) > α0, f(x) ≤ α

, for α ∈ [0, 1).

By using the fuzzy granule RT xλ, the granular representationof the fuzzy lower and upper approximation operators in FVPRSis described as follows.

Definition 3.2: Given FD = (U,R ∪ D) and A ∈ F (U), thefuzzy lower and upper approximations of FVPRS are, respec-tively, defined as follows.

D3.2.1) Rϑ αA = ∪RT xλ : (RT xλ)α ⊆ A;

D3.2.2) RT αA = ∪RT xλ : xλ ⊆ AN (α).

where

AN (α) =

N(α), A(x) ≥ N(α)

A(x), otherwise.

D3.1.1) and D3.1.2) can be seen as the membership functionrepresentation of Rϑ α

A and RT αA. D3.2.1) and D3.2.2) can

seen as the granular representation of Rϑ αA and RT α

A. Now,the task is to prove these two types of representation ways areequivalent. As preliminaries, we present a theorem about thegranular representation of Rϑ α

A and RT αA as follows.

Theorem 3.1:T3.1.1) Let γ = (∪RT xλ : (RT xλ)α ⊆ A)(x) for x ∈

U , then the following statements hold: (RT xγ )α ⊆



A always holds; (RT xβ )α ⊆ A does not hold forany β > γ.

T3.1.2) ∪RT xλ : xλ ⊆ AN (α) = ∪RT xλ′ : λ′ =

AN (α)(x).Proof:

T3.1.1) By the condition of γ = (∪RT xλ : (RT xλ)α ⊆A)(x), we get for any x ∈ U there ex-ist t ∈ (0, 1] and y ∈ U satisfying γ = RT yt(x)and (RT yt)α ⊆ A. Then, the two statements,T (R(y, x), t) = γ and if T (R(y, z), t) > α, thenT (R(y, z), t) ≤ A(z) for any z ∈ U , hold. Thus, thestatement ∀z ∈ U , RT xγ (z) = T (R(x, z), γ) =T (R(x, z), T (R(y, x), t)) ≤ T (R(y, z), t) holds. IfT (R(x, z), t) > α, then T (R(y, z), t) > α. Thus, wehave T (R(x, z), t) ≤ A(z). Hence, (RT xγ )α ⊆ A.

Assume (RT xβ )α ⊆ A for β > γ, thenγ ≥ RT xβ (x) = β by γ = (∪RT xλ : (RT xλ)α ⊆A)(x). That is to say, γ ≥ β. This contradicts thecondition β > γ. Thus, for any β > λ, the formula(RT xβ )α ⊆ A does not hold.

T3.1.2) If xλ ⊆ AN (α) , then λ ≤ AN (α)(x); ∀x, z ∈ U ,RT xλ(z) = T (R(x, z), λ). By the monotonic-ity of the T -norm, we have RT xλ ⊆ RT xλ

′ for∀xλ ⊆ AN (α) and λ′ = AN (α)(x). Thus, (∪RT xλ :xλ ⊆ AN (α)) ⊆ (∪RT xλ

′ : λ′ = AN (α)(x)).Hence, ∪RT xλ : xλ ⊆ AN (α) = ∪RT xλ

′ : λ′ =AN (α)(x).

By employing Theorem 3.1, we describe and prove the fol-lowing two theorems.

Theorem 3.2: Given FD = (U,R ∪ D) and A ∈ F (U), thefollowing two statements are equivalent.

T3.2.1) Rϑ αA(x) = inf

A(u)≤αϑ(R(x, u), α) ∧ inf

A(u)>αϑ(R(x, u),

A(u)),∀x ∈ U ;T3.2.2) Rϑ α

A = ∪RT xλ : (RT xλ)α ⊆ A.Proof: ∀x ∈ U

Rϑ αA(x) = inf

A(u)≤αϑ(R(x, u), α) ∧ inf

A(u)>αϑ(R(x, u), A(u))

= infA(u)≤α

(supc ∈ [0, 1], T (R(x, u), c) ≤ α)

∧ infA(u)>α

(supc∈[0, 1], T (R(x, u), c)≤A(u))

= infA(u)≤α

(supc ∈ [0, 1], (RT xc)α (u) ≤ A(u))

∧ infA(u)>α

(supc ∈ [0, 1], (RT xc)α (u) ≤ A(u))

= infu∈U

(supc ∈ [0, 1], (RT xc)α (u) ≤ A(u)).

Let λ = infu∈U λu =infu∈U (supc∈ [0, 1], (RT xc)α (u) ≤A(u)), here λu = supc ∈ [0, 1], (RT xc)α (u) ≤ A(u).Then, for any u ∈ U , (RT xλ)α (u) ≤ (RT xλu

)α (u) ≤ A(u)holds. For any γ > λ, γ ∈ [0, 1], ∃u′ ∈ U , (RT xγ )α (u) >A(u) exists. That is to say, λ is the maximum value tomake (RT xλ)α (u) ≤ A(u) hold for any u ∈ U . Then,(RT xλ)α ⊆ A holds for any x ∈ U , and for any β > λ,(RT xβ )α ⊆ A does not hold. By Theorem 3.1, we get

(∪RT xλ : (RT xλ)α ⊆ A)(x) = λ. Thus,

infA(u)≤α


ϑ(R(x, u), A(u))

= (∪RT xλ : (RT xλ)α ⊆ A)(x)holds.

Theorem 3.3: Given FD = (U,R ∪ D) and A ∈ F (U), thefollowing two statements are equivalent.

T3.3.1) RT αA(x) = supA(u)≥N (α)

T (R(x, u), N(α))

∨ supA(u)<N (α)

T (R(x, u), A(u)),∀x ∈ U ;

T3.3.2) RT αA = ∪RT xλ : xλ ⊆ AN (α).Proof: By Theorem 3.1, RT α

A = ∪RT xλ′ : λ′=

AN (α)(x). Then,

RT αA(x)

= (∪RT uλ′ : λ′ = AN (α)(u))(x)

= supRT uλ′(x) : λ′ = AN (α)(u)

= supA(u)≥N (α)

RT uN (α)(x) ∨ supA(u)<N (α)

RT uA(u)(x)

= supA(u)≥N (α)

T (R(x, u), N(α)) ∨ supA(u)<N (α)

T (R(x, u), A(u))

i.e.,

supA(u)≥N (α)

T (R(x, u), N(α)) ∨ supA(u)<N (α)

T (R(x, u), A(u))

= (∪RT xλ : xλ ⊆ AN (α))(x).

Theorems 3.2 and 3.3 show that the membership functionrepresentation and granular representation of Rϑ α

A and RT αA

are equivalent. They are the key theorems for the constructionof discernibility matrix.

From Theorem 3.2, we find that a fuzzy cut set is introducedinto the lower approximation operators in FVPRS, and the con-dition RT xλ ⊆ A in the granular representation of FRS is re-laxed by the condition (RT xλ)α ⊆ A. We can illustrate this factby some demonstration graphs (see Fig. 4). Assuming α = 0.2,we need to check whether the fuzzy cut sets (f1)0.2 , (f2)0.2 , and(f3)0.2 are included in A0.2 (since (RT xλ)α ⊆ A is equivalentto (RT xλ)α ⊆ Aα ).

Fig. 4 shows that (f1)0.2 is not included in A0.2 , whereas(f2)0.2 and (f3)0.2 are included in A0.2 . That is to say, whenwe ignore some small membership degrees, some more fuzzysets (e.g., f2) are then selected to construct the fuzzy lowerapproximation. As a result, the positive region becomes largerand the knowledge representation power becomes stronger.

In FVPRS, we omit some misclassification or small perturba-tion by controlling the threshold α in the fuzzy cut set (RT xλ)α .As a result, the values of set approximations are flexible and lesssensitive to misclassification and perturbation, and further, theattribute reduction results are robust. In a word, FVPRS is lesssensitive to misclassification and perturbation.

Since POSRα(D)(x) = ∪z∈U Rϑ α

[z]D (x), we know thatPOSRα

(D)(x) can get its maximum value at cer-tain Rϑ α

[z]D (x). The following theorem implies that



Fig. 4. Inclusion relation. (a) (f1 )0 .2 and A0 .2 . (b) (f2 )0 .2 and A0 .2 . (c)(f3 )0 .2 and A0 .2 .

POSRα(D)(x) always takes Rϑ α

[x]D (x) as its maximumvalue in a fuzzy decision table with the decision symbolicattributes.

Lemma 3.1: Rϑ αand RTα

are monotone.Proof: Suppose A ⊆ B ⊆ U ; if (RT xλ)α ⊆ A, then

(RT xλ)α ⊆ B. By Rϑ αA = ∪RT xλ : (RT xλ)α ⊆ A, we

get Rϑ αA ⊆ Rϑ α

B. Hence, Rϑ αis monotone. By RTα

A =∪RT xλ : xλ ⊆ A1−α and A1−α ⊆ B1−α , we get RTα

A ⊆RTα

B. Hence, RTαis monotone.

Lemma 3.2: If (RT xλ)α ⊆ [z]D , then (RT xλ)α ⊆ [x]D .Proof: If [z]D = [x]D , it is straightforward to get (RT xλ)α ⊆

[x]D . If [z]D = [x]D , RT xλ(x) ≤ α, i.e., λ ≤ α holds (by thedefinition of fuzzy cut set). Thus, ((RT xλ)α = φ) ⊆ [x]D al-ways holds for λ ≤ α.

Theorem 3.4: Given FD = (U,R ∪ D) with the decisionsymbolic attributes and ∀x ∈ U , the following statement holds:POSRα

(D)(x) = Rϑ α[x]D (x).

Proof: Since POSRα(D)(x) = ∪z∈U Rϑ α

[z]D (x), we getthat POSRα

(D)(x) can get its maximum value at certainRϑ α

[z]D (x). Let λ = POSRα(D)(x), then Rϑ α

[z]D (x) = λ

and Rϑ α[y]D (x) ≤ λ for every y ∈ U . By Theorem 3.1,

(RT xλ)α ⊆ [z]D holds. By Lemma 3.2, (RT xλ)α ⊆ [x]Dholds. Then, λ ≤ Rϑ α

[x]D (x) holds. Thus, λ = Rϑ α[x]D (x),

i.e., POSRα(D)(x) = Rϑ α

[x]D (x). By Theorem 3.4, we know that in a fuzzy decision table

with the decision symbolic attributes, POSRα(D)(x) always

gets its value at its lower approximation value of [x]D , i.e.,Rϑ α

[x]D (x). Thus, we conclude that keeping the positive re-gion value POSRα

(D)(x) invariant is equivalent to keepingRϑ α

[x]D (x) invariant.Theorem 3.5: For two T -similarity relations R and P , if

R ⊆ P , then the following hold.T3.5.1) Rϑ α

A ⊇ Pϑ αA;

T3.5.2) RT αA ⊆ PT α

A;T3.5.3) POSRα

D ⊇ POSPαD.

Proof:T3.5.1) R⊆P ⇒∀x, u∈U,R(x, u) ≤ P (x, u) ⇒ ∀x, u ∈ U,

infA(u)≤α


ϑ(R(x, u), A(u)) ≥inf

A(u)≤αϑ(P (x, u), α) ∧ inf

A(u)>αϑ(P (x, u), A(u)) ⇒

Rϑ αA ⊇ Pϑ α

A.T3.5.2) R⊆P ⇒∀x, u∈U,R(x, u) ≤ P (x, u)⇒∀x, u ∈ U,

supA(u)≥1−α

T (R(x, u), 1 − α) ∧ supA(u)<1−α

T (R(x, u), A(u))

≤ supA(u)≥1−α

T (P (x, u), 1 − α) ∧ supA(u)<1−α

T (P (x, u),

A(u)) ⇒ RT αA ⊆ PT α

A.T3.5.3) R ⊆ P ⇒ Rϑ α

A ⊇ Pϑ αA [by T3.5.1)] ⇒

POSRαD ⊇ POSPα

D (by Theorem 3.4). It is easy to see from Theorem 3.5 that the smaller the fuzzy

T -similarity relation is, the larger the fuzzy lower approximationand fuzzy positive region are.

B. Attribute Reduction With FVPRS

In this section, we propose a general concept of attributereduction by employing the fuzzy approximation operators inFVPRS. A fuzzy decision table with condition fuzzy attributesand decision symbolic attributes is adopted as the platform ofattribute reduction since it occurs in most practical datasets.

Definition 3.3: Given FD = (U,R ∪ D), P ⊆ R, and α ∈[0, 1), if the following two statements hold: POSRα

(D) =POSPα

(D) and P is α-independent in R, then P is calledthe α-reduction of R.

It is easy to see that P is a minimal subset of R to keep thepositive region invariant.

The α-core of R is the collection of the attributes that areα-indispensable in R.

By Theorem 3.4, we have the following theorem to charac-terize the α-reduction.



Theorem 3.6: Given FD = (U,R ∪ D), P ⊂ R, and α ∈[0, 1). The following two statements are equivalent: T3.6.1)P ⊂ R contains one α-reduction of R and T3.6.2) ∀x ∈ U,T (P (x, z), λ) ≤ α for every z /∈ [x]D and λ = Rϑ α

[x]D (x).Proof: T3.6.1)⇒ T3.6.2) If P contains one α-reduction of R,

then POSRα(D) = POSPα

(D). Let λ = POSRα(D)(x) =

POSPα(D)(x) ∀x ∈ U . By Theorems 3.1 and 3.4, we have

λ = Rϑ α[x](x) = Pϑ α

[x](x) and (PT xλ)α ⊆ [x]D . By thedefinition of fuzzy cut set, we have T (P (x, z), λ) ≤ α for everyz /∈ [x]D .

T3.6.2) ⇒ T3.6.1) By Theorem 3.5, λ = Rϑ α[x]D (x) ≥

Pϑ α[x]D (x) holds. If T (P (x, z), λ) ≤ α for every

z /∈ [x]D , then (PT xλ)α ⊆ [x]D . By Rϑ αA = ∪RT xλ :

(RT xλ)α ⊆ A, Pϑ α[x]D (x) ≥ (PT xλ)α (x) = λ holds.

Thus, λ = Rϑ α[x](x) = Pϑ α

[x](x). By Theorem 3.4,λ = POSRα

(D)(x) = POSPα(D)(x) holds. This shows that

P ⊂ R contains one α-reduction of R. Clearly, P is a α-reduction in FD = (U,R ∪ D) if and

only if P is a minimal subset satisfying the conditions inTheorem 3.6. Now, we can construct a α-discernibility matrix.

Suppose U = x1 , x2 , x3 , . . . , xn, by Mα (U,R,D) we de-note an n × n matrix (cij ), called the α-discernibility matrix of(U,R ∪ D), such that

M3.1) cij = a : T (a(xi, xj ), λ) ≤ α,λ = Rϑ α

[xi ]D (x) for D(xi, xj ) = 0;M3.2) cij = φ, for D(xi, xj ) = 1.Theorem 3.7: CoreDα

(R) = a : cij = a for some 1 ≤i, j ≤ n.

Proof: a ∈ CoreDα(R)⇔POSRα

D = POS(R −a)αD⇔

there exists xi ∈ U such that POSRαD(xi) >

POS(R −a)αD(xi) ⇔ Rϑ ([xi ]D )(xi) > (R − a)ϑ

([xi ]D )(xi) ⇔ Let λ(xi) = Rϑ ([xi ]D )(xi) andt(xi) = (R − a)ϑ ([xi ]D )(xi), then λ(xi) > t(xi),RT (xi)λ(xi ) ⊆ [xi ]D , (R − a)T (xi)t(xi ) ⊆ [xi ]D , and

(R − a)T (xi)λ(xi ) ⊂ [xi ]D (by Theorem 3.1) ⇔ thereexists xj ∈ U such that T ((R − a)(xi, xj ), λ(xi)) > αand T (R(xi, xj ), λ(xi)) ≤ α ⇔ for any b ∈ R − a,T (b(xi, xj ), λ(xi)) > α, and T (a(xi, xj ), λ(xi)) ≤ α hold ⇔cij = a.

The statement cij = a implies that a is the unique attributeto maintain T (R(xi, xj ), λ(xi)) ≤ α.

This theorem shows that by using α-discernibility matrix, wecan find the α-core.

Theorem 3.8: If P ⊂ R, then P contains α-reduction of R ifand only if P ∩ cij = φ for every cij = φ.

The proof is straightforward by Theorem 3.6 and the defini-tion of cij .

Corollary 3.2: If P ⊂ R, then P is one α-reduction of R ifand only if P is a minimal subset of R satisfying P ∩ cij = φfor every cij = φ.

Theorem 3.8 and Corollary 3.2 show that by using α-discernibility matrix, we can find the α-reduction.

A discernibility function fα (FD) for FD = (U,R ∪ D) isa Boolean function with m Boolean variables a1 , . . . , am cor-responding to the attributes a1 , . . . , am , respectively, and is de-fined as fα (FD)(a1 , . . . , am ) = ∧∨(cij ) : 1 ≤ j < i ≤ n,

where ∨(cij ) is the disjunction of all variables a suchthat a ∈ cij . Let gα (FD) be the reduced disjunctive formof fα (FD) obtained from fα (FD) by applying the mul-tiplication and absorption laws as many times as possi-ble. Then, there exist l and Rk ⊆ R for k = 1, . . . , l suchthat gα (FD) = (∧R1) ∨ · · · ∨ (∧Rl) where every elementin Rk only appears one time. We have the followingtheorem.

Theorem 3.9: RedDα(R) = R1 , . . . , Rl; here, RedDα

(R)is the collection of all the α-reductions of R.

The proof is omitted since this theorem is similar to the onein [16].

Theorem 3.9 shows that we can find all reductions of R byusing an α-discernibility matrix.

An algorithm to compute all α-reductions for a fuzzy decisionsystem is then designed (see Algorithm 3.1).

Algorithm 3.1: To find all α- reductions of FVPRS:Step 1: Compute the similarity relation of the set of all condi-

tion attributes: SIM (R);Step 2: Compute Rϑ α

([x]D ) for every x ∈ U ;Step 3: Compute cij : cij = α : T (a(xi, xj ), λ) ≤ α, λ =

Rϑ α[xi ]D (xi) for D(xi, xj ) = 0; otherwise Cij = φ;

Step 4: Compute CoreDα(R) = a : cij = a; delete

those cij with nonempty overlap with CoreDα(R);

Step 5: Define fα (FD) = ∧∨(cij ) with nonemtpy cij

and computing gα (FD) = (∧Rl) ∨ · · · ∨ (∧Rl) byfα (DS);

Step 6: Output RedDα(R) = CoreDα

(R) ∪ Rl, . . . , Core ∪Rl .

This algorithm to find all α-reductions is an NP-completeproblem [30]. In practical applications, it is enough to employa near-optimal reduction to address the real problems. Thus, wedesign a heuristic algorithm to find one near-optimal reductionby using the following steps to replace steps 5 and 6 in Algorithm3.1:Step 5: Let Reduct = CoreDα

(R);Step 6: Add the element a with maximum of frequency of

occurrence into Reduct, and delete those cij withnonempty overlap with Reduct;

Step 7: If there still exist some cij = φ, go to step 6; otherwise,go to step 8;

Step 8: If Reduct is not independent, delete the redundant ele-ments in Reduct;

Step 9: Output Reduct.

IV. EXPERIMENTAL COMPARISONS AND SCALABILITY

ANALYSIS

In this section, we compare and analyze FVPRS with sev-eral dimension reduction methods. These methods can beroughly divided into two types. One type focuses on severalRS-based methods. They are RS, FRS, VPRS, VPRS-FRS,and VPFRS. The other type focuses on two-feature (i.e., at-tributes) dependency-based methods. They are MRMR and CA-CLF [31], [42].

Before we conduct comparison and analysis, two issuesshould be specified. One issue is that in this section, some



Fig. 5. Procedure and scalability analysis of attribute reduction in FVPRS.

methods (i.e., RS, VPRS, and FRS) are designed to find thereductions by using the discernibility matrix approach. Others(i.e., VPRS-FRS and VPFRS) are designed by keeping the de-pendency function invariant. Another issue is that the algorithmmentioned in this section is the heuristic to find a near-optimalreduction, since it is NP-hard to find the optimal one.

A. Scalability Analysis and Comparison

In this section, we illustrate the procedure of attribute reduc-tion in FVPRS (see Fig. 5). The procedure of FVPRS is similarto the procedures of RS, VPRS, and FRS. The main differ-ence among them lies in the steps covered by “Preprocessing.”In the fuzzy cases, i.e., FRS and FVPRS, the “Preprocessing”step means computing similarity relation and lower approxi-mation, whereas in crisp cases, i.e., RS and VPRS, it meansthe discretization of the original datasets. As a result, the com-putational complexity in the fuzzy cases is O(|U |2 × |A|) (thesimilarity relation) and O(|U |2 × |D|) (the lower approxima-tion), whereas the computational complexity in the crisp casesdepends on the selection of discretization method (the compu-tational complexity of the selected discretization method in thispaper is O(|U |2 × |A|) [43]). By the aforementioned analysis, itis easy to see that FVPRS and FRS spend similar time and spaceto find one reduction, whereas RS and VPRS spend similar timeand space.

In Fig. 5, |U | is the size of the universe, |A| is the number ofattributes, |D| is the number of the decision classes, and |k| isthe number of selected attributes in the obtained reduction.

Since, up to now, there was no algorithm that used discerni-bility matrix to find the reductions for VPRS-FRS and VPFRS,we need design some heuristic algorithms by keeping depen-dency function invariant for them. In the following, we designa heuristic algorithm by keeping dependency function invariantfor FVPRS (see Algorithm 4.1). If we replace step 2 in Algo-rithm 4.1 by “computing the lower approximation of VPRS-FRS

TABLE IINFORMATION OF SOME DATASETS FROM UCI MACHINE

LEARNING REPOSITORY

or VPFRS,” the algorithms to find one near-optimal reductionof VPRS-FRS or VPFRS are then obtained.

From the aspect of computational complexity, we find thatthe main difference of FVPRS, VPRS-FRS, and VPFRS liesin the lower approximations. The computational complexity tocompute the lower approximation of FVPRS and VPRS-FRS issimilar (i.e., O(|U |2 × |D|) in crisp decision case and O(|U |3)in real-valued decision case); the computational complexity ofVPFRS is O(|U |4 × |D|) in crisp decision case (O(|U |5) inreal-valued decision case). From the aforementioned analysis,we find that the real-valued decision case spends more timeand space than the crisp decision case to compute the lowerapproximation.

Algorithm 4.1 (heuristic):Step 1: Compute SIM (R);Step 2: Compute Rϑ α

([x]D ) for every x ∈ U ;Step 3: Compute POSRα

(D)=∪x∈U Rϑ α[x]D and γRα

(D)=|POSRα

(D)| / |U |;Step 4: Add the attribute with maximum γα∪Reductα

(D)into Reduct and delete it from R;

Step 5: If γReductα(D) = γRα

(D), go to step 4; otherwise,go to step 6;

Step 6: If Reduct is not independent, delete the redundant ele-ments in Reduct;

Step 7: Output Reduct.

B. Experimental Setup

The experiments in this section are set up as follows.Dataset: Ten datasets from University of California, Irvine

(UCI) Machine Learning Repository [35] are used (see Table I).Classifier: Nearest neighbor algorithm in [36] is used to be

the classifier. [Note that decision tree is used in the regressionproblems (i.e., those datasets with continuous decision attributessuch as forest fires and housing).]



Fig. 6. Trend of the number of selected attributes varying with the threshold.

Dataset Split: In the process of classification, the dataset afterattribute reduction is split into two parts. The randomly chosen50% of objects are used as the training set and the remainderas the testing set. The classification result is the average of 20times training and testing.

Indexes: They are: 1) the number of selected attributes and2) the classification accuracy of the reduction.

Parameter Specification: In FVPRS, VPRS-FRS, andVPFRS, we try the threshold from 0 to 1 (not including 1)with step 0.01. In VPRS, we try the threshold from 0 to 0.5(not including 0.5) with step 0.01. In MRMR, we try theparameter from 1 to the number of attributes in the correspond-ing dataset.

Triangular norm: “Lukasiewicz” T -norm is selected to con-struct the lower approximations of these RS-based generalizedmethods, i.e., FRS, FVPRS, VPRS-FRS, and VPFRS.

The first six datasets with different target classes (multiclassor two-class) are used in studying the effect of the threshold inFVPRS. What is more, these six datasets are used as the datasetswith crisp target class in comparing FVPRS with other RS-basedmethods. The last two datasets are used as the datasets with real-valued decision in comparing FVPRS, VPFRS, and VPRS-FRS.The remaining two datasets are used in comparing FVPRS withCACLF.

Table I shows that the selected ten datasets differ greatly fromsample size, feature number, data distribution, and target class

(multiclass or two-class or continuous). We study the differentdimension reduction methods on these datasets and we believethat these datasets suit for the dimension reduction methodsunder different conditions.

C. Effect of Threshold in FVPRS

To clearly show the effect of the threshold in FVPRS, we setthe threshold from 0.9 to 1 with the step 0.005 and summarize theresults in Figs. 6 and 7, where the horizontal axis represents thethreshold in FVPRS and the vertical axes represent the numberof attributes and the accuracy, respectively. The line within Fig. 7 is the accuracy of the original dataset, which providesa baseline to evaluate the performance of the reductions.

Fig. 6 shows that the number of attributes moves downwardalong the threshold. That is to say, the larger the threshold is,the smaller the number of selected attributes is. Fig. 7 showsthat the accuracy after reduction is close to the accuracy beforereduction at the beginning, but it suddenly becomes smallerthan the accuracy before reduction when the threshold is largeenough. All these give us a guideline on how to specify thethreshold for the further applications of FVPRS, i.e., the largerthe threshold is, the smaller the number of selected attributeis, whereas if the threshold is too large, the accuracy becomessmaller than the accuracy before reduction.



Fig. 7. Trend of classification accuracy varying with the threshold.

D. Experimental Comparison and Analysis of FVPRS- andRS-Based Methods

In this section, we first compare FVPRS with RS, FRS, andVPRS [17], [24], [41], and then we compare FVPRS with VPRS-FRS, VPFRS, and VQRS.

1) Experimental Comparison of FVPRS With RS, FRS, andVPRS: We would like to point out the main differences of thesemethods as follows: 1) RS and VPRS are the methods to dealwith the datasets with symbolic values, whereas FVPRS andFRS are the methods to handle the datasets with real values and2) in the preprocessing step, RS and VPRS need to discretize thereal-valued datasets, whereas FVPRS and FRS need to computethe similarity relation and the fuzzy lower approximations.

Discretizaton of real-valued attributes is an important stepfor RS and VPRS since their attribute reduction performancesdepend on this step. In this paper, the discretization methodproposed by Nguyen and Nguyen is chosen since it not onlypreserves the discernibility information in the original datasetsbut also is fast in discretization [43].

We summarize the comparison results in Table II. First, wefind that RS has comparable classification power (0.8438) withFVPRS (0.8351 and 0.8471), but the number of selected at-tributes in RS (8.5) is much larger than FVPRS (4.33 and 5).

TABLE IICOMPARISON OF FVPRS, RS, FRS, AND VPRS

These show that RS has maintained the information of the orig-inal datasets, but its attribute reduction result is not compacterthan FVPRS because RS was sensitive to misclassification.

Next, Table II shows that FVPRS often finds compacter at-tribute reductions than FRS does. That is to say, the dimensionreduction power of FRS on the datasets with real values isrelatively weak. The reason is that FRS was sensitive to mis-classification and perturbation.



Finally, Table II shows that VPRS has a better reduction powerthan FVPRS (from the viewpoint of the number of selectedattributes), whereas it misses some useful information. It is easyto see that the classification accuracy of VPRS (0.7646) is farless than the accuracy of FVPRS (0.8351 and 0.8471). Thisis because VPRS was not flexible enough for the real-valuedattributes. In contrast, FVPRS is a flexible model.

2) Comparisons of FVPRS With VPRS-FRS, VPFRS, andVQRS: Before comparison, we highlight some differencesamong these methods.

One main difference of these methods is on the concepts oflower approximations. VPRS-FRS was proposed by introducingthe β-precision aggregation into the set approximations of onespecified FRS framework; VPFRS was proposed by introducingone crisp cut set into the set approximations of a general FRSframework; FVPRS is proposed by incorporating a fuzzy cutset into a general FRS framework; VQRS was proposed byintroduced a fuzzy quantifier into the set approximations.

Another difference is that VPRS-FRS and VPFRS are ef-fective to handle misclassification but sensitive to perturbation,whereas FVPRS and VQRS are less sensitive to both misclas-sification and perturbation.

A third difference is about the applications of these methods,i.e., attribute reduction. In VPRS-FRS and VPFRS, attributereduction was not yet mentioned; FVPRS not only proposesthe concepts of attribute reduction, but also designs some de-tailed algorithms of attribute reduction; VQRS could not designthe algorithm of attribute reduction by keeping approximationquality invariant since the monotonicity of quality measure didnot hold in VQRS. Note that although monotonicity of qualitymeasure no longer holds, it is still possible to define attributereductions in VQRS [51]. Since the key idea of keeping approx-imation quality measure invariant is adopted in this paper, weomit the comparison of attribute reduction between FVPRS andVQRS.

The last difference among these methods is about the compu-tational complexity. We omit this point since we have mentionedit in Section IV-A.

In the following, we conduct the detailed experimental com-parison among these methods (except VQRS). The experimentalcomparison is twofold. One is experimental comparison on thedatasets with the crisp decision while the other is on the datasetswith the real-valued decision.

a) Experimental comparison on the databases with thecrisp decision: The experimental comparison results are sum-marized in Table III and Fig. 8. Fig. 8 summarizes the distri-bution of those reductions that have the classification accuracycomparable to or better than the accuracy of the original dataset.In each subfigure, the horizontal axis represents the number ofselected attributes, and each line with some points represents thereduction distribution of a certain method. For example, the linewith diamonds represents the reduction distribution of FVPRS.

Fig. 8 shows that FVPRS distributes more densely thanVPRS-FRS and VPFRS on the left side of each subfigure. Thatis to say, FVPRS often finds some compacter reductions thanVPRS-FRS and VPFRS do. The reason is that FVPRS can ef-fectively handle the problems with both misclassification and

TABLE IIICOMPARISON AMONG FVPRS, VPFRS, AND VPRS-FRS ON DATASETS WITH

THE CRISP DECISION

perturbation, whereas VPRS-FRS and VPFRS are sensitive toperturbation.

Two types of reductions are selected to compare the classifi-cation accuracy: one type has the classification accuracy com-parable to or better than the accuracy of the original dataset, assummarized in Table III(b), while the other has the best clas-sification accuracy in each method on the specified dataset, assummarized in Table III(a).

Table III shows that the numbers of selected attributes inVPFRS are obviously smaller than the ones in VPRS-FRS inmost cases, and the numbers of selected attributes in FVPRS aresignificantly smaller than the ones in VPFRS and VPRS-FRS.What is more, FVPRS can find the attribute subset with high-est classification and smallest number of selected features [seeTable III(a)]. All these show that FVPRS has a better perfor-mance of dimension reduction than VPFRS and VPRS-FRSbecause FVPRS is less sensitive to both misclassification andperturbation, whereas VPFRS and VPRS-FRS were only lesssensitive to misclassification.

b) Experimental comparison on the datasets with the real-valued decision: The comparison results have been summarizedin Table IV, which shows that under the condition with similarregression results, FVPRS has the smallest number of selectedattributes; VPFRS has a smaller number of selected attributesthan VPRS-FRS. We then draw a conclusion that VPFRS istime-consuming (we have mentioned this in Section IV-A);VPRS-FRS has a poor reduction performance; FVPRS is anacceptable choice to find the compacter attribute subset withinthe reasonable time and space.



Fig. 8. Reduction distribution of FVPRS, VPRS, and VPRS-FRS.

TABLE IVCOMPARISON AMONG FVPRS, VPFRS, AND VPRS-FRS ON THE DATASETS

WITH THE REAL-VALUED DECISION

E. Comparison of FVPRS With Two-Feature Dependency-Based Methods: MRMR and CACLF

In this section, we would like to compare FVPRS with MRMRand CACLF.

1) Comparison of FVPRS With MRMR: MRMR was pro-posed as a well-designed feature selection method based on themutual information [31]. Its key idea to find the feasible subsetof attributes was to minimize feature redundancy and maxi-mizing the feature dependency of the target classes, which wassimilar to FVPRS. Now, it is necessary to compare FVPRS withMRMR.

It is already pointed out in [44] and [45] that the ensembles ofreductions have a better performance than a single reduction. Asa result, it is necessary to conduct the ensembles of reductions tocompare with the classifiers constructed on a single reduction.

In the following, we conduct the comparisons among FVPRS(ensemble), FVPRS (single), and MRMR, as summarized in

Table V. Here, we randomly choose n (n < 10) reductions toconstruct the ensembles of reductions.

Table V shows that under the condition with the similar num-ber of attributes, FVPRS (ensemble) often has the better clas-sification accuracy than FVPRS (single). FVPRS (single) hasthe comparable classification accuracy with MRMR. For exam-ple, the average of classification accuracy in FVPRS (ensem-ble) (0.861, 0.855, 0.847) is often higher than FVPRS (single)(0.836, 0.829, 0.816) and MRMR (0.829, 0.826, 0.818). Thecomputational complexity of MRMR increases with the numberof selected attributes. The computational complexity of FVPRSincreases with the square of the number of attributes at the worstcase. All these show that MRMR and FVPRS suit for differentapplications: MRMR is fast to find feature selection, whereasFVPRS has a better classification performance (especially theensemble approach).

Note that to further improve the ensemble, a genetic algo-rithm, as mentioned in [44], can be used to find the best ensem-ble of reductions. Also, the method proposed in [45] by using theaccuracy-guided forward searching strategy to find the best en-semble of reductions is another choice. Since these two methodsare computationally consuming, we omit them here and inter-ested readers are referred to [44] and [45] for details.

2) Comparison of FVPRS With CACLF: In this section, wecompare FVPRS with CACLF, which is one method of dimen-sion reduction by considering the feature dependency. Before



TABLE VCOMPARISON AMONG FVPRS (ENSEMBLE), FVPRS (SINGLE), AND MRMR

TABLE VICOMPARISON BETWEEN FVPRS AND CACLF

experimental comparison, we would like to point out some dif-ferences between them. The main difference between them isthat FVPRS is a feature selection method, whereas CACLFwas a feature extraction method since it found new features bycombining some original features. Another difference is that inCACLF, the number of extracted features can be specified in ad-vance, whereas FVPRS cannot. What is more, we find that thecomputational complexity of FVPRS is O(|U |2 × |A| × |K|),whereas the computational complexity of CACLF is decided bythe optimal problems of finding the suitable parameters for thelinear combination. Here, we compare these two methods onthe datasets that have been analyzed in CACLF and summarizethem in Table VI.

Table VI shows that under the condition with similar num-ber of features, CACLF has better classification accuracy thanFVPRS, whereas if we relax this condition, it is easy to findthat the classification accuracy of FVPRS is significantly bet-

ter than CACLF. For example, under the condition with similarnumber of features, the accuracy of FVPRS is comparable tothe accuracy of CACLF with the measure Q1 . If we relax thiscondition, FVPRS can find the feature subset with better accu-racy than CACLF with the measure Qmod in most cases. All theaforementioned comparison shows that these two methods aresuitable for different applications: FVPRS can find the reduc-tions with better classification performance, whereas CACLFcan find the compacter feature subset.

Remarks: FVPRS has its strengths in comparing with othermethods. However, it still has some limitations. One limitationof FVPRS is that there is a parameter, i.e., the threshold infuzzy lower approximation operators, to be specified. Althoughthe specification of the parameter depends on the domain ofproblems and the requirements of the classification accuracy, itis necessary to point out the extent of the reasonable parameter.Now one guide of specifying this parameter is that the larger thethreshold is, the smaller the number of selected attributes is.

V. CONCLUSION

In this paper, we have set up a framework named FVPRS bycombining FRS and VPRS. In FVPRS, a controlled threshold isintroduced into knowledge representation of FRS (in the sight ofgranular computing, a fuzzy cut set is introduced into FRS), andthen, the concepts of the fuzzy lower and upper approximationswith variable precision were proposed. We briefly list severalcharacterizations of FVPRS as follows.

1) FVPRS takes the existing FRS as a special case when thethreshold is set to zero.

2) FVPRS can find all the reductions by employing the dis-cernibility matrix approach.

3) The methodology of FVPRS is less sensitive to misclassi-fication and perturbation.

ACKNOWLEDGMENT

The authors would like to thank the reviewers for their valu-able suggestions. They would also like to thank M. Kyle for hishelp in the presentation of this paper.

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Suyun Zhao (S’06) received the Bachelor’s andMaster’s degrees from the School of Mathematicsand Computer Science, Hebei University, Baoding,China, in 2001 and 2004, respectively. She is cur-rently working toward the Ph.D. degree in the Depart-ment of Computing, Hong Kong Polytechnic Univer-sity, Kowloon, Hong Kong.

Her current research interests include machinelearning, pattern recognition, fuzzy sets, and roughsets.

Ms. Zhao is a Student Member of the IEEE Sys-tem, Man, and Cybernetics Society.

Eric C. C. Tsang (A’04–M’04) received the B.Sc.degree in computer studies from the City Universityof Hong Kong, Kowloon, Hong Kong, in 1990, andthe Ph.D. degree in computing from the Hong KongPolytechnic University, Hong Kong, in 1996.

His current research interests include fuzzyrough sets, fuzzy neural networks, genetic algorithm,fuzzy support vector machine, and multiple classifiersystems.

Degang Chen received the M.S. degree from theNortheast Normal University, Changchun, China, in1994, and the Ph.D. degree from Harbin Institute ofTechnology, Harbin, China, in 2000.

He was a Postdoctoral Fellow at Xi’an JiaotongUniversity, Xi’an, China (from 2000 to 2002), andat Tsinghua University, Beijing, China (from 2002to 2004). Since 2006, he has been a Professor atthe North China Electric Power University, Beijing,China. His current research interests include fuzzygroup, fuzzy analysis, rough sets, and support vector

machine (SVM).



Short Papers

Intermediate Variable Normalization for Gradient DescentLearning for Hierarchical Fuzzy System

Di Wang, Xiao-Jun Zeng, and John A. Keane

Abstract—When applying gradient descent learning methods to hierar-chical fuzzy systems, there is great difficulty in handling the intermediatevariables introduced by the hierarchical structures, as the intermediatevariables may go outside their definition domain that makes gradient de-scent learning invalid. To overcome this difficulty, this paper proposes alearning scheme that integrates a normalization process for intermediatevariables into gradient descent learning. This ensures that gradient descentmethods are applicable to, and correctly used for, learning general hier-archical fuzzy systems. Benchmark datasets are used to demonstrate thevalidity and advantages of the proposed learning scheme over other existingmethods in terms of better accuracy, better transparency, and fewer fuzzyrules and parameters.

Index Terms—Fuzzy systems, gradient descent method, hierarchicalfuzzy systems, learning.

I. INTRODUCTION

Standard fuzzy systems have been widely and successfully appliedin function approximation [1]–[3], system control [4]–[6], classifica-tion [7], and clustering [8]. However, when fuzzy systems are appliedto more complex and high-dimensional systems, the “curse of dimen-sionality” becomes increasingly apparent as a bottleneck to their widerapplication. To overcome this, hierarchical fuzzy systems were pro-posed in the early 1990s by Raju and Zhou [9] and have attracted muchattention in recent years. In hierarchical fuzzy systems, instead of us-ing a standard (flat) high-dimensional fuzzy system, a number of lowerdimensional fuzzy subsystems are linked in a hierarchical manner.

The main topics of hierarchical fuzzy systems research have beenconstruction, learning, and analysis (see the survey by Torra [10]).Wang and his colleagues [5], [11] proposed a kind of hierarchicalfuzzy system with one fuzzy subsystem in each layer that has oneoriginal input and one input from the output of the lower subsystem.They [5], [11] applied the gradient descent method for learning andanalyzed the relative importance of input variables as a criterion for thehierarchical structure construction. Although their proposed hierarchi-cal fuzzy systems managed to decrease the exponential growth of fuzzyrules, the exponential growth of parameters remains inherent. Chungand Duan [12] discussed how to design a hierarchical structure basedon the correlated or coupled relationship between input variables andshowed the applicability of their approach by simulation. Campello andAmaral [13] developed a method to construct hierarchical fuzzy sys-tems by using Gaussian membership functions. Joo and Lee [14], [15]proposed a scheme to construct another kind of general hierarchicalfuzzy system, in which the outputs of lower layers are used only in

Manuscript received April 27, 2007; revised April 4, 2008 and September 29,2008; accepted December 16, 2008. First published February 10, 2009; currentversion published April 1, 2009. This work is supported by the U.K. Engineeringand Physical Sciences Research Council (EPSRC) under Grant EP/C513355/1.

D. Wang is with ThinkAnalytics Ltd., Glasgow G3 7QF, U.K. (e-mail:[email protected]).

X.-J. Zeng and J. A. Keane are with the School of Computer Sci-ence, University of Manchester, Manchester M13 9PL, U.K. (e-mail:[email protected]; [email protected]).


the THEN part instead of the IF part of upper layers. However, theirmethod needs many parameters while obtaining less accurate results.Zeng and Keane [16] investigated the approximation capability andhave theoretically shown that hierarchical fuzzy systems can achievebetter approximation accuracy with fewer parameters and rules for alarge class of systems.

Gradient descent methods are widely used in parameter learning.Despite the aforementioned research results for hierarchical fuzzy sys-tems, there is no effective scheme to handle intermediate variables—afeature that does not occur in gradient descent learning for flat fuzzysystems. However, in order to make the gradient decent learning ap-plicable to, and effective for, hierarchical fuzzy systems, it is nec-essary and important to handle these intermediate variables properlyfor two reasons. First, in order to define the membership functions ofthe upper-layered fuzzy subsystems, it is necessary to know the def-inition domain of intermediate variables which are the outputs of thelower-layered fuzzy subsystems and the inputs to the upper-layeredfuzzy subsystems. Unfortunately, it is impossible to know the domainof intermediate variables before the final hierarchical fuzzy system isdetermined. Second, after the initial definition of membership func-tions in the gradient descent learning process, often the intermediatevariables go outside their definition domain, which results in no activa-tion, and hence, no parameter updating is done for the current traininginstance. If many training instances cause no parameter updating in thisway, then the gradient descent learning method is invalid. Hence, thepurpose of this research is to address this gap by introducing a normal-ization process to handle intermediate variables in the gradient descentlearning method to make it applicable to, and effective for, hierarchicalfuzzy systems.

In this paper, a gradient descent learning scheme integrated with thenormalization of intermediate variables for general hierarchical fuzzysystems is proposed to overcome the particular difficulty in determiningthe definition domain of intermediate variables. Further, it is shown that,although some extra errors may be introduced by the normalizationprocess for intermediate variables, such errors can be corrected in thefollowing learning iterations. The proposed learning scheme leads toadvantages such as better accuracy, better transparency, and fewer rulesand parameters. These advantages are validated both theoretically inthe Appendix and experimentally through simulation in Section III.

This paper is organized as follows. The learning algorithm with thenormalization of intermediate variable is proposed and validated inSection II. In Section III, the proposed algorithm is applied to bench-mark problems to demonstrate its advantages over existing methods forhierarchical fuzzy systems. Then, conclusions are given in Section IV.

II. GRADIENT DESCENT LEARNING WITH INTERMEDIATE

VARIABLE NORMALIZATION

A. Overview of Gradient Descent Learning for General HierarchicalFuzzy Systems

In a general multiple-input and single-output (MISO) hierarchicalfuzzy system (see Fig. 1), there may be multiple layers and multi-ple fuzzy subsystems in each layer. Outputs of lower-layered fuzzysubsystems form as inputs to their neighboring upper-layered fuzzysubsystems. These are termed as intermediate variables. The inputs tothe lowest layer are all original input variables. The inputs to the lth (l>1) layer are the combination of its lower-layered (the (l−1)th layer)outputs and some (or none) of the original input variables.

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Fig. 1. General structure for hierarchical fuzzy systems.

In general, a fuzzy subsystem Fl,p (the pth fuzzy subsystem in thelth layer) is represented in the form as

yl ,p = fl,p (yl−1 ,p ,1 , yl−1 ,p ,2 , . . . , yl−1 ,p ,P l−1 , p,

xl−1 ,p ,1 , . . . , xl−1 ,p ,Q l−1 , p) (1)

where yl ,p is the output of fuzzy subsystem Fl,p , yl−1 ,p ,j is the outputfrom fuzzy subsystem Fl−1 ,p ,j (the jth fuzzy subsystem of the (l−1)thlayer) to Fl,p , where Pl−1 ,p is the total number of outputs from the(l−1)th layer to Fl,p , xl−1 ,p ,j is the jth original input variable to Fl,p ,and Ql−1 ,p is the total number of original input variables to Fl,p .

Assuming L is the total number of layers (there is only one fuzzysubsystem in the top layer, the Lth layer, for an MISO system), thefinal output (the output of the Lth layer) is represented by

y = yL ,1 = fL ,1 (yL−1 ,1 ,1 , yL−1 ,1 ,2 , . . . , yL−1 ,1 ,PL −1 , 1 ,

xL−1 ,1 ,1 , . . . , xL−1 ,1 ,Q L −1 , 1 ). (2)

Uniquely, for the first (lowest) layer, the output of the pth fuzzysubsystem F1 ,p is

y1 ,p = f1 ,p (x0 ,p ,1 , . . . , x0 ,p ,Q 0 , p). (3)

In the proposed algorithm, for each fuzzy subsystem Fl,p , acommonly used defuzzifier, center-average defuzzifier, is appliedbased on Mamdani reasoning [17] (4) as shown at the bottomof this page, where j1 j2 · · · jP l−1 , p

i1 i2 · · · iQ l−1 , pis the index of

rules for the Fl,p , µjkl,p ,k (yl−1 ,p ,k ) is the membership function for

yl−1 ,p ,k , vikl ,p ,k (xl−1 ,p ,k ) is the membership function for xl−1 ,p ,k , and

yj1 j2 ···jP l−1 , p

i1 i2 ···iQ l−1 , p

l ,p is the parameters to be learned during gradi-ent descent learning.

Fig. 2. Example of triangular membership function.

If

Ul,p =

P l , 1 , p∏k=1

ujkl,p ,k (yl−1 ,p ,k ) (5)

and

Vl,p =

Q l−1 , p∏k=1

vikl ,p ,k (xl−1 ,p ,k ) (6)

then (4) can be represented as (7) as follows. For a simpler representa-tion, we use Ul,p and Vl,p in the remainder of this paper

yl ,p =∑

j1 j2 ···jP l−1 , pi1 i2 ···iQ l−1 , p

Ul,p Vl,p∑j1 j2 ···jP l−1 , p

i1 i2 ···iQ l−1 , p

Ul,p Vl,p

× y

j1 j2 ···jP l−1 , pi1 i2 ···iQ l−1 , p

l ,p . (7)

In the aforementioned discussion, µjkl,p ,k (yl−1 ,p ,k ) and

vikl ,p ,k (xl−1 ,p ,k ) in (5) and (6) are the membership functions,

which can be defined by many types of functions, such as Gaussian,triangular, trapezoid, or bell-shape. Specially, triangular functions arechosen as the membership functions in this paper. Furthermore, theedge of one triangular membership function is chosen to intersect to themiddle point of its neighboring triangular membership functions (asshown in Fig. 2). Detailed reasons for the use of triangle membershipfunctions are given in Remark 2.1 at the end of this section. Basedon the analysis before, the membership functions µ

jkl,p ,k (yl−1 ,p ,k )

are designed as µjkl,p ,k (yl−1 ,p ,k ) : Γl−1 ,p ,k ⊂ R → Γl ,p ⊂ R,

where Γl−1 ,p ,k = [αl−1 ,p ,k , βl−1 ,p ,k ]. If yl−1 ,p ,k is evenly par-titioned with Nl−1 ,p ,k , then Γl−1 ,p ,k = [αl−1 ,p ,k , βl−1 ,p ,k ] =⋃N l , 1 , p , k −1

i=0 [eil−1 ,p ,k , ei+1

l−1 ,p ,k ], and ejk +1l−1 ,p ,k − e

jkl−1 ,p ,k = (αl−1 ,p ,k

− βl−1 ,p ,k )/Nl−1 ,p ,k .

yl,p=fl,p(yl−1,p,1 , yl−1,p,2 , . . . , yl−1,p,Pl−1 , p, xl−1,p,1 , . . . , xl−1,p,Ql−1 , p

)

=∑

j1 j2 ···jP l−1 , pi1 i2 ···iQ l−1 , p

∏Pl−1 , p

k=1 µjk

l,p,k (yl−1,p,k )∏Ql−1 , p

k=1 vik

l,p,k (xl−1,p,k )∑j1 j2 ···jP l−1 , p

i1 i2 ···iQ l−1 , p

∏Pl−1 , p

k=1 µjk

l,p,k (yl−1,p,k )∏Ql−1 , p

k=1 vik

l,p,k (xl−1,p,k )

yj1 j2 ···jP l−1 , p

i1 i2 ···iQ l−1 , p

l,p

(4)



The triangular membership function µjkl,p ,k (yl−1 ,p ,k ) for yl−1 ,p ,k is

defined as follows:

µjkl,p ,k (yl−1 ,p ,k )

=

yl−1 ,p ,k − ejk −1l−1 ,p ,k

ejkl−1 ,p ,k − e

jk −1l−1 ,p ,k

, ejk −1l−1 ,p ,k ≤ yl−1 ,p ,k < e

jkl−1 ,p ,k

ejk +1l−1 ,p ,i − yl−1 ,p ,k

ejk +1l−1 ,p ,k − e

jkl−1 ,p ,k

, ejkl−1 ,p ,k ≤ yl−1 ,p ,k < ejk +1

l−1 ,p ,k

0, otherwise.

(8)

In a similar way, vikl ,p ,k (xl−1 ,p ,k ) is defined as

vikl ,p ,k (xl−1 ,p ,k )

=

xl−1 ,p ,k − eik −1l−1 ,p ,k

eikl−1 ,p ,k − e

ik −1l−1 ,p ,k

, eik −1l−1 ,p ,k ≤ xl−1 ,p ,k < e

ikl−1 ,p ,k

eik +1l−1 ,p ,i − xl−1 ,p ,k

eik +1l−1 ,p ,k − eik

l−1 ,p ,k

, eikl−1 ,p ,k ≤ xl−1 ,p ,k < e

ik +1l−1 ,p ,k

0, otherwise.

(9)

For the aforementioned membership functions, only 2n rules areactivated each time, where n is the number of input variables. Asshown by [3, Th. 2] ∑

j1 j2 ···jP l−1 , pi1 i2 ···iQ l−1 , p

Ul,p Vl,p = 1 (10)

then (7) can be simplified as

yl ,p =∑

j1 j2 ···jP l−1 , pi1 i2 ···iQ l−1 , p

Ul,p Vl,p yj1 j2 ···jP l−1 , p

i1 i2 ···iQ l−1 , p

l ,p .

(11)Remark 2.1: There are several important reasons which motivate

the use of triangular membership functions in this paper. First, fuzzysystems with triangular membership functions are more transparentand interpretable than fuzzy systems with other types of membershipfunctions. Second, fuzzy systems with triangular membership func-tions lead to a much simpler gradient decent learning algorithm fortwo reasons: one is the much simpler mathematical expression possi-ble, as shown in (11) where the complicated denominator terms in (7)disappear based on (10); the other is that only 2n rules are activatedfor each training sample that leads to much simpler calculation in eachiteration of the training process. This is significantly different to fuzzysystems with other types of membership functions, especially Gaus-sian membership functions where all rules are active for each trainingsample. Due to these two reasons, the corresponding gradient decentalgorithm takes much less time to learn and is much quicker to con-verge. Such simplification and improved performance are especiallyimportant for hierarchical fuzzy systems that have a much more com-plex system structure than (single layered) flat fuzzy systems and arethus much more complicated to learn. Third, fuzzy systems with trian-gular membership functions are one of the most widely used classesof fuzzy systems and have shown better performance than other typesof membership functions in many applications (for example, [19]). Inparticular, a valuable property of fuzzy systems with triangular mem-bership functions is that they can reproduce (i.e., represent with noerror) many important and widely used functions and systems such aslinear functions, multilinear functions, and piecewise linear functions,as shown in [18]. Fourth, fuzzy systems with triangular membership

functions have the decomposition property [18], i.e., by properly di-viding the input space into subinput spaces, a general fuzzy systemis decomposed into several fuzzy subsystems that are the simplestfuzzy systems in the subinput spaces. This feature is particular usefulin advanced techniques for fuzzy control design and analysis such aspiecewise Lyapounov methods [20].

In the remainder of this section, a gradient descent learning schemeintegrated with normalization of intermediate variables for general hi-erarchical fuzzy systems is proposed. For this purpose, it is assumedthat the hierarchical structure has been determined, and the discussionwill focus on the parameter learning process for general hierarchicalfuzzy systems.

B. Normalization for the Intermediate Variables

The definition domain for each variable must be known in orderto define triangular membership functions given in (8) and (9). Thedefinition domain of the original input variables can be obtained directlyfrom the training data. However, the possible values of the intermediatevariables are unknown. As a result, it is impossible to define the domainof [αl−1 ,p ,k , βl−1 ,p ,k ] for intermediate variable yl−1 ,p ,k in hierarchicalfuzzy systems before training.

In Wang’s research [11], the domain for the intermediate variable(y1 in his simulation example) is defined as [0, 1] during training.However, the value of the intermediate variable y1 is, in fact, notalways between zero and one during gradient descent learning. On thecontrary, y1 might be outside of the range [0, 1]. A value of y1 outsideof the range [0, 1] will not activate any membership function. In suchcases, the parameters (dlij and yp q

1 [11]) will not be updated during thegradient descent learning process. As a result, such gradient descentlearning is not valid for hierarchical fuzzy systems with triangularmembership functions. This might be the reason why Wang’s systemin [11] is less accurate than ours, despite using more parameters. Thisis illustrated by simulation comparison in Section III.

To bridge this gap, a normalization process is proposed to overcomethis particular difficulty in determining the definition domain of in-termediate variables. Our research has found that there is a valuableproperty related to intermediate variables: if a hierarchical fuzzy sys-tem can achieve a desired approximation accuracy within a definitiondomain [αl−1 ,p ,k , βl−1 ,p ,k ] for an intermediate variable yl−1 ,p ,k , thenby choosing its definition domain to be any interval [α′

l−1 ,p ,k , β ′l−1 ,p ,k ],

the same approximation accuracy can be achieved by an associated hi-erarchical fuzzy system. In other words, the approximation accuracy ofhierarchical fuzzy systems is independent of the choice of the defini-tion domains for intermediate variables. As a result, we can always findthe optimal hierarchical fuzzy system by restricting the intermediatevariables to any definition domain of [α′

l−1 ,p ,k , β ′l−1 ,p ,k ]. Particularly,

in our work, optimal hierarchical fuzzy systems are constructed byrestricting the definition domain of all intermediate variables to [0, 1].A normalization process is introduced to restrict the definition domainof intermediate variables to [0, 1], which is integrated into the gradi-ent descent learning algorithm. We now give the following theorem(Theorem 2.1) that provides the theoretical justification of this normal-ization process of intermediate variables to [0, 1].

Theorem 2.1 (Approximation Accuracy Preservation Prop-erty of Normalization): Define, in a hierarchical fuzzy sys-tem H(X), the pth fuzzy subsystem in the lth layer asFl,p :yl ,p = fl,p (Yl−1 ,p , Xl−1 ,p ), where yl ,p is the output, Xl−1 ,p =(xl−1 ,p ,1 , xl−1 ,p ,2 , . . . , xl−1 ,p ,Q l−1 , p

) are original inputs to Fl,p ,and Yl−1 ,p = (yl−1 ,p ,1 , yl−1 ,p ,2 , . . . , yl−1 ,p ,P l−1 , p

) are outputs of theconnected fuzzy subsystems in the (l−1)th layer and yl ,p ,j ∈[αl−1 ,p ,i , βl−1 ,p ,i ] (i = 1, . . . , Ql,p ). Under this definition, for a



given function G(X), if there exists a hierarchical fuzzy system H(X)defined by (1)–(4)

H(X) = fL ,1 (fL−1 ,1. . . [f2 ,1 (f1 ,1 (X0 ,1 )

, . . . , f1 ,P 1 (X0 ,P 1 ), X1 ,1 ), . . .] . . .

, . . . , fL−1 ,PL −1 . . . , XL−1 ,1 ) (12)

such that, for any given input vector X , |G(X) − H(X)| ≤ ε (whereε is a small constant representing the approximation error), then thereexists an associated hierarchical fuzzy system H ′(X) defined by (1)–(4), where y′

l ,p ,j ∈ [0, 1], (l < L), which has the same hierarchicalstructure as H(X) and is given by

H ′(X) = f ′L ,1 (f

′L−1 ,1. . . [f ′

2 ,1 (f′1 ,1 (X0 ,1 )

, . . . , f ′1 ,P 1

(X0 ,P 1 ), X1 ,1 ), . . .] . . .

, . . . , f ′L−1 ,PL −1

. . . , XL−1 ,1 ) (13)

such that H(X) = H ′(X) and |G(X) − H ′(X)| ≤ ε.The proof of Theorem 2.1 can be found in Appendix A.Remark 2.2: As shown in Theorem 2.1, for a given hierarchical

fuzzy system H(X) with the approximation error ε, there exists anassociated hierarchical fuzzy system H ′(X)with its all intermediatevariables yl,p = f ′

l ,p ∈ [0, 1](l < L) that can achieve the same accu-racy as H(X). This is why the aforementioned theorem is called theapproximation accuracy preservation property of normalization. Fur-ther, by the same proof, it can be shown that the theorem still holds if[0, 1] is replaced by any interval. For simplicity, in our research, thedefinition domain of all intermediate variables is restricted to [0, 1].

C. Learning Based on Gradient Descent Algorithm

In this section, the formula of gradient descent learning for hier-archical fuzzy systems with the integrated normalization process isgiven.

In gradient descent learning, the error of the final output is propa-gated back from upper layers to lower ones. The parameter updatingof the lower layers is based on the errors propagated back from theirupper layers. The objective is to minimize total error.

The error between the actual output y(k) and the model output yL (k)at time k is defined as

eL (k) = yL (k) − y(k). (14)

For simplicity, consider two connected fuzzy subsystems in twoconsequent layers, fuzzy subsystem q and fuzzy subsystem p, where qis the neighboring upper-layered fuzzy subsystem of fuzzy subsystemp (as shown in Fig. 3).

Further, for simplifying the expressions, in this section, the variablesin discussion and equations are omitted. For example, we representµ

jpq ,p (yq ,p ,j , k) by µ

jpq ,p (k), where k means “at time k.”

Then, the error propagated back from fuzzy subsystem q to p isdefined as

ep (k) = eq (k) × ∂yq (k)∂yp (k)

(15)

Fig. 3. Illustration of two neighboring fuzzy subsystems.

where ep (k) is the error of fuzzy subsystem p, which is propagatedfrom its neighboring upper-layered fuzzy subsystem q, and eq (k) isthe propagated error of fuzzy subsystem q.

Similar to (5) and (6), if we define

Uq (k) =Pq∏i=1

ujiq ,i (k) (16)

Vq (k) =Q q∏i=1

νjiq ,i (xp,i ) (17)

then,

∂yq (k)∂yp (k)

=∑

j1 j2 ···jP q , p i1 i2 ···iQ q , p

yj1 j2 ···jP q , p i1 i 2 ···iQ q , pq

∂(Uq (k)Vq (k))∂yp (k)

(18)where

∂(Uq (k)Vq (k)∂yp (k)

=Uq (k)Vq (k)

µjpq ,p (k)

× ∂(µjpq ,p (k))

∂yp (k). (19)

From (5), we have

∂(µjpq ,p (k))

∂yp (k)=

1, e

jp −1p (k) ≤ yp (k) < e

jpp (k)

−1, ejpp (k) ≤ yp (k) < e

jp + 1p (k)

0, otherwise.

(20)

Hence, (21) shown at the bottom of this page.

The formula to update the parameters yj1 j2 ···jP q , p i1 i2 ···iQ q , pp (k) in

gradient descent learning is

yj1 j2 ···jP q , p i1 i2 ···iQ q , pp (k + 1)

= yj1 j2 ···jP q , p i1 i2 ···iQ q , pp (k) − η × Uq (k)Vq (k) × ep (22)

where η is the learning rate.As discussed, we aim to assign the definition domain for the inter-

mediate variables as [0, 1]. However, the outputs of a fuzzy subsystem

yl ,p computed by (11) are dependent on the parameters yj1 j2 ···jn q

l ,p andcannot always be guaranteed to be between zero and one. To force the

ep(k) =

eq (k) ×∑

j1 j2 ···jP q , p i1 i2 ···iQ q , p

(y

j1 ,j2 ···jP q , p i1 i2 ···iQ q , pq (k)Uq (k)Vq (k)

µjpq ,p(k)

), if e

jp −1p (k) ≤ yp(k) < e

jpp (k)

eq (k) ×∑

j1 j2 ···jP q , p i1 i2 ···iQ q , p

(y

j1 j2 ···jP q , p i1 i2 ···iQ q , pq (k)Uq (k)Vq (k)

µjpq ,p(k)

), if e

jpp (k) ≤ yp(k) < e

jp + 1p (k)

0, otherwise.

(21)



intermediate variables to be in the range [0, 1], normalization has to beapplied by using (23), shown at the bottom of this page.

After this normalization, all yl ,p are always between zero and one.This can be formally expressed in Theorem 3.1.

Theorem 3.1: After the normalization process by (23), for any fuzzysubsystem Fl,p (i.e., the pth fuzzy subsystem of the lth layer), all the

THEN part yj1 j2 ···jP q , p i1 i2 ···iQ q , p

l ,p (say, the j1 j2 · · · jPq , p i1 i2 · · · ithQ q , p

rule of Fl,p ) is between zero and one (i.e., yj1 j2 ···jP q , p i1 i2 ···iQ q , p

l ,p ∈[0, 1]). Thus, the output of Fl,p , yl ,p , is also between zero and one (i.e.,yl ,p ∈ [0, 1]).

The proof of Theorem 3.1 can be found in Appendix B.However, the normalization process, according to (23), introduces

extra error; hence, the parameters may no longer be optimal after nor-malization. The error introduced by this normalization process can beefficiently corrected in the following gradient descent learning itera-tions. This is shown by the following theorem.

Theorem 3.2: In the proposed learning algorithm, the extra errorintroduced in the normalization process [according to (23)] can beeliminated by its following gradient descent training process.

The proof of Theorem 3.2 is given in Appendix C.

III. SIMULATION AND COMPARISON

In this section, we illustrate the advantages of our proposed methodover other methods for hierarchical fuzzy system learning using twosimulation examples.

Simulation 1: This is a function approximation problem referred byWang [11]

g(x1 , x2 , x3 ) =1

1 + sin2 (πx1 ) + sin2 (πx2 ) + sin2 (πx3 )

on U = [0, 1]3 .

For comparison, 63 input–output pairs (xij k0 ; g(xij k

0 )), wherexij k

0 = [0.2(i − 1), 0.2(j − 1), 0.2(k − 1)] for i, j, k = 1, 2, . . . , 6,are created for training [11], and 63 uniformly distributed input–outputpairs are used for testing. Wang used six equally spaced triangularmembership functions over [0, 1] on both the inputs x1 , x2 , x3 , and theintermediate variable.

In Wang’s method, there is one fuzzy subsystem in each layer, andthere are two inputs to each fuzzy subsystem created by using a gridpartition. Hence, there are (n−1) layers altogether, and N 2 fuzzy rulesfor each layer, where N is the partition for each variable (both theoriginal and intermediate variables) and n is the input dimension (i.e.,the total number of the original input variables). In Wang’s scheme,the conclusion part for lowest layer is a constant, i.e., the number ofparameters for the lowest layer is equal to the number of fuzzy rulesN 2 . For the other layers, the conclusion part of a fuzzy rule is an(N−1) order polynomial (hence, N parameters for each fuzzy rule) ofits lower layer outputs. Therefore, the total number of parameters usedin Wang’s method is (n − 2) ∗ N 3 + N 2 .

In this simulation, N = 6 and n = 3, so the total number of param-eters used is 63 + 62 = 252 for Wang’s method. On the contrary, our

TABLE ICOMPARISON WITH WANG’S METHOD [11]

TABLE IICOMPARISON WITH JOO’S WORK 2002 (300 CASES)

proposed method is based on Mamdani reasoning and the number of pa-rameters is equal to the number of fuzzy rules, i.e., only 62 + 62 = 72parameters are needed in our proposed scheme for the same hierarchi-cal structure and partitions. Therefore, far fewer parameters are used inour scheme than Wang’s, and, in turn, much better accuracy is obtained,as shown in Table I, where it can be seen that by using 72 parameters,we obtain an accuracy of average percentage error (APE) = 0.44067%for the training data, and an accuracy of APE = 2.9721% for the testingdata in comparison with 10% by Wang’s method [11]. More results interms of APE by using different number of parameters and fuzzy rulesin our scheme are shown in Table I. All these results show that ourproposed scheme constructs a hierarchical fuzzy system with far fewerparameters and better accuracy than Wang’s method.

Joo and Lee [14], [15] have proposed a hierarchical fuzzy system[14], where the outputs of the previous layer are not used in the IFparts, but only the THEN parts of the fuzzy rules of the current layer.Using Simulation 2 (used in [14] and [15]), we illustrate the betterperformance of our proposed algorithm over Joo and Lee’s approach.

yj1 j2 ···jP q , p i1 i2 ···iQ q , p

l,p =y

j1 j2 ···jP q , p i1 i2 ···iQ q , p

l,p − minj1 j2 ...jn q

(yj1 j2 ···jP q , p i1 i2 ···iQ q , p

l,p )

maxj1 j2 ···jn q

(yj1 j2 ···jP q , p i1 i2 ···iQ q , p

l,p )− minj1 j2 ···jn q

(yj1 j2 ···jP q , p i1 i2 ···iQ q , p

l,p )

(23)



TABLE IIICOMPARISON WITH JOO’S WORK 2005 (3000 CASES)

Simulation 2: This simulation is to stabilize a ball and beam controlsystem. The control law obtained is as follows:

u ∗ (x)=4BGx4 cos x3+6BG sin x3−4x2−x1−BGx2

4 sin x3

−BG cos x3

with B and G being 0.7143 and 9.81, respectively. So, the task of thissimulation is to find a hierarchical fuzzy system to approximate theaforementioned control function.

The simulation results are compared with Joo and Lee’s method [14],[15], as shown in Table II [14] and Table III [15] in terms of Ju, which

is defined as Ju = (1/N )√∑N

i=1 (u∗k − uk )2 , where u∗

k is the kth

objective output and uk is the kth model output. Our proposed schemeobtains better accuracy of Ju = 0.0167 while also using fewer fuzzyrules and fewer parameters (12 fuzzy rules and 12 parameters) than Jooand Lee’s method with an accuracy of Ju = 0.02 (27 fuzzy rules and 45parameters). When more fuzzy rules are used in our proposed scheme,better accuracy is obtained, e.g., 54 fuzzy rules and parameters result inan accuracy of Ju = 0.0035. So, this shows that our proposed schemehas better performance than Joo and Lee’s method, in terms of bothaccuracy and the number of parameters.

IV. CONCLUSION

Hierarchical fuzzy systems have been shown by a number of papersto be an effective approach to overcoming the “curse of dimensional-ity” in flat grid-based fuzzy systems. When applying gradient descentmethods to learn hierarchical fuzzy systems, there is a great difficultyin handling the intermediate variables introduced by the hierarchicalstructures, as the intermediate variables often go outside their defini-tion domain that makes gradient descent learning invalid. In order toovercome this difficulty, this paper has proposed a gradient descentlearning scheme integrated with the normalization process of interme-diate variables for general hierarchical fuzzy systems.

The main contribution of this paper is the introduction of a nor-malization process for the intermediate variables during the learningiterations, to make the gradient descent learning method applicable to,and effective for, general hierarchical fuzzy systems. We theoreticallyproved the validity of this normalization process and verified that theproposed normalization process does not damage the optimization ofthe final solution. The advantages of the proposed learning schemeover existing methods [5], [11], [14], [15] are shown by benchmarksimulations in terms of better accuracy, better transparency, and fewerfuzzy rules and parameters.

Nonetheless, there remain some related and open problems for learn-ing hierarchical fuzzy systems, such as how to identify and learn theirassociated hierarchical structure. Developments in these areas shouldresult in wider application of hierarchical fuzzy systems and, in turn,help to extend fuzzy systems to successfully solve more complicatedand high-dimensional problems.

APPENDIX

A. Proof of Theorem 2.1

To prove Theorem 2.1, a lemma is proved first as follows.Lemma A.1: Let F (X) = f1 [f0 ,1 (X1 ), f0 ,2 (X2 ), . . . , f0 ,p

(Xp ), X0 ] be a function defined on X = [X1 , X2 , . . . ,Xp , X0 ] ∈ U = U1 × U2 × · · · × Up × U0 and yi = f0 , i (Xi )∈ [αi , βi ] for Xi ∈ Ui with αi = minX i ∈U i

f0 , i (Xi ) andβi = maxX i ∈U i

f0 , i (Xi ) (i = 1, 2, . . . , p). Then, there ex-ists an associated function F ′(X) = f ′

1 [f ′0 ,1 (X1 ), f ′

0 ,2(X2 ), . . . , f ′

0 ,p (Xp ), X0 ] defined on U = U1 × U2 × · · · × Up × U0

such that F (X) = F ′(X) for all X ∈ U and y′i = f ′

0 , i (Xi ) ∈ [0, 1]for Xi ∈ Ui with minX i ∈U i

f ′0 , i (Xi ) = 0 and maxX i ∈U i

f ′0 , i (Xi ) =

1 (i = 1, 2, . . . , p).Proof: Define y′

i = f ′0 , i (Xi ) = (f0 , i (Xi ) − αi )/(βi − αi ) (i =

1, 2, . . . , p) for Xi ∈ Ui , then y′i = f ′

0 , i (Xi ) ∈ [0, 1] for Xi ∈Ui with minX i ∈U i

f ′0 , i (Xi ) = 0 and maxX i ∈U i

f ′0 , i (Xi ) = 1 (i =

1, 2, . . . , p).Now, define

f ′1 (y

′1 , y

′2 , . . . , y

′p , X0 ) = f1 [(β1 − α1 )y′

1 + α1 , (β2 − α2 )y′2

+ α2 , . . . , (βp − αp )y′p + αp , X0 ]

for Y ′ = [y′1 , y

′2 , . . . , y

′p ] ∈ [0, 1] × [0, 1] × · · · × [0, 1], then for all

X = [X1 , X2 , . . . , Xp , X0 ] ∈ U = U1 × U2 × · · · × Up × U0 ,

F ′(X) = f ′1 [f

′0 ,1 (X1 ), f ′

0 ,2 (X2 ), . . . , f ′0 ,p (Xp ), X0 ]

= f1 [(β1 − α1 )f ′0 ,1 (X1 )

+ α1 , (β2 − α2 )f ′0 ,2 (X2 )

+ α2 , . . . , (βp − αp )f ′0 ,p (Xp ) + αp , X0 ]

= f1

[(β1 − α1 )

f0 ,1 (X1 ) − α1

β1 − α1

+ α1 , (β2 − α2 )f0 ,2 (X2 ) − α2

β2 − α2

+ α2 , . . . , (βp − αp )f0 ,p (Xp ) − αp

βp − αp

+ αp , X0

]= f1 [f0 ,1 (X1 ), f0 ,2 (X2 ), . . . , f0 ,p (Xp ), X0 ] = F (X).

This ends the proof.Proof of Theorem 2.1: Based on Lemma 1, which is the case for

a two-level hierarchical fuzzy system and mathematical induction,Theorem 2.1 can be obtained immediately. Here, we omit the detailedsteps to apply mathematical induction due to the space limitation.

B. Proof of Theorem 3.1

Proof of Theorem 3.1: We prove that the conclusion part for anyfuzzy subsystem yl ,p = fl,p (Yl−1 ,p , Xl−1 ,p ) is between zero and one



(i.e., yl ,p ∈ [0, 1]). First, we have from (4) that

y l,p =

∑j1 j2 ···jP l−1 , p

i1 i2 ···iQ l−1 , p

U (yl−1 ,p )V (xl−1 ,p )∑

j1 j2 ···jn l , p

U (yl−1 ,p )V (xl−1 ,p )

× yj1 j2 ···jP l−1 , p

i1 i2 ···iQ l−1 , p

l ,p

(B1)

where U (yl−1 ,p ) and V (xl−1 ,p ) are defined as

U (yl−1 ,p ) =

P l−1 , p∏k=1

µjkl,p ,k (yl−1 ,p ,k ) (B2)

V (xl−1 ,p ) =

Q l−1 , p∏k=1

vikl ,p ,k (xl−1 ,p ,k ). (B3)

For a simple representation, we define a normalization factor as

N (yl−1 ,p ,xl−1 ,p ) =∑

j1 j2 ···jn l , p

U (yl−1 ,p )V (xl−1 ,p ). (B4)

Then, the following two inequalities are obtained as shown (B5) and(B6) at the bottom of this page.This immediately implies yl ,p ∈ [0, 1].

C. Proof of Theorem 3.2

Proof of Theorem 3.2: Consider the hierarchical fuzzy system givenin (4) or (11). To simplify the expression, in the following, we omit theindex of layers and the index of fuzzy subsystems in that layer. Then,for a given training instance, the output of any given fuzzy subsystemcan be presented as

y = o =R∑

r=1

(yr ×

m∏p=1

µr (yp ) ×n 0∏

q=1

µr (xq )

)(C1)

where yp (p = 1, 2, . . . , m) are the associated intermediate variables,xq (q = 1, 2, . . . , n0 ) are the original input variables, y is the output,yr is the THEN part of the rth rule, µr (yp ) is the membership ofthe pth intermediate variable, and µr (xq ) is membership of the qth

original variable. There are m intermediate variables and n0 originalinput variables for the given subsystem.

Now, define the output for the rth rule for the given fuzzy subsystemas or

or = yr ×m∏

p=1

µr (yp ) ×n 0∏

q=1

µr (xq ). (C2)

Then,

y = o =R∑

r=1

or . (C3)

The overall error of the given fuzzy subsystem caused by normal-ization of the intermediate variables yp (where p = 1,. . ., m) is

e =R∑

r=1

(yr ×

m∏p=1

µr (yp ) ×n 0∏

q=1

µr (xq )

)

−R∑

r=1

(yr ×

m∏p=1

µr (yp ) ×n 0∏

q=1

µr (xq )

)

=R∑

r=1

yr ×

n 0∏q=1

µr (xq ) ×

[m∏

p=1

µr (yp ) −m∏

p=1

µr (yp )

](C4)

where

yp =yp − minp

maxp −minp

. (C5)

If we define the error caused by the rth rule by the normalization ofthe intermediate variables as er

er = yr ×n 0∏

q=1

µr (xq ) ×

[m∏

p=1

µr (yp ) −m∏

p=1

µr (yp )

]. (C6)

Then,

e =R∑

r=1

er . (C7)

As (C6) and (C7) show that the overall error e is the sum of the errorcaused by each fuzzy rule, this implies that, to prove that the overallerror caused by the normalization process is eliminated, it is sufficientto prove the error caused by each fuzzy rule is eliminated, which isgiven later.

yl,p =∑

j1 j2 ···jP l−1 , pi1 i2 ···iQ l−1 , p

[U(yl−1,p)V (xl−1,p)N(yl−1,p ,xl−1,p)

]y

j1 j2 ···jP l−1 , pi1 i2 ···iQ l−1 , p

l,p

≤∑

j1 j2 ···jP l−1 , pi1 i2 ···iQ l−1 , p


]max

j1 j2 ···jn l , p

(y

j1 j2 ···jP l−1 , pi1 i2 ···iQ l−1 , p

l,p

)= 1

(B5)

yl,p =∑

j1 j2 ···jP l−1 , pi1 i2 ···iQ l−1 , p


]y

j1 j2 ···jP l−1 , pi1 i2 ···iQ l−1 , p

l,p

≥∑

j1 j2 ···jP l−1 , pi1 i2 ···iQ l−1 , p


]min

j1 j2 ···jn l , p

(y

j1 j2 ···jP l−1 , pi1 i2 ···iQ l−1 , p

l,p

)= 0 (B6)



Consider the error caused by the rth rule as given in (C6) and define

∆ =m∏

i=1

µr (yp ) −m∏

i=1

µr (yp ). (C8)

Then, (C6) can be rewritten as

er = yr ×n 0∏

q=1

µr (xq ) ×

[m∏

p=1

µr (yp ) −m∏

p=1

µr (yp )

]

= yr ×n 0∏

q=1

µr (xq ) × ∆ (C9)

which implies

∂or

∂yr=

m∏p=1

µr (yp ) ×n 0∏

q=1

µr (xq ) (C10)

yr (k + 1) = yr (k) − η × ∂or

∂yr× er

= yr (k) − η ×m∏

p=1

µr (yp ) ×n 0∏

q=1

µr (xq ) × er .

(C11)

Equation (C9) is the error caused by the rth rule of the given fuzzysubsystem and (C11) is used for parameter updating. Based on (C8)–(C11), we consider each of the three possible cases as follows:

1) If∏m

p=1 µr (yp ) = 0 and∏m

p=1 µr (yp ) = 0, or∏m

p=1

µr (yp ) =∏m

p=1 µr (yp ) > 0, i.e., ∆ =∏m

i=1 µr (yp ) −∏m

i=1 µr (yp ) = 0, then based on (C9), er = 0, i.e., noerror is introduced by the normalization for this rule.Further, no action is taken to adjust the parameters, asyr (k + 1) = yr (k) − η × (∂or /∂yr ) × er = yr (k) based on(C11).

2) If (∏m

i=1 µr (yp ) = 0 and∏m

i=1 µr (yp ) > 0), or (∏m

i=1µr (yp ) > 0,

∏m

i=1 µr (yp ) > 0 and∏m

i=1 µr (yp ) −∏m

i=1µr (yp ) > 0), then ∆ =

∏m

i=1 µr (yp ) −∏m

i=1 µr (yp ) > 0.Then, there are two possible cases that are considered next.

a) If yr (k) > 0, then from (C9), er (k) > 0, which im-plies η ×

∏m

p=1 µr (yp ) ×∏n 0

q=1 µr (xq ) × er (k) > 0and yr (k + 1) < yr (k) from (C11). Given aproper small value of the learning rate η in(C11), then yr (k) and yr (k + 1) have thesame sign, i.e., yr (k + 1) > 0. Then, based on(C9), er (k + 1) = yr (k + 1) ×

∏n 0q=1 µr (xq ) × ∆ > 0.

Further, ∆er = er (k + 1) − er (k) = [yr (k + 1) −yr (k)] ×

∏n 0q=1 µr (xq ) × ∆ < 0 as yr (k + 1) < yr (k).

Based on er (k + 1) > 0, er (k) > 0, ander (k + 1) < er (k), we have |er (k + 1)| < |er (k)|.

b) Else if yr (k) < 0, then from (C9), er (k) < 0, whichimplies η ×

∏m

p=1 µr (yp ) ×∏n 0

q=1 µr (xq ) × er (k) < 0and yr (k + 1) > yr (k) from (C11). Given aproper small value of the learning rate η in(C11), then yr (k) and yr (k + 1) have thesame sign, i.e., yr (k + 1) < 0. Then, based on(9), er (k + 1) = yr (k + 1) ×

∏n 0q=1 µr (xq ) × ∆ < 0.


∏n 0q=1 µr (xq ) × ∆ > 0 as yr (k + 1) > yr (k).

Based on er (k + 1) < 0, er (k) < 0, ander (k + 1) > er (k), we have |er (k + 1)| < |er (k)|.

3) If (∏m

i=1 µr (yp ) > 0 and∏m

i=1 µr (yp ) = 0) or (∏m

i=1µr (yp ) > 0,

∏m

i=1 µr (yp ) > 0, and∏m

i=1 µr (yp ) −∏m

i=1 µr (yp ) < 0), then ∆ =∏m

i=1 µr (yp ) −∏m

i=1µr (yp ) < 0. Then, there are two possible cases that areconsidered next.

a) If yr (k) > 0, then from (C9), er (k) < 0, whichimplies η ×

∏m

p=1 µr (yp ) ×∏n 0

q=1 µr (xq ) × er (k) < 0and yr (k + 1) > yr (k) from (C11). Given a proper smallvalue of the learning rate η in (C11), then yr (k)and yr (k +1) have the same sign, i.e., yr (k + 1) > 0. Then, basedon (C9), er (k + 1) = yr (k + 1) ×

∏n 0q=1 µr (xq ) × ∆ <

0. Further, ∆er = er (k + 1) − er (k) = [yr (k + 1) −yr (k)] ×

∏n 0q=1 µr (xq ) × ∆ < 0 as yr (k + 1) > yr (k).

Based on er (k + 1) > 0, er (k) > 0, and er (k + 1) <er (k), we have |er (k + 1)| < |er (k)|.

b) Else if yr (k) < 0, then from (C9), er (k) > 0, whichimplies η ×

∏m

p=1 µr (yp ) ×∏n 0

q=1 µr (xq ) × er (k) > 0and yr (k + 1) < yr (k) from (C11). Given aproper small value of the learning rate η in(C11), then yr (k) and yr (k + 1) have thesame sign, i.e., yr (k + 1) < 0. Then, based on(9), er (k + 1) = yr (k + 1) ×

∏n 0q=1 µr (xq ) × ∆ > 0.


∏n 0q=1 µr (xq ) × ∆ > 0 as yr (k + 1) < yr (k).

Based on er (k + 1) < 0, er (k) < 0, ander (k + 1) > er (k), we have |er (k + 1)| < |er (k)|.

By combining the conclusion of 1)–3), it is implied that|er (k + 1)| ≤ |er (k)|, where er (k) and er (k + 1) are errors causedby the normalization process. Then, we say the error introduced bythe normalization process for the considered fuzzy subsystem can becorrected in the following training process. This conclusion can then beapplied to arbitrary fuzzy subsystems involving intermediate variables.

Remark C.1: The aforementioned proof is based on that, for a giventraining data point, the error introduced by the normalization for thisdata point in some iteration will be corrected by the same data pointin subsequent iterations. In the practical execution of the proposedalgorithm, the error introduced by the normalization of a training datapoint, in fact, can occur with the same iteration, if, in the trainingdataset, there are input–output pairs which are very similar to the givendata point. This can be verified by using similar arguments to thosegiven in the aforementioned proof.

ACKNOWLEDGMENT

The authors would like to thank the Associate Editor and reviewersfor their detailed constructive comments and valuable suggestions thathave greatly helped improve the presentation and quality of the paper.

REFERENCES

[1] H. Ying, “Sufficient conditions on general fuzzy systems as function ap-proximators,” Automatica, vol. 30, pp. 521–525, 1994.

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[3] X. J. Zeng and M. G. Singh, “Approximation accuracy analysis of fuzzysystem as function approximators,” IEEE Trans. Fuzzy Syst., vol. 4, no. 1,pp. 44–63, Feb. 1996.

[4] J. J. Buckley, “Universal fuzzy controller,” Automatica, vol. 28, pp. 1245–1248, 1992.

[5] L. X. Wang, “Fuzzy systems are universal approximators,” in Proc. Conf.Fuzzy Syst., San Diego, CA, 1992, pp. 1163–1170.

[6] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its ap-plications to modelling and control,” IEEE Trans. Syst., Man Cybern.,vol. SMC-15, no. 1, pp. 116–132, Feb. 1985.



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Observer-Based Relaxed H∞ Control for Fuzzy SystemsUsing a Multiple Lyapunov Function

Sung Hyun Kim and PooGyeon Park

Abstract—This short paper proposes a method of designing a fuzzyobserver-based H∞ controller for discrete-time Takagi–Sugeno (T–S) fuzzysystems. To enhance the applicability of the output-feedback controller andimprove its performance, this short paper first builds a set of fuzzy controlrules with premise variables different from those of the T–S fuzzy system,and sets the overall controller to be dependent on not only the current timebut also the one-step-past information on the estimated fuzzy weightingfunctions. Then, based on the fuzzy control rules, this short paper estab-lishes a less conservative H∞ stabilization condition incorporated witha multiple Lyapunov function dependent on the estimated fuzzy weight-ing functions. Through a two-step design procedure, the H∞ stabilizationcondition is formulated in terms of parameterized linear matrix equalities(PLMIs), which are reconverted into LMIs with the help of an efficient andeffective relaxation scheme.

Index Terms—Fuzzy weighting-dependent Lyapunov function(FWDLF), H∞ performance, observer-based fuzzy control, non-paralleldistributed compensation (non-PDC) scheme, parameterized linear matrixinequalities (PLMIs), relaxation scheme.

I. INTRODUCTION

For a systematic control design of nonlinear systems, the Takagi–Sugeno (T–S) fuzzy model [1], [27] has been a popular choice not onlyin consumer products but also in industrial processes due to its abilityto represent the nonlinear system only from input–output data withoutcomplex mathematical equations. Thus, based on the T–S fuzzy model,various kinds of fuzzy control methods have been developed underthe so-called parallel distributed compensation (PDC) scheme [2], [4],[15]. Recently and notably, one has focused on developing the fuzzycontrol method associated with a fuzzy weighting-dependent (multiple)Lyapunov function (FWDLF) [11], [20], [24], which is because, for alarge number of fuzzy rules, the use of the common quadratic Lyapunovfunction (CQLF) [2], [4], [22], [25] leads to overconservative designsolutions. In spite of this trend, to the best of our knowledge, there hasbeen almost no results using the FWDLF approach when designingfuzzy output-feedback controllers, except for [24].

Practically, all states are not fully measurable; hence, it is necessaryin this area to design a fuzzy output-feedback controller, such as staticoutput feedback [14], [22], [25], dynamic output feedback [15], [24],and observer-based output feedback [5], [7], [9], [12], [13], [16], [21],[23], [28]. Especially in the case where the premise variables of the T–Sfuzzy system are related to the immeasurable state, one needs to designa fuzzy observer estimating the state. Of course, under the assumptionthat the premise variables of the fuzzy controller are same as those ofthe T–S fuzzy system, Ma et al. [5] and Yoneyama et al. [7] proved theseparation principle for the observer-based fuzzy controller; Xiaodongand Qingling [13] proposed two sufficient linear matrix inequalities(LMI) conditions guaranteeing the existence of the observer-based H∞control; and Lo and Lin [16] proposed the method of designing an

Manuscript received February 11, 2008; revised May 30, 2008 and October2, 2008; accepted December 6, 2008. First published December 22, 2008; cur-rent version published April 1, 2009. This work was supported in part by theMinistry of Knowledge Economy, Korea, under the Information TechnologyResearch Center (ITRC) support program supervised by the Institute of In-formation Technology Advancement (IITA): IITA-2008-C1090-0801-0037 andIITA-2008-C1090-0801-0004.

The authors are with the Division of Electrical and Computer Engineer-ing, Pohang University of Science and Technology, Pohang 790-784, Korea([email protected]; [email protected]).


1063-6706/$25.00 © 2009 IEEE



observer-based robust H∞ control. Besides, Lam and Zhou [24] alsoapplied the FWDLF approach under the assumption. However, it isworth noting that the assumption imposes a strict constraint on theapplicability of the designed controller in practical applications [12],[21].

This short paper aims at enhancing the applicability of the output-feedback controller while improving its H∞ performance. To this end,we first build a set of fuzzy control rules with premise variables differ-ent from those of the T–S fuzzy system and set the overall controllerto be dependent on not only the current time but also the one-step-pastinformation on the estimated fuzzy weighting functions. Then, basedon the fuzzy control rules, we establish a less conservative H∞ stabi-lization condition incorporated with a multiple Lyapunov function. Inthe derivation, the H∞ stabilization conditions are formulated in termsof parameterized LMIs (PLMIs) through a two-step design proceduresimilar to [12], [13], [16], and [21], which are reconverted into LMIswith the help of an efficient and effective relaxation scheme [11]. Sincethe proposed stabilization condition incorporates some additional con-ditions on fuzzy weighting functions into the interactions among thefuzzy subsystems, more powerful and useful results can be obtained asthe number of the fuzzy rules involved increases.

Notation: The Lebesgue space L2+ = L2 [0,∞) consists of square-integrable functions on [0,∞). The notations X ≥ Y and X > Y meanthat X − Y is positive semidefinite and positive definite, respectively.The notation A ⊕ B stands for the Kronecker sum of two matrices Aand B. In symmetric block matrices, (∗) is used as an ellipsis for termsthat are induced by symmetry.

II. SYSTEM DESCRIPTION AND PRELIMINARIES

Consider a discrete-time T–S fuzzy model described by fuzzy IF–THEN rules of the following form:

Plant rule i: IF η1 (k) is Fi1 and · · · and ηs (k) is Fis , THEN

xk+1 = Aixk + B1 iwk + B2 i uk

zk = C1 i xk + D11 iwk + D12 i uk

yk = C2 i xk + D21 iwk , for i = 1, 2, . . . , r (1)

where the consequent subsystems (1) represent linear systems in localoperating regions, Fij denotes a fuzzy set, η1 (k), . . . , ηs (k) denote thepremise variables of the plant, r denotes the number of IF–THEN rules,and xk ∈ Rn x , uk ∈ Rn u , yk ∈ Rn y , wk ∈ Rn w , and zk ∈ Rn z de-note the state, the control input, the measured output, the disturbance,and the performance output, respectively. For the premise variables,η1 (k), . . . , ηs (k), we assume that the following assumptions are valid.

A1) The premise variables do not explicitly depend on the controluk and the disturbance wk ∈ L2+ .

A2) The premise variables are determined by the unmeasurable stateof the plant.

Based on A1), the overall fuzzy model is inferred as follows:

xk+1 = A(Θk )xk + B1 (Θk )wk + B2 (Θk )uk

zk = C1 (Θk )xk + D11 (Θk )wk + D12 (Θk )uk

yk = C2 (Θk )xk + D21 (Θk )wk (2)

where A(Θk ) B1 (Θk ) B2 (Θk )

C1 (Θk ) D11 (Θk ) D12 (Θk )

C2 (Θk ) D21 (Θk ) 0

=r∑

i=1

θi (η(k))

Ai B1 i B2 i

C1 i D11 i D12 i

C2 i D21 i 0

(3)

θi (η(k)) =gi (η(k))∑r

i=1 gi (η(k)), gi (η(k)) =

s∏j=1

fij (ηj (k)). (4)

The notation η(k) = [η1 (k), . . . , ηs (k)]T ∈ Rs , fij (ηj (k)) denotesthe grade of membership of ηj (k) in Fij , and Θk ∈ Rr stands for avector of time-varying fuzzy weighting functions θi (η(k)). Moreover,let gi (η(k)) ≥ 0, for i = 1, . . . , r, and

∑r

i=1 gi (η(k)) > 0 for all timek. Then, θi (η(k)) ≥ 0, for i = 1, . . . , r,

∑r

i=1 θi (η(k)) = 1 for alltime k. Here, if the state is directly unmeasurable, it is impossible fora fuzzy control law to share the same premise variables as those of thesystem (2) since the premise variables are generally associated withthe state of the system (2). Thus, we consider a set of fuzzy controlIF–THEN rules such as the following.

Control Rule i: IF η1 (k) is Fi1 and · · · and ηs (k) is Fis , THEN

xk+1 = Ai xk + B2 i uk + L(Θk )(yk − yk ), yk = C2 i xk

uk = F (Θk−1 , Θk )xk , for i = 1, 2, . . . , r (5)

where η1 (k) · · · ηs (k) denote the premise variables of the controller,xk ∈ Rn x and yk ∈ Rn y denote the estimated state and measurementoutput, respectively, and the notation Θk (or Θk−1 ) denotes a vectorof time-varying fuzzy weighting functions θi (η(k)) [or θi (η(k − 1))]with respect to (4). Here, L(Θk ) and F (Θk−1 , Θk ) are not assumed tobe affine in time-varying θi (η(k)) and θi (η(k − 1)).

Remark 1: Differently from [21], the proposed observer-based fuzzycontroller is designed to be dependent not only on the current time esti-mated fuzzy weighting function vector Θk but also on the one-step-pastestimated fuzzy weighting function vector Θk−1 for time k, which isafterward implemented by nonparallel distributed compensation (non-PDC) scheme.

Remark 2: A reason for employing both Θk−1 and Θk in (5) is touse the transition information existing between Θk−1 and Θk as well asthe instant information Θk when performing the control action (referto [11] and [18]).

As in (2), the fuzzy control system (5) can be represented as follows:

xk+1 = A(Θk )xk + B2 (Θk )uk + L(Θk ) (yk − yk )

yk = C2 (Θk )xk

uk = F (Θk−1 , Θk )xk (6)

where A(Θk )=∑r

i=1 θi (η(k))Ai , B2 (Θk )=∑r

i=1 θi (η(k))B2 i ,

and C2 (Θk )=∑r

i=1 θi (η(k))C2 i . Hence, the closed-loop fuzzy sys-tem under the control law (6) is given by (7), as shown at the bottomof this page.

xk+1

xk+1

zk

=

A(Θk ) + B2(Θk )F (Θk−1 , Θk ) − L(Θk )C2(Θk ) L(Θk )C2(Θk ) L(Θk )D21(Θk )

B2(Θk )F (Θk−1 , Θk ) A(Θk ) B1(Θk )

D12(Θk )F (Θk−1 , Θk ) C1(Θk ) D11(Θk )

xk

xk

wk

. (7)



Now, let us consider an appropriate similarity transform matrixyielding

TS

[xk

xk

]=

[I 0

−I I

][xk

xk

]=

[xk

ek

](8)

where ek denotes the estimation error ek = xk − xk . Then, the equiv-alent representation of (7) becomes[

xk+1

zk

]=

[A(Θk , Θk , Θk−1 ) B(Θk , Θk )

C(Θk , Θk , Θk−1 ) D11 (Θk )

][xk

wk

](9)

where xTk = [xT

k eTk ] denotes the augmented state, and the element

matrices are given by

A(Θk , Θk , Θk−1 ) =

[A11 L(Θk )C2 (Θk )

A21 A22

]B(Θk , Θk ) =

[L(Θk )D21 (Θk )

B1 (Θk ) − L(Θk )D21 (Θk )

]C(Θk , Θk , Θk−1 ) = [ C11 C1 (Θk ) ]

in which A11= A(Θk ) + B2 (Θk )F (Θk−1 , Θk ) + L(Θk ) (C2 (Θk ) −C2 (Θk )), A21 = (A(Θk ) − A(Θk )) + (B2 (Θk )−B2 (Θk ))F (Θk−1 ,Θk )−L(Θk )(C2 (Θk )−C2 (Θk )), A22 = A(Θk )−L(Θk )C2 (Θk ),and C11 = C1 (Θk ) + D12 (Θk )F (Θk−1 , Θk ).

Remark 3: From (9), we can observe that if wk ≡ 0 and Θk ≡ Θk ,then the closed-loop system becomes xk+1 = A(Θk , Θk−1 )xk , where

A(Θk , Θk−1 ) =

[AF (Θk−1 , Θk ) L(Θk )C2 (Θk )

0 AL (Θk )

](10)

where AF (Θk−1 , Θk )= A(Θk ) + B2 (Θk )F (Θk−1 , Θk ) and

AL (Θk )= A(Θk ) − L(Θk )C2 (Θk ). In this sense, Ma et al. [5]

proved, based on the vector comparison principle, that the design ofthe controller can be carried out independently of the design of theobserver, and Sala et al. [27] also mentioned on it.

Finally, note that θi (η(k)) is generally subject to the follow-ing constraints for all time k: C1) 0 ≤ θi (η(k)) ≤ 1 ∀i ∈ 1, . . . , r;C2)

∑r

i=1 θi (η(k)) = 1; C3) |θi (ηk ) − θi (ηk )| ≤ δi < 1, from whichwe shall derive a relaxed H∞ stabilization condition based on LMIs.

III. H∞ STABILIZATION BASED ON PLMI CONDITIONS

In general, the H∞ norm boundedness of the transfer function fromw to z, Gz w , is denoted as ‖Gz w ‖∞ < γ, i.e., ‖zk ‖2 < γ‖wk ‖2 for allnonzero wk ∈ L2+ , where the upper bound γ represents the disturbancerejection capability [3]. This short paper focuses on how to design anobserver-based H∞ output-feedback controller (6) that achieves themaximal disturbance rejection while satisfying the following design

specification (DS): the closed-loop system is asymptotically stable forall admissible grades Θk , Θk , and Θk−1 when wk ≡ 0, i.e., this shortpaper aims at solving the following optimization problem:

min γ subject to ‖zk ‖2 < γ‖wk ‖2 and DS

for all nonzero wk ∈ L2+ . (11)

Consider a Lyapunov function V (xk ) of the following form:

V (xk ) = xTk P (Θk−1 )xk , P (Θk−1 ) > 0 (12)

whose forward difference along the closed-loop system trajectories

∆V (xk )= V (xk+1 ) − V (xk ) is given by

∆V (xk ) = xTk+1P (Θk )xk+1 − xT

k P (Θk−1 )xk . (13)

Then, the following two statements are equivalent (see [11] and [16]).1) The closed-loop system (9) is stable with the H∞ performance

γ.2) There exist P (Θk−1 ) and P (Θk ) such that

0 <

P (Θk−1 ) 0 (∗) (∗)0 γ2I (∗) (∗)

−−−−−−−−−−−−−−−−−−−−−A(Θk , Θk , Θk−1 ) B(Θk , Θk ) P −1 (Θk ) 0

C(Θk , Θk , Θk−1 ) D11 (Θk ) 0 I

. (14)

As is well known, (14) is directly derived by 0 < γ2wTk wk − zT

k zk −∆V (xk ).

Remark 4: To enhance the causality between the controllerF (Θk−1 , Θk ) and the difference ∆V (xk ), we set the Lyapunov func-tion (12) to be dependent not on Θk but on Θk−1 (refer to [11] and [18]).

Let us partition matrices P (Θk−i ) and P (Θk−i )= P −1 (Θk−i ), for

i = 0, 1, in the form

P (Θk−i ) =

[X(Θk−i ) 0

0 Z(Θk−i )

]P (Θk−i ) =

[X(Θk−i ) 0

0 Z(Θk−i )

](15)

where all element matrices are positive definite, X(Θk−i ) =X−1 (Θk−i ), and Z(Θk−i ) = Z−1 (Θk−i ). And define the matrices

Q(Θk−1 )= diag(Q(Θk−1 ), I) and R

= diag(I, R), where Q(Θk−1 ) ∈

Rn x ×n x and R ∈ Rn x ×n x are positive definite. Then, pre- and post-multiplying T T and T

= diag(Q(Θk−1 ), I, R, I) > 0 on the right-hand side of (14) yields (16), as shown at the bottom of this page, whichis guaranteed by (17), as shown at the bottom of this page, because, for

0 <

QT (Θk−1)X(Θk−1)Q(Θk−1) ⊕ Z(Θk−1) 0 (∗) (∗)

0 γ2I (∗) (∗)RT A(Θk , Θk , Θk−1)Q(Θk−1) RT B(Θk , Θk ) X(Θk ) ⊕ RT Z(Θk )R 0

C(Θk , Θk , Θk−1)Q(Θk−1) D11(Θk ) 0 I

(16)

0 <

(QT (Θk−1) + Q(Θk−1) − X(Θk−1)) ⊕ Z(Θk−1) 0 (∗) (∗)

0 γ2I (∗) (∗)RT A(Θk , Θk , Θk−1)Q(Θk−1) RT B(Θk , Θk ) X(Θk ) ⊕ (RT + R − Z(Θk )) 0

C(Θk , Θk , Θk−1)Q(Θk−1) D11(Θk ) 0 I

(17)



any matrix Q, it always holds that QT XQ ≥ QT + Q−X−1 ,X > 0.Here, note that (17) naturally implies Q(Θk−1 ) > 0 and R > 0 underthe following conditions:

0 < X(Θk−1 ) and 0 < Z(Θk ). (18)

Based on (9), the element matrices in (17) are formulated as follows:

RT A(·)Q(Θk−1 ) =

[(1, 1)a L(Θk )C2 (Θk )

(2, 1)a (2, 2)a

](1, 1)a =

(A(Θk ) + L(Θk )

(C2 (Θk ) − C2 (Θk )

))× Q(Θk−1 ) + B2 (Θk )F (Θk−1 , Θk )

(2, 1)a = RT(A(Θk ) − A(Θk )

)Q(Θk−1 )

+ RT(B2 (Θk ) − B2 (Θk )

)F (Θk−1 , Θk )

− L(Θk )(C2 (Θk ) − C2 (Θk )

)Q(Θk−1 )

(2, 2)a = RT A(Θk ) − L(Θk )C2 (Θk )

RT B(Θk , Θk ) =

[L(Θk )D21 (Θk )

(2, 1)b

](2, 1)b = RT B1 (Θk ) − L(Θk )D21 (Θk )

C(·)Q(Θk−1 ) = [ (1, 1)c C1 (Θk ) ]

(1, 1)c = C1 (Θk )Q(Θk−1 ) + D12 (Θk )F (Θk−1 , Θk )

where F (Θk−1 , Θk )= F (Θk−1 , Θk )Q(Θk−1 ) and L(Θk )

=

RT L(Θk ). In spite of employing P (Θk ) and P (Θk ) of the form (15),the observer-based H∞ stabilization problem (17) is still nonconvex.Thus, similarly to [12], [13], [16], and [21], we shall propose a propertwo-step procedure of converting the stabilization problem (17) intotwo PLMI problems, based on the fact that (17) implies

0 <

Z(Θk−1 ) 0 (∗) (∗)

0 γ2I (∗) (∗)(2, 2)a (2, 1)b RT + R − Z(Θk ) 0

C1 (Θk ) D11 (Θk ) 0 I

. (19)

Proposition 1: Following is the two-step procedure.Step 1: For a prescribed γ > 0, solve (19) and 0 < Z(Θk ). Then,

from its solutions, we can reconstruct the observer gain L(Θk ) asfollows: L(Θk ) = R−T L(Θk ).

Step 2: Substitute the obtained R and L(Θk ) into (17), and solvethe following optimization problem:

γC = minX (Θ k ) ,X (Θ k −1 ) ,Z (Θ k ) ,Z (Θ k −1 ) ,Q (Θ k −1 ) , F (Θ k −1 ,Θ k )

γ

subject to (17) and (18).

Then, from its solutions, we can reconstruct the control gainF (Θk−1 , Θk ) as follows:

F (Θk−1 , Θk ) = F (Θk−1 , Θk )Q−1 (Θk−1 ). (20)

Remark 5: Solving the PLMIs in Proposition 1 is equivalent tosolving an infinite number of LMIs. Thus, to numerically solve theoptimization problems, one needs to find a finite number of solvableLMIs from the PLMIs by selecting an appropriate structure for eachfuzzy weighting-dependent variable, i.e., X(Θk ), X(Θk−1 ), Z(Θk ),Z(Θk−1 ), Q(Θk−1 ), L(Θk ), and F (Θk−1 , Θk ).

Remark 6: To reduce the number of decision variables, we can putQ(Θk−1 ) = X(Θk−1 ) in (17), but it leads to a more conservative resultthan when the slack variable Q(Θk−1 ) is used (refer to [17] and [26]).

IV. H∞ STABILIZATION CONDITIONS BASED ON LMIS

To obtain a finite number of LMIs from the derived PLMIs, we spe-cially select the structures of the fuzzy weighting-dependent variablesas follows:

X(Θk−i ) =r∑

=1

θ (η(k − i))X , for i = 0, 1

Z(Θk−i ) =r∑

=1

θ (η(k − i))Z , for i = 0, 1

Q(Θk−1 ) =r∑

=1

θ (η(k − 1))Q , L(Θk ) =r∑

=1

θ (η(k))L

F (Θk−1 , Θk ) = F1 (Θk−1 ) + F2 (Θk ) (21)

where F1 (Θk−1 ) =∑r

=1 θ (η(k − 1))F1 and F2 (Θk ) =∑r

=1θ (η(k))F2 , i.e., all variables are assumed to be affinely depen-dent on θ (η(k)) or θ (η(k − 1)). Henceforth, for shortening theexpression, we use the following notations: θi = θi (η(k)), θi =θi (η(k)), and θ−

i = θi (η(k − 1)).

A. Step 1: Observer Design Conditions

Based on (21), the PLMI conditions (19) and 0 < Z(Θk ) can be,respectively, rewritten as

0 <

r∑i ,j,=1

θi θj θ− Lij (22)

0 <

r∑j=1

θj Zj (23)

where

Lij =

Z 0 (∗) (∗)0 γ2I (∗) (∗)

(2, 2)a (2, 1)b RT + R − Zj 0C1 i D11 i 0 I

in which (2, 2)a = RT Ai − Lj C2 i and (2, 1)b = RT B1 i − Lj D21 i .

The following lemma presents a set of LMI conditions for step 1 inProposition 1.

Lemma 4.1: For a prescribed γ > 0, suppose that there exist Zi > 0,Li , for i ∈ [1, r], and R such that

0 < Lij ∀i, j, ∈ [1, r]. (24)

Then, the observer gain L(Θk ), for k ≥ 0, can be online recon-structed as follows: L(Θk ) = R−T

∑r

i=1 θi Li . Moreover, a min-imized H∞ performance (observer) can be obtained by γO =min γ subject to (24).

Proof: Obviously, (22) and (23) are guaranteed by (24) that naturallyimplies 0 < Zj , j ∈ [1, r].

Remark 7: In (17), the reason for using a common R is tomake L(Θk ) affinely dependent on θi , i.e., L(Θk ) =

∑r

i=1 θiLi ,

Li= R−T Li . The matrices Li will be used for the LMI formulation

of step 2.

B. Step 2: Relaxed Control Design Conditions

Using (21), we can rewrite (17) as follows: 0 <∑r

,m =1

θ θ−m Gm (Θk ), as shown (25), at the bottom of next page, whose

element matrices are given by (1, 1)a = A(Θk )Qm + R−T L(Θk )(C2 − C2 (Θk ))Qm + B2 (Θk )(F1m + F2 (Θk )), (1, 2)a = R−T L



(Θk )C2 , (2, 1)a = RT (A − A(Θk ))Qm + RT (B2 − B2 (Θk ))(F1m + F2 (Θk )) − L(Θk )(C2 − C2 (Θk ))Qm , (2, 2)a = RT A −L(Θk )C2 , (1, 1)b = R−T L(Θk )D21 , (2, 1)b = RT B1 − L(Θk )D21 , and (1, 1)c = C1Qm + D12 (F1m + F2 (Θk )). Furthermore,by the S-procedure, (17) subject to C3) can be written as follows:

0 <

r∑,m =1

θ θ−m Gm (Θk) +

r∑i=1

(θi− θi− δi)Ui+r∑

i=1

(−δi− θi+ θi)Qi

(26)where Ui > 0 and Qi > 0. Note that from

∑r

=1 θ =∑r

m =1 θ−m = 1, it follows that

∑r

i=1 (θi − θi − δi )Ui =∑r

,m =1 θ θ−m (U −

∑r

i=1 δiUi −∑r

i=1 θiUi ) ≤ 0 and∑r

i=1 (−δi − θi + θi )Qi =∑r

,m =1 θ θ−m (−Q −

∑r

i=1 δiQi +∑r

i=1 θiQi ) ≤ 0. Then, we can obtain the following conditionguaranteeing (26):

0 < Gm (Θk ) +

(U −Q −

r∑i=1

δiUi −r∑

i=1

δiQi

)

+r∑

i=1

θi (Qi − Ui ) ∀, m ∈ [1, r] (27)

which can be rewritten as follows:

0 < G(m )0 +

(U −Q −

r∑i=1

δiUi −r∑

i=1

δiQi

)

+r∑

i=1

θi

(G(m )

i + G(m )Ti + Qi − Ui

)+

r∑i=1

θ2i G

(m )ii

+r∑

i=1

(i−1∑j=1

θi θj G(m )ij +

r∑j= i+1

θi θj G(m )Tij

)(28)

where G(m )0 , G(m )

i , G(m )ii , and G(m )

ij are defined at the bottom ofnext page.

Based on (28), the following theorem presents a relaxed LMI con-dition set for step 2 of Proposition 1.

Theorem 1: Let the matrices R > 0 and Li , for i ∈ [1, r], be given,and separate Qi = Qi + QT

i and Ui = Ui + UTi . Suppose that there

exist Xi > 0, Zi > 0, Qi , Ui , Qi , F1 i , F2 i , Λi , Si , S0 , Ξij , and γ > 0such that

0 < Lm=

Γ(m )0 (∗) (∗) · · · (∗)

−−−−−−−−−−−−−−−−−−Γ(m )

1 ∆(m )1 (∗) · · · (∗)

Γ(m )2 Φ(m )

21 ∆(m )2

. . ....

......

. . .. . . (∗)

Γ(m )r Φ(m )

r 1 · · · Φ(m )r (r−1) ∆(m )

r

∀, m ∈ [1, r] (29)

0 < Xi , 0 < S0 + ST0 , 0 < Λi + ΛT

i (30)

0 < Qi + QTi , 0 < Ui + UT

i ∀i ∈ [1, r] (31)

0 < Ξij ∀i, j ∈ [1, r], j = i (32)

where Λi , Si , S0 , and Ξij are in Rn c ×n c , nc= 4nx + nw + nz

Γ(m )0

= G(m )

0 +

(U −Q −

r∑i=1

δiUi −r∑

i=1

δiQi

)−(S0 + ST

0

)∆(m )

i

= G(m )

ii +(Λi + ΛT

i

)+(Si + ST

i

)Γ(m )

i

= G(m )

i + Qi − Ui − Λi + S0 − Si

Φ(m )ij

= Gm

ij + (Si + Sj ) − Ξij − Ξj i .

Then, the closed-loop fuzzy system (9) with the constraints C1)–C3) isasymptotically stabilizable with H∞ performance γ for all admissiblegrades Θk , Θk , and Θk−1 , and the corresponding fuzzy controller gainF (Θk−1 , Θk ), for k ≥ 0, can be online reconstructed as follows:

F (Θk−1 , Θk ) =

(r∑

i=1

θ−i F1 i +

r∑i=1

θi F2 i

)(r∑

i=1

θ−i Qi

)−1

.

(33)Moreover, the minimized H∞ performance can be obtained by thefollowing optimization problem:

γC = min γ subject to (29)–(32).

Proof: By the S-procedure [3] and Finsler’s lemma [10], [19], (28)subject to C1)–C2) can be written as follows:

0 < Gm (Θk ) +

(U −Q −

r∑i=1

δiUi −r∑

i=1

δiQi

)

+r∑

i=1

θi (Qi − Ui ) −N (Θk ) ∀, m ∈ [1, r] (34)

where 0 ≤ N (Θk ) is given by

N (Θk ) = C1 + CT1 +

r∑i=1

C2 i (Λi + ΛTi )

+r∑

i=1

r∑j=1 ,j = i

C3 ij

(Ξij + ΞT

ij

)(35)

0 = C1=

I

θ1I

...

θr I

T

I

−I...

−I

ST0

ST1

...

STr

T

I

θ1I

...

θr I

0 ≤ C2 i

= − θ2

i + θi 0 ≤ C3 ij= θi θj .

Gm (Θk )=

(QT

m + Qm − Xm

)⊕ Zm 0 (∗) (∗)

0 γ2I (∗) (∗)[(1, 1)a (1, 2)a

(2, 1)a (2, 2)a

] [(1, 1)b

(2, 1)b

]X(Θk ) ⊕ (RT + R − Z(Θk )) 0

[ (1, 1)c C1 ] D11 0 I

(25)



Here, C1 , C2 i , and C3 ij are from C1)–C2), respectively, and the mul-tiplier variables Λi , S0 , Si , and Ξij are in Rn c ×n c and should satisfythe conditions 0 < Λi + ΛT

i and 0 < Ξij + ΞTj i in (30) and (32). With

some algebraic manipulations, (35) can be represented as follows:

N (Θk ) = N0 +r∑

i=1

θi

(Ni + NT

i

)+

r∑i=1

θ2i Nii

+r∑

i=1

(i−1∑j=1

θi θj Nij +r∑

j= i+1

θi θj NTij

)(36)

where N0 = S0 + ST0 , Ni = Λi − S0 + Si , Nii = −

(Λi + ΛT

i

)−(

Si + STi

), and Nij = − (Si + Sj ) + (Ξij + Ξj i ). Hence, the H∞

stabilization condition (34) becomes, for all , m ∈ [1, r]

0 < Γ(m )0 +

r∑i=1

θi

(Γ(m )

i + Γ(m )Ti

)+

r∑i=1

θ2i ∆(m )

i

+r∑

i=1

(i−1∑j=1

θi θj Φ(m )ij +

r∑j= i+1

θi θj Φ(m )Tij

)(37)

where Γ(m )0 , Γ(m )

i , ∆(m )i , and Φ(m )

ij are defined in Theorem 1. Con-sequently, (37) boils down to

0 < [ I | θ1I · · · θr I ]Lm [ I | θ1I · · · θr I ]T ∀, m ∈ [1, r].

Meanwhile, if (29) holds, then (34) also holds, which im-plies 0 < Gm (Θk ) because 0 ≤ N (Θk ) and 0 ≥ (U −Q −∑r

i=1 δiUi −∑r

i=1 δiQi ) +∑r

i=1 θi (Qi − Ui ). Thus, from (25), it

follows that 0 < Zm , i.e., 0 < Z(Θk ). And, the condition 0 < Xi in(30) assures 0 < X(Θk−1 ).

Remark 8: In order to reduce the number of decision vari-ables, we can follow the constraint-elimination method in [11]:S0 = −Λ, Si = Λ, i ∈ [1, r], and Ξij = 0, i, j ∈ [1, r], j = i. How-ever, the method [11] produces a more conservative result thanTheorem 1.

Corollary 1: A method to reduce the computational complexity ofTheorem 1 is to set Qi = Xi together with Remark 8, i.e., to useX(Θk−1 ) instead of Q(Θk−1 ).

Now, we shall consider some possible relaxation forms for the H∞stabilization condition (27), based on, as shown (G(m )

ij ), at the bottomof next page, where (1, 1)a = AiQm + R−T Li (C2 − C2j )Qm +B2 i (F1m + F2j ), (1, 2)a = R−T LiC2 , (2, 1)a = RT (A − Ai )Qm + RT (B2 − B2 i )(F1m + F2j )− Li (C2 −C2j )Qm , (2, 2)a =

G(m )0

=

QTm + Qm − Xm 0 0 (∗) (∗) (∗)

0 Zm 0 (∗) (∗) (∗)0 0 γ2I (∗) (∗) (∗)0 0 0 0 0 0

RT AQm + RT B2 F1m RT A RT B1 0 RT + R 0

C1Qm + D12 F1m C1 D11 0 0 I

G(m )i

=

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

AiQm + R−T LiC2Qm + B2i F1m R−T LiC2 R−T LiD2112Xi 0 0

−RT AiQm − RT B2i F1m + RT B2 F2i − LiC2Qm −LiC2 −LiD21 0 −12Zi 0

D12 F2i 0 0 0 0 0

G(m )ii

=

0 0 0 (∗) (∗) 0

0 0 0 0 0 0

0 0 0 0 0 0

−R−T LiC2iQm + B2i F2i 0 0 0 0 0

−RT B2i F2i + LiC2iQm 0 0 0 0 0

0 0 0 0 0 0

G(m )ij

=

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

−R−T LiC2jQm + B2i F2j − R−T LjC2iQm + B2j F2i 0 0 0 0 0

−RT B2i F2j + LiC2jQm − RT B2j F2i + LjC2iQm 0 0 0 0 0

0 0 0 0 0 0

.



RT A − LiC2 , (1, 1)b = R−T LiD21 , (2, 1)b = RT B1 − LiD21 ,and (1, 1)c = C1Qm + D12 (F1m + F2 i ).

Relaxation A [6]: For all , m, i ∈ [1, r], j ∈ (i, r], min γ subjectto G(m )

ii > 0, G(m )ij + G(m )

j i > 0.Relaxation B [8]: For all , m, i ∈ [1, r], j ∈ (i, r], min γ subject

to 2G(m )ii ≥ G(m )

ii ,G(m )ij + G(m )

j i ≥ G(m )ij , 0 < [G(m )

ij ]r×r , where

G(m )j i = G(m )

ij and G(m )Tij = G(m )

ij , i ≤ j.Relaxation C [13]: For all , m, i ∈ [1, r], j ∈ (i, r], min γ

subject to G(m )ii ≥ G(m )

ii ,G(m )ij + G(m )

j i ≥ G(m )ij + G(m )

j i , 0 <

[G(m )ij ]r×r , where G(m )

j i = G(m )T

ij , i ≤ j.Relaxation D [29]: For all , m, i, j ∈ [1, r], min γ subject to

G(m )ii > 0, 1/(r − 1)G(m )

ii + (1/2)(G(m )ij + G(m )

j i ) > 0, i = j.Relaxation E [32]: For all , m, h, i ∈ [1, r], j ∈ (i, r], min γ

subject to Xi > 0, Tijh ≥ 0, Mij = MTij

0 <

G(m )

11 − C1h G(m )12h · · · G(m )

1r h

G(m )21h G(m )

22 − C2h · · · G(m )2r h

......

. . ....

G(m )r 1h G(m )

r 2h · · · G(m )22 − C2h

Cih =

Mih , if i < h

Mhi , if i > h

0, if h = i

where

G(m )ij h =

12(G(m )

ij + G(m )j i ) − Tijh

+ (Hijh − HTijh ) +

12Wijh ,

if i < j

12(G(m )

ij + G(m )j i ) − Tj ih

+ (HTj ih − Hjih ) +

12Wjih ,

if i > j

Wijh =

Mij , if i = h or j = h

0, if i = h and j = h.

Relaxation F [33]: Assume 0 ≤ θi θj ≤ βij . For all , m, i ∈[1, r], j ∈ (i, r], min γ subject to G(m )

ii + Yii − U ≥ G(m )ii , G(m )

ij +

G(m )j i + Yij − 2U ≥ G(m )

ij + G(m )j i , and 0 < [G(m )

ij ]r×r , where

U =∑r

i=1

∑i≤j≤r

βijYij , G(m )j i = G(m )T

ij , and 0 < YTij = Yij ,

i ≤ j.Relaxation G [22]: min γ subject toG(m )

ii > G(m )iii for all i ∈ [1, r],

G(m )ii + G(m )

ij + G(m )j i ≥ G(m )

iij + G(m )ij i + G(m )T

iij for all i, j ∈[1, r], j = i, G(m )

ij + G(m )ih + G(m )

j i + G(m )j h + G(m )

h i + G(m )h j ≥

G(m )ij h + G(m )

ih j + G(m )j ih + G(m )T

ij h + G(m )Tih j + G(m )T

j ih for all i ∈[1, r − 2], j ∈ [i + 1, r − 1], h ∈ [j + 1, r], and 0 < [G(m )

ih j ]r×r forall h ∈ [1, r].

Remark 9: Apart from Relaxations A–G, various results on the re-laxation have appeared in the literature [30], [31], [35]. The first two

give asymptotic exactness of the relaxation, considering no knowledgeon the membership functions, and the last one proposes a method thatcan reduce the number of LMI conditions by taking advantage of theredundancy of descriptor systems.

Remark 10: In [34], it has been known that the conditions in [32]are more relaxed than those in [29]. And it was proved in [13] that theconditions in [13] admit a great deal more freedom than those in [6]and [8]. Particularly for r ≥ 3, more relaxed conditions were proposedin [22] than those in [13] and [32]. Meanwhile, Sala and Arino [33]mentioned that their own method does not provide better results thanin [13] for βij = 0.25, i.e., the method with normal partitions is usefulonly in fuzzy models with three or more rules.

Remark 11: The derived BMI problem can be solved by other pos-sible iterative algorithms, such as cone complementarity linearizationalgorithm or D-K iteration algorithm. However, our interest is not inmaking a comparison between our algorithm and other algorithms butin introducing a simple method of solving the BMI problem within twosteps. Of course, one can use this method as a procedure for determiningan initial point of such iterative algorithm.

Remark 12: As a simple attempt to improve the H∞ performance,we can add a small variation to R when incorporating R (obtained byLemma 4.1) in Theorem 1.

Remark 13: Similar to [23] and [28], we can also convert the PLMIcondition into a quasi-LMI condition with some tuning parameters.However, since this kind of approach yields a large matrix inequalitycondition unsuitable for our relaxation scheme, we did not employ itin our study.

V. NUMERICAL EXAMPLE

To verify the performance of the proposed observer-basedH∞ fuzzycontrol method, we consider a problem of determining the duty ratioof the pulsewidth modulation (PWM) buck converter (refer to [21]):

x1 (t) =β

L(x1 (t) − x1d )2 − 1

Lx2 (t) −

1L

Vl (t)

+1L

(Vin + VD − RM x1 (t))τ (t) +1L

x2d

x2 (t) =1C

x1 (t) −1

RC(x2 (t) − β(x1 (t) − x1d )2 )

− 1C

Il (t) y(t) = x2 (t) (38)

where x1 and x2 denote the inductor current and the voltage of thecapacitor, x1d and x2d = R x1d denote the desired state, and the dutyratio τ (t) corresponds to the control input (see Fig. 1). By the map-ping ζ1 (t) = x1 (t) − x1d and ζ2 (t) = x2 (t) − x2d , the system (38) isconverted into

ζ1 (t) =β

Lζ2

1 − 1L

ζ2 (t) −1L

Vl (t)

+1L

(Vin + VD − RM (ζ1 (t) + x1d ))τ (t)

G(m )ij

=

(QTm +Qm − Xm ) ⊕ Zm 0 (∗) (∗)

(∗) (∗) (∗) (∗)[(1, 1)a (1, 2)a

(2, 1)a (2, 2)a

] [(1, 1)b

(2, 1)b

]Xi ⊕ (RT +R−Zi) 0

[ (1, 1)c C1 ] D11 0 I

+ (U − Ui) + (Qi −Q) −r∑

i=1

δiUi −r∑

i=1

δiQi



Fig. 1. Equivalent circuit of a PWM buck converter.

TABLE ICOMPARISON OF γC ACCORDING TO VARIOUS β

ζ2 (t) =

(1C

+β

RCζ1 (t)

)ζ1 (t) −

1RC

ζ2 (t) −1C

Il (t)

z(t) = ζ2 (t) + 0.4Vl (t) y(t) = ζ2 (t) + 0.3Vl (t) (39)

where Vl (t) and Il (t) denote the leakage voltage and current. In (39),we can naturally regard the unmeasurable state ζ1 as a premise vari-able. Thus, given the assumption that ζ1 (t) ∈ [ρ1 , ρ2 ], the premisevariable ζ1 (t) can be rewritten as ζ1 (t) =

∑2i=1 θi (ζ1 (t))ρi , where

θ1 = (ζ1 (t) − ρ2 )/(ρ1 − ρ2 ), θ2 = (ρ1 − ζ1 (t))/(ρ1 − ρ2 ), and theresulting system model can be described as follows:

ζ(t) =2∑

i=1

θi (ζ1 (t))(Aiζ(t) + B1w(t) + B2 i τ (t))

where

Ai =

β

Lρi − 1

L

1C

+β

RCρi − 1

RC

B1 =

[−1/L 0

0 −1/C

]

B2 i =

[ 1L

(Vin + VD − RM (ρi + x1d ))

0

].

After all, we can obtain a discrete-time fuzzy system for (x1d , x2d ) =(2, 12) from the bilinear transformation with fast sampling time Ts =0.0003 s, where L = 0.09858 mH, C = 0.2025 mF, R = 6 Ω, VD =0.82 V, Vin = 30 V, ρ1 = −2, ρ2 = 2, RM = 0.5 Ω, and the current-dependent source parameter β is changed from 0.04 to 0.12. Based onthe discrete-time fuzzy system, we first obtain R and Li , i ∈ [1, r], bysolving the LMI-based problem in Lemma 4.1, where (β, γO ) = (0.04,1.0975), (0.043, 1.1043), (0.046, 1.1114), (0.048, 1.1163), (0.05,1.1211), (0.1, 1.264), and (0.12, 1.3345). Next, based on the obtainedLi and R, we solve the LMI-based optimization problem in Theorem 1,and then compare its result with those of Relaxations C–E (seeTable I). Of course, since the solved problem is a nonconvex opti-mization problem whose global optimal solution cannot be obtainedeasily by our two-step algorithm, we cannot make an absolute compari-son from the result shown in Table I. However, from Table I, we observethat under the same BMI solver, our relaxation method (Theorem 1)is more effective at increasing the feasibility of the H∞ stabilization

condition than Relaxations C–E. Here, the reason for not using Re-laxations F and G is that these cannot apply to the case of r < 3 (seeRemark 10).


In this short paper, we addressed the problem of designing a fuzzyobserver-based H∞ controller for discrete-time T–S fuzzy systems. Bybuilding a set of fuzzy control rules with premise variables differentfrom those of the T–S fuzzy system, we enhanced the applicability ofthe output-feedback controller.

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