less conservative criteria for fault accommodation of time-varying delay systems using adaptive...

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2010; 24:322–334Published online 7 August 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/acs.1138

Less conservative criteria for fault accommodation of time-varying delaysystems using adaptive fault diagnosis observer

Bin Jiang1,∗,†, Ke Zhang1 and Peng Shi2,3,4

1College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People’s Republic of China2Faculty of Advanced Technology, University of Glamorgan, Pontypridd CF37 1DL, U.K.3School of Engineering and Science, Victoria University, Melbourne, Vic 8001, Australia

4School of Mathematics and Statistics, University of South Australia, Mawson Lakes 5095, Australia

SUMMARY

This paper studies the problem of fault accommodation of time-varying delay systems using adaptive fault diagnosis observer.Based on the proposed fast adaptive fault estimation (FAFE) algorithm using only a measured output, a delay-dependentcriteria is first established to reduce the conservatism of the design procedure, and the FAFE algorithm can enhance theperformance of fault estimation. On the basis of fault estimation, the observer-based fault-tolerant tracking control is thendesigned to guarantee tracking performance of the closed-loop systems. Furthermore, comprehensive analysis is presentedto discuss the calculation steps using linear matrix inequality technique. Finally, simulation results of a stirred tank reactormodel are presented to illustrate the efficiency of the proposed techniques. Copyright q 2009 John Wiley & Sons, Ltd.

Received 5 October 2007; Revised 4 May 2009; Accepted 1 July 2009

KEY WORDS: fast fault estimation; fault accommodation; adaptive observer; time-varying delay

1. INTRODUCTION

The demand for increased productivity leads to morechallenging operating conditions for many modern

∗Correspondence to: Bin Jiang, College of Automation Engi-neering, Nanjing University of Aeronautics and Astronautics,Nanjing 210016, People’s Republic of China.

†E-mail: [email protected]

Contract/grant sponsor: National Natural Science Foundation ofChina; contract/grant number: 60811120024Contract/grant sponsor: Graduate Innovation Research Founda-tion of Jiangsu Province; contract/grant number: CX08B 090ZContract/grant sponsor: Doctoral Innovation Foundation ofNanjing University of Aeronautics and Astronautics;contract/grant number: BCXJ08-03Contract/grant sponsor: Engineering and Physical SciencesResearch Council of U.K.; contract/grant number: EP/F029195

engineering systems, fault detection and isolation(FDI) algorithms and their application to a widerange of industrial and commercial processes havebeen the subjects of intensive investigation over thepast two decades. In order to improve efficiency, thereliability can be achieved by fault-tolerant control(FTC), which relies on early detection of faults, usingFDI procedures. So FDI has become a popular topicand received considerable attention during the pasttwo decades. Fruitful results can be found in severalexcellent books [1–3], survey papers [4–7] and thereferences therein.

In general, fault tolerance can be achieved in twoways: (1) passively, using feedback control laws thatare robust with respect to possible system faults [8, 9]or (2) actively, using an FDI module and accommo-dation technique. Active FTC is obtained by fault

Copyright q 2009 John Wiley & Sons, Ltd.

FAULT ACCOMMODATION OF TIME-VARYING DELAY SYSTEMS USING AFDO 323

accommodation [10–14], which controls the faultysystem, or by system reconfiguration, which controlsthe healthy part of the system. Therefore, in faultaccommodation, FDI module must detect and isolatethe faults, as well as estimate them. Once a faultoccurs, fault accommodation can be activated to ensuresystem performance. FDI is the first step in faultaccommodation to monitor the system and determinethe location of the fault. Then, fault estimation isutilized to determine online the magnitude of thefault. Finally, using the obtained fault information, anadditive controller can be designed to compensate forthe fault.

Some researchers pay more attention to adaptivefault diagnosis observer (AFDO) approach and adap-tive observer techniques have found wide practicalapplication in many areas [13, 15–18]. The mainadvantage of AFDO is that the full state vector esti-mation and actuator fault estimation can be achievedsimultaneously. However, the existing main problemin the use of AFDO is performance requirements offault estimation, which is useful for fault accommo-dation, cannot be satisfied because the conventionaladaptive fault estimation (CAFE) algorithm is onlysuitable for the constant fault case. But in prac-tical situations, the faults are indeed time-varyingand sometimes may be fast time-varying. Therefore,obtaining an effective solution to overcome the abovedifficulty is necessary, which motivates us for thisstudy.

On the other hand, it is well known that time delayfrequently occurs in many practical systems, suchas manufacturing systems, telecommunication andeconomic systems. Therefore, time-delay systems havebeen a hot research area for the last few decades. Thedesign methods can be classified into two categories:delay-dependent and delay-independent methods.Delay-independent condition does not take the delaysize into consideration, and thus is often conservativefor systems with small delay. Therefore, in recent years,much attention has been drawn to the developmentof delay-dependent conditions, and many importantresults have been reported in the literature [19–22].Since delay-dependent methods make use of informa-tion on the length of delay, they are generally lessconservative than delay-independent ones when the

size of the delay is small. However, these results areonly about delay-dependent stability analysis, and fewresults are obtained to design the delay-dependentAFDO.

Moreover, to the best of our knowledge, only fewresults have appeared to investigate the delay-dependentAFDO design of time-varying delay systems, whichputs focus on fault estimation. In [13, 15, 16], faultestimation of linear and nonlinear system withouttime-delay is only considered. Some fault estimationresults of time-delay systems are obtained in [12, 23],but the delay-independent case is considered, whichleads to the unavoidable conservatism. In particular,Jiang et al. [23] studies the problem of fault estimationfor a class of time-varying delay systems, but only slowtime-varying delay systems can be considered becausethe used fault estimation algorithm is the conven-tional adaptive one. Whereas the work in [24–26]only considers fault detection of uncertain time-delaysystems, fault estimation is not included. New resultson H∞ filter design of time-varying delay systemsare obtained in recent papers [27, 28], but the studiedsystems are fault-free. Therefore, the aim of this paperis to develop a less conservative criteria not only toimprove the performance of fault estimation, but alsoto deal with fast time-varying delay systems. Thisextends earlier results of fault estimation using adaptiveobserver and stability conclusions of time-varying delaysystems.

The main contributions of this paper are threeaspects. First, a fast adaptive fault estimation (FAFE)algorithm only using the measurable output vector isproposed to enhance the performance of fault esti-mation, where the adaptive estimator composed ofa proportional term and an integral one can guar-antee both satisfactory dynamical and steady-stateperformance. Second, based on the proposed FAFEalgorithm, a delay-dependent criterion for fault esti-mation of time-varying delay systems is obtained byintroducing free matrices such that fast time-varyingdelay systems can be dealt with, which widen the appli-cation range of AFDO for time-varying delay systems.Third, detailed design steps of the proposed strategyare given based on linear matrix inequality (LMI)technique, which is convenient to calculate the designparameters.

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2010; 24:322–334DOI: 10.1002/acs

324 B. JIANG, K. ZHANG AND P. SHI

The rest of this paper is organized as follows.System description and the construction of AFDO arepresented in Section 2. In Section 3, a delay-dependentcriteria for time-varying delay systems is presentedusing the proposed FAFE algorithm, and the designsteps are discussed in detail. In Section 4, based onthe on-line obtained fault information, observer-basedfault-tolerant tracking control is designed to achievefault accommodation based on the sliding mode theory.Simulation results of a practical example are given inSection 5, followed by some concluding remarks inSection 6.

2. SYSTEMS DESCRIPTION

Consider the following linear time-varying delaysystem:

x(t)= Ax(t)+Adx(t−d(t))+Bu(t)+E f (t) (1)

y(t)=Cx(t) (2)

x(t)=�(t), t ∈[−h,0] (3)

where x(t)∈ Rn is the unmeasurable state vector,u(t)∈Rm is the measurable input vector, y(t)∈ Rp

is the measurable output vector and f (t)∈ Rr repre-sents the unknown actuator fault. A, Ad , B, E andC are known constant real matrices of appropriatedimensions, the matrix E is of full column rank, i.e.rank(E)=r , the pair (A,C) is observable. d(t) isthe time-varying delay of the system and satisfies0<d(t)�h, d(t)��, where h and � are known constantscalars. �(t) is a continuous initial function. It isassumed that the derivative of f (t) with respect to timeis norm-bounded, i.e. ‖ f (t)‖� f1, where 0� f1<∞.

The AFDO is constructed as

˙x(t) = Ax(t)+Ad x(t−d(t))+Bu(t)+E f (t)−L(y(t)

−y(t))−H(y(t−d(t))− y(t−d(t))) (4)

y(t)=Cx(t) (5)

where x(t)∈ Rn is the observer state vector, y(t)∈Rp is the observer output vector, f (t)∈ Rr is an

estimate of actuator fault f (t). Since it has beenassumed that the pair (A,C) is observable, the observergain matrix L can be selected such that (A−LC) is astable matrix.

Denote

ex (t) = x(t)−x(t), ey(t)= y(t)− y(t)

ef(t) = f (t)− f (t)(6)

then the error dynamics is described by

ex (t) = (A−LC)ex (t)+(Ad −HC)ex (t−d(t))

+Eef(t) (7)

ey(t)=Cex (t) (8)

Remark 1It is supposed that the time-varying delay d(t) is knownfor the AFDO design. If the time-varying delay cannotbe obtained accurately, robust fault diagnosis will needto be involved.

3. MAIN RESULTS

3.1. Modified adaptive fault estimation algorithm

Owing to f (t) �=0 in many practical systems, in thispaper, we consider time-varying faults rather thanconstant faults. The derivative of ef(t) with respect totime is

ef(t)= ˙f (t)− f (t) (9)

Now we are ready to present our main results in thispaper. A novel algorithm will be proposed not only toevidently improve performances of fault estimation, butalso to obtain less conservative results.

Theorem 1If there exist symmetric positive-definite matricesP,Q, Z ∈ Rn×n , M ∈ Rr×r , and matrices Y1,Y2∈ Rn×p,N1,N2∈ Rn×n , F ∈ Rr×p such that the following



conditions hold⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�11 �12 −(ATP−CTY T1 )E hN1 h(ATP−CTY T

1 )

∗ �22 −(ATd P−CTY T

2 )E hN2 h(ATd P−CTY T

2 )

∗ ∗ −2ETPE+M 0 hETP

∗ ∗ ∗ −hZ 0

∗ ∗ ∗ ∗ −hPZ−1P

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

<0 (10)

ETP=FC (11)

then the FAFE algorithm

˙f (t)=−�F(ey(t)+ey(t)) (12)

can realize estimation errors of both state and faultuniformly ultimately bounded, where

�11=PA+ATP−Y1C−CTY T1 +Q+N1+NT

1

�12= PAd −Y2C−N1+NT2

�22=−(1−�)Q−N2−NT2

Y1= PL , Y2= PH , ∗ denotes the symmetric elementsin a symmetric matrix and the symmetric positive-definite matrix �∈ Rr×r is the learning rate.

ProofConsider the following Lyapunov function

V (t)=V1(t)+V2(t)+V3(t)+V4(t) (13)

where

V1(t)=eTx (t)Pex (t)

V2(t)=∫ t

t−d(t)eTx (s)Qex (s)ds

V3(t)=∫ 0

−h

∫ t

t+�eTx (s)Zex (s)ds d�

V4(t)=eTf (t)�−1ef(t)

The derivatives of V1(t), V2(t) and V3(t) with respectto time are

V1(t) = eTx (t)Pex (t)+eTx (t)Pex (t)

= eTx (t)(P(A−LC)+(A−LC)TP)ex (t)

+2eTx (t)P(Ad −HC)ex (t−d(t))

+2eTx (t)PEef(t) (14)

V2(t) = eTx (t)Qex (t)−(1−d(t))eTx (t−d(t))

×Qex (t−d(t))

� eTx (t)Qex (t)−(1−�)eTx (t−d(t))

×Qex (t−d(t)) (15)

V3(t)=heTx (t)Zex (t)−∫ t

t−heTx (s)Zex (s)ds (16)

Using (11) and (12), one obtains

V4(t) = −2eTf (t)F(ey(t)+ey(t))−2eTf (t)�−1 f (t)

= −2eTf (t)ETP(ex (t)+ex (t))−2eTf (t)�−1 f (t)

= −2eTf (t)ETP(A−LC)ex (t)

−2eTf (t)ETP(Ad −HC)ex (t−d(t))

−2eTf (t)ETPEef(t)−2eTf (t)ETPex (t)

−2eTf (t)�−1 f (t) (17)



For a symmetric positive-definite matrix M , it is easyto show that

−2eTf (t)�−1 f (t) � eTf (t)Mef(t)

+ f T(t)�−1M−1�−1 f (t) (18)

� eTf (t)Mef(t)

+ f 21 �max(�−1M−1�−1) (19)

By Newton–Leibniz formula, we have∫ t

t−d(t)ex (s)ds=ex (t)−ex (t−d(t)) (20)

then the following equation is true for any matrices N1and N2 with appropriate dimensions

2[eTx (t)N1+eTx (t−d(t))N2]×[ex (t)−ex (t−d(t))

−∫ t

t−d(t)ex (s)ds

]=0 (21)

Calculating the derivative of V (t) and adding the leftside of (21) into it yield

V (t) � eTx (t)(P(A−LC)+(A−LC)TP)ex (t)

+2eTx (t)P(Ad −HC)ex (t−d(t))

+eTx (t)Qex (t)−(1−�)eTx (t−d(t))Qex (t−d(t))

+heTx (t)Zex (t)−∫ t

t−heTx (s)Zex (s)ds

−2eTf (t)ETP(A−LC)ex (t)

−2eTf (t)ETP(Ad −HC)ex (t−d(t))

−2eTf (t)ETPEef(t)

+eTf (t)Mef(t)+ f 21 �max(�−1M−1�−1)

+2[eTx (t)N1+eTx (t−d(t))N2]

×[ex (t)−ex (t−d(t))−

∫ t

t−d(t)ex (s)ds

]

= �T(t)[�+hN Z−1NT+h ATZ A]�(t)

−∫ t

t−d(t)[�T(t)N+ eTx (s)Z ]Z−1[NT�(t)

+Zex (s)]ds+ f 21 �max(�−1M−1�−1)

� �T(t)[�+hN Z−1NT+h ATZ A]�(t)+� (22)

where

�(t) =⎡⎢⎣

ex (t)

ex (t−d(t))

ef(t)

⎤⎥⎦

� =

⎡⎢⎢⎣

�11 �12 −(A−LC)TPE

∗ �22 −(Ad −HC)TPE

∗ ∗ −2ETPE+M

⎤⎥⎥⎦

A =

⎡⎢⎢⎣

(A−LC)T

(Ad −HC)T

ET

⎤⎥⎥⎦

N =⎡⎢⎣N1

N2

0

⎤⎥⎦ , �= f 21 �max(�

−1M−1�−1)

So if (�+hN Z−1NT+h ATZ A)<0, which isequivalent to (10) by the Schur complement, thenV (t)�−ε‖�(t)‖2+�, where ε is the minimum eigen-value of −(�+hN Z−1NT+h ATZ A). It followsthat V (t)<0 for ε‖�(t)‖2>�, which means thatthe trajectory of �(t) that is outside of the set�={�(t)|‖�(t)‖2��/ε} will converge to this set �according to the Lyapunov stability theory, so the �(t)is uniformly ultimately bounded. �

Next, the CAFE algorithm is presented with thepurpose of comparing with the FAFE one. For theCAFE algorithm, if the constant fault (i.e. f (t)=0) isonly considered, then the derivative of ef(t)with respectto time can be written as

ef(t)= ˙f (t) (23)



First, two assumptions [29] and one lemma [30] aregiven.

Assumption 1rank(CE)=r .

Assumption 2Invariant zeros of (A,E,C) lie in open left half plane.

Lemma 1Assumptions 1–2 hold if and only if there exist asymmetric positive definite matrix P and matrices F ,L such that (11) and the following condition hold:

P(A−LC)+(A−LC)TP<0 (24)

Theorem 2Under Assumptions 1–2, if there exist symmetricpositive-definite matrices P,Q∈ Rn×n and matricesY1,Y2∈ Rn×p, F ∈ Rr×p such that (11) and thefollowing condition hold[

PA+ATP−Y1C−CTY T1 +Q PAd−Y2C

∗ −(1−�)Q

]<0 (25)

then the CAFE algorithm

˙f (t)=−�Fey(t) (26)

can realize limt→∞ ex (t)=0 and limt→∞ ef(t)=0,where Y1= PL , Y2= PH , ∗ denotes the symmetricelements in a symmetric matrix and the symmetricpositive-definite matrix �∈ Rr×r is the learning rate.

ProofThe proof can refer to that of Theorem 1 and [23], thusis omitted here. �

Remark 2In [13, 15, 16, 23], it can be seen that for the CAFEalgorithm, existence conditions of the AFDO have notbeen given explicitly, so we introduce Lemma 1 to theAFDO design. Note that (24) appears as one part ofsub-block (1,1) in (25), and (11) has also been used inthe proof of Theorem 2, so it can be readily concludedthat assumptions 1–2 are necessary conditions for itsfeasibility.

Remark 3From (26), the fault estimate using the CAFEalgorithm is

f (t)=−�F∫ t

tfey(�)d� (27)

where tf denotes the instant when fault occurs. In fact,this method is only a pure integral term in essence.Although it guarantees that the estimation for constantfaults is unbiased, it fails to deal with time-varyingfaults. Also, when a large learning rate is chosen,rapid fault estimation can be achieved, but a big over-shoot is unavoidable. On the other hand, when a smalllearning rate is selected, the overshoot can be over-come at the cost of slow responses. Therefore, we aremotivated to improve the conventional adaptive algo-rithm such that time-varying faults can be consideredwith satisfactory accuracy and rapidity using AFDO.

Remark 4From (12), we can see that if f (t)=0 (i.e. f1=0) theproposed adaptive algorithm can achieve asymptoticestimation for constant faults, which indicates that thecharacteristic feature of the CAFE algorithm is reservedin the new one. Also, it is easy to show that the faultestimate using the proposed FAFE algorithm combinesproportional term with integral one

f (t)=−�F

(ey(t)+

∫ t

tfey(�)d�

)(28)

The introduction of the proportional term plays a majorrole to improve the rapidity of fault estimation. In addi-tion, fault estimate using another algorithm is proposedin [30].

f (t)=−�Fey(t) (29)

This method is a pure proportional term, which canasymptotically estimate fault to any desired degree ofaccuracy by taking � sufficiently large. Therefore, wecan draw the conclusion that the proposed FAFE algo-rithm is a combination of (26) and (29), which canenhance both constant and time-varying fault estima-tion performances including accuracy and rapidity.

Remark 5By comparing sub-block (2,2) in (10) and (25), wecan see that if ��1, there are no solutions to (25).



On the contrary, the requirement of �<1 can beremoved in (10) because of the introduced freeweighting matrices N1,N2 of appropriate dimensions,which is suitable to a wider class of time-varyingdelay systems. The main reason is that there existsthe quadratic term of ef(t) in the proof of Theorem 1such that

∫ 0−h

∫ tt+� e

Tx (s)Zex (s)ds d� can be added,

while for Theorem 2, there is no quadratic term ofef(t) in the proof and its stability is guaranteed by theLyapunov–LaSalle theory.

3.2. Calculation steps

Now, how to solve conditions in Theorem 1 is studied.Since (10) is no longer LMI condition because of theterm −hPZ−1P , we need to find a feasible algo-rithm. Usually, there are two methods to deal with theproblem. A simple method to obtain LMI condition isto set Z =�P(�>0). The other way is to solve LMIbased on nonlinear minimization problem using thecone complementary linearization algorithm [19, 31].

Define a new matrix S satisfying

PZ−1P�S (30)

then the condition (10) can be written as⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

�11 �12 −(ATP−CTY T1 )E hN1 h(ATP−CTY T

1 )

∗ �22 −(ATd P−CTY T

2 )E hN2 h(ATd P−CTY T

2 )

∗ ∗ −2ETPE+M 0 hETP

∗ ∗ ∗ −hZ 0

∗ ∗ ∗ ∗ −hS

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

<0 (31)

Equation (30) is equivalent to P−1Z P−1�S−1, whichis expressed as [

S−1 P−1

P−1 Z−1

]�0 (32)

by the Schur complement. Thus, by introducing newmatrices U , J , R, the original condition is representedas (31) and[U J

J R

]�0, U = S−1, J = P−1, R= Z−1 (33)

Using a cone complementary problem, this problemis converted to following nonlinear minimizationproblem, which is based on LMI:

Minimize trace(SU+PJ+ZR) subject to (31) and[U J

J R

]�0,

[S I

I U

]�0

[P I

I J

]�0,

[Z I

I R

]�0 (34)

Note that (10) can be used as a stop condition since itis very difficult to obtain the optimal solution such thattrace(SU+PJ+ZR)=3n. In Section 5, we will showthat the above method can provide quite satisfactoryresults.

Subsequently, the above suboptimal method can beutilized to solve (10) , but the solving difficulty is addedbecause of Equation (11). This point is not mentionedin [13, 15, 23]. Fortunately, we can transform (11) inTheorems 1 and 2 into the following optimizationproblem [30]:

Minimize subject to[I ETP−FC

(ETP−FC)T I

]�0 (35)

When we choose sufficiently small, (11) can be satis-fied by solving (35).

Finally, the integrated steps for solving the conditionsof Theorem 1 are presented as follows:

Step 1: Choose a prescribed sufficiently small scale>0.



Step 2: Find a feasible set (S0,U0, P0, J0, Z0, R0,Q0,

M0,N10,N20,Y10,Y20,F0) satisfying (31), (34) and (35).Set k=0.

Step 3: Solve the following inequations for the vari-ables S, U , P , J , Z , R, Q, M , N1, N2, Y1, Y2, F .Minimize trace(SkU+SUk+Pk J+P Jk+Zk R+Z Rk)

subject to (31), (34) and (35).Step 4: If the condition (10) holds or is not satis-

fied within a specified number of iterations, then exit.Otherwise, set k=k+1 and go to Step 3.

4. FAULT ACCOMMODATION

In this section, our task is to design observer-basedfault-tolerant tracking control such that the output y(t)tracks a reference signal in the presence of actuatorfault. Suppose that the reference signal yr (t) is gener-ated by

xr (t)= Ar xr (t)+r(t) (36)

yr (t)=Cr xr (t) (37)

where yr (t) has the same dimension of y(t). xr (t),r(t)∈ Rnr are the reference state and the boundedreference input, respectively. Ar and Cr are knownconstant real matrices of appropriate dimensions withAr Hurwitz.To achieve the control objective, the tracking control

is designed based on the sliding mode theory [32–34].Denote the tracking error

er (t)= y(t)− yr (t) (38)

and the switching surface is also defined as S(t)=er (t).

Assumption 3Matrix (CB) is invertible.

The observer-based normal tracking control isdesigned as

uh(t)=ueq(t)+us(t) (39)

where ueq(t) is the equivalent control and us(t) is theswitching control.

Since the state vectors x(t), x(t−d(t)) are unavail-able, the estimation values x(t), x(t−d(t)) are substi-tuted for x(t), x(t−d(t)). Therefore, the equivalent

control ueq(t) and the switching control us(t) areconstructed, respectively, as follows:

ueq(t) = −(CB)−1[CAx(t)+CAd x(t−d(t))− yr (t)]= −(CB)−1[CAx(t)+CAd x(t−d(t))

−Cr Ar xr (t)−Crr(t)] (40)

and

us(t)=−(CB)−1[kS(t)+

S(t)

‖S(t)‖]

(41)

where k,>0. is the switching gain.

Assumption 4rank(B,E)= rank(B).

Remark 6From Assumption 4, it is easy to show that the vectorspace spanned by the columns of E is a subset ofthe space spanned by the column vectors of B, i.e.span(E)⊆span(B), which is equivalent to the existenceof B∗ such that

(I −BB∗)E=0 (42)

The residual ey(t) can be treated as a signal tomonitor the system. Once a fault occurs, based onthe accurate and rapid fault estimates, the followingobserver-based FTC is activated to compensate for theeffect of the fault.

u(t)=uh(t)−B∗E f (t) (43)

From (43), we can see that based on the obtainedfault information f (t), the designed active fault-tolerantcontrol is utilized to compensate for effects of faultsand guarantee the system stability in the presence offaults.

Substituting (43) into (1) yields

x(t) = Ax(t)+Adx(t−d(t))−B(CB)−1

×[CAx(t)+CAd x(t−d(t))− yr (t)+kS(t)

+S(t)

‖S(t)‖]−BB∗E f (t)+E f (t)



= [A−B(CB)−1CA]x(t)+[Ad−B(CB)−1CAd ]×x(t−d(t))−B(CB)−1CAex (t)

−B(CB)−1CAdex (t−d(t))+B(CB)−1

×[yr (t)−kS(t)−

S(t)

‖S(t)‖]−E f (t)+E f (t)

−BB∗E f (t)+E f (t)

= [A−B(CB)−1CA]x(t)+[Ad−B(CB)−1CAd ]×x(t−d(t))−B(CB)−1CAex (t)

−B(CB)−1CAdex (t−d(t))+B(CB)−1

×[yr (t)−kS(t)−

S(t)

‖S(t)‖]−Eef(t) (44)

Theorem 3Under Assumptions 3–4, if the switching gain satisfies>‖v(t)‖, where v(t)=−CAex (t)−CAdex (t−d(t))−CEef(t), then the observer-based FTC (43) can realizeasymptotic output tracking of system (1).

ProofConsider the following Lyapunov function:

V (t)= ST(t)S(t) (45)

Then the derivative of V (t) with respect to time is

V (t) = 2ST(t)S(t)

= 2ST(t)(Cx(t)− yr (t))

= 2ST(t)

[−CAex (t)−CAdex (t−d(t))+ yr (t)

−kS(t)−S(t)

‖S(t)‖ −CEef(t)− yr (t)

]

= −2kST(t)S(t)−2‖S(t)‖+2ST(t)v(t)

� −2kST(t)S(t)−2‖S(t)‖+2‖S(t)‖·‖v(t)‖ (46)

where v(t)=−CAex (t)−CAdex (t−d(t))−CEef(t).Then based on the definition of the error vector

�(t) in Theorem 1, v(t) can be rewritten as v(t)=−C[A, Ad ,E]�(t). Owing to the boundedness of �(t),

it is concluded that v(t) also is uniformly ultimatelybounded. Therefore, if the switching gain satisfies>‖v(t)‖, we can get V (t)<0 such that tracking errorer (t) asymptotically converges to zero. �

5. SIMULATION RESULTS

In this section, a practical example is presented to showthe effectiveness of the theoretical results obtainedin this paper. Consider a stirred tank reactor modelborrowed from [35, 36], which has been linearizedaround the operating point.

x(t)= Ax(t)+Adx(t−d(t))+Bu(t)

y(t)=Cx(t)

where x(t)=[x1(t), x2(t)]T with x1(t) and x2(t) beingthe conversion rate of the reaction and the dimension-less temperature, respectively, the operating point is[0.1440,0.8862]T and

A=[−1.4274 0.0757

−1.4189 −0.9442

], Ad =

[0.25 0

0 0.25

]

B=[

0

0.3

], C=[0 1]

The pair (A,C) is observable. Under the assumption ofE= B, it is easy to verify that rank(CE)=1, (A,E,C)

has a stable invariant zero z=−1.4274, so the proposedmethod is applicable.

First, we consider fast time-varying delay case.It is assumed that time-varying delay is d(t)=0.5+0.2sin(6t), then we can readily get h=0.7, �=1.2.It is obvious that the method using the CAFE algo-rithm in Theorem 2 is not applied here becauseof �>1. However, by taking =1×10−9 and solving theconditions in Theorem 1, one can obtain trace(SU+P J+Z R)=6.0000 and the following solutions afteriteration:

S =[7.1723 4.4429

4.4429 7.2048

], U=

[0.2256 −0.1391

−0.1391 0.2246

]

P =[9.4572 0.0000

0.0000 10.0124

], J =

[0.1057 0.0000

0.0000 0.0999

]



Z=[1.5341 0.0012

0.0012 0.9791

], R=

[0.6518 −0.0008

−0.0008 1.0213

]

Q=[0.0159 0.0110

0.0110 1.4193

], N1=

[−2.1916 −0.0017

−0.0017 −1.3988

]

N2 =[2.1916 0.0017

0.0017 1.3988

], L=

[1.2271

0.8683

]

H =[−0.0001

0.2497

], M=0.0040, F=3.0037

Second, we consider slow time-varying delay case inorder to compare the performance of fault estimationusing the FAFE algorithm with CAFE one. Supposethat time-varying delay is d(t)=0.5+0.2sin(3t), soh=0.7, �=0.6. After iteration, we can get trace(SU+P J+Z R)=6.0000 and the solutions as follows:

S=[7.1873 4.4521

4.4521 7.1978

], U=

[0.2256 −0.1395

−0.1395 0.2252

]

P =[9.4415 0.0000

0.0000 10.0146

], J =

[0.1059 0.0000

0.0000 0.0999

]

0 1 2 3 4 5 6 7 8 9 10

0

0.05

0.1

0.15

0.2

0.25

t/ s

y r(t

) an

d y(

t)

Figure 1. Reference signal yr (t) (dotted) and actual outputy(t) (solid) under f1(t) using the FAFE algorithm.

Z=[1.5326 0.0012

0.0012 0.9791

], R=

[0.6525 −0.0008

−0.0008 1.0213

]

Q=[0.0623 0.0781

0.0781 3.1149

], N1=

[−2.1895 −0.0017

−0.0017 −1.3987

]

N2 =[2.1895 0.0017

0.0017 1.3987

], L=

[1.2300

0.8616

]

H =[−0.0012

0.2539

], M=0.0043, F=3.0044

Suppose that the reference signal is generated by

xr (t)=−5xr (t)+r(t)

yr (t)= xr (t)

where r(t)=sign(sin(0.5�)). Then the equivalentcontrol ueq(t) can be designed according to (40), whilethe switching control us(t) is constructed as

us(t)=−(CB)−1[4S(t)+0.01

S(t)

‖S(t)‖]

Owing to rank(B,E)= rank(B), B∗ can be calculatedto meet (42).

B∗ =[0 3.3333]

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t/ s

f 1(t

) an

d es

timat

e of

f 1(t

)

Figure 2. Constant fault f1(t) (dotted) and its estimatef1(t)(solid) using the FAFE algorithm.



0 1 2 3 4 5 6 7 8 9 10

0

0.05

0.1

0.15

0.2

0.25

t/ s

y r(t

) an

d y(

t)

Figure 3. Reference signal yr (t) (dotted) and actual outputy(t) (solid) under f2(t) using the FAFE algorithm.

0 1 2 3 4 5 6 7 8 9 10

0

0.1

0.2

0.3

0.4

t/ s

f 2(t

) an

d es

timat

e of

f 2(t

)

Figure 4. Time-varying fault f2(t) (dotted) and its estimatef2(t) (solid) using the FAFE algorithm.

By taking the learning rate �=50 and the samplingtime 0.01 s, we compare the two algorithms using thefollowing simulations to show that the proposed algo-rithm is superior to the conventional one. Assume thatthe constant fault f1(t) and time-varying fault f2(t)are, respectively, created as

f1(t)={0, 0�t�3

0.4, 3<t�10

0 1 2 3 4 5 6 7 8 9 10

0

0.05

0.1

0.15

0.2

0.25

t/ s

y r(t

) an

d y(

t)

Figure 5. Reference signal yr (t) (dotted) and actual outputy(t) (solid) under f1(t) using the CAFE algorithm.

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t/ s

f 1(t

) an

d es

timat

e of

f 1(t

)

Figure 6. Constant fault f1(t) (dotted) and its estimate f1(t)(sold) using the CAFE algorithm.

f2(t)={0, 0�t�3

0.3sin(5t−15), 3<t�10

For constant fault f1(t) and time-varying fault f2(t),based on the FAFE algorithm, simulation resultsfor output tracking responses and fault estimationare shown, respectively, in Figures 1–4. We can seethat the proposed FAFE algorithm can improve the



0 1 2 3 4 5 6 7 8 9 10

0

0.05

0.1

0.15

0.2

0.25

t/ s

y r(t

) an

d y(

t)

Figure 7. Reference signal yr (t) (dotted) and actual outputy(t) (solid) under f2(t) using the CAFE algorithm.

performance of fault estimation evidently, and based onthe obtained online fault estimation, the fault-tolerantcontrol can recover system performance rapidly in thepresence of faults. For the sake of comparison, simula-tion results are presented using the CAFE algorithm inFigures 5–8. Although the CAFE algorithm can realizeasymptotical estimation of constant fault, its dynamicresponse is not good. Meanwhile, the CAFE algorithmcannot achieve time-varying fault estimation withsatisfactory performance. Also, we also compare thesimulation results about FTC using the proposed FAFEalgorithm with CAFE one. From the above simulationresults, we can conclude that the performance of faultestimation using the FAFE algorithm is better than thatof the CAFE algorithm.

6. CONCLUSION

In this paper, based on the proposed FAFE algorithm,a less conservative criterion is established for faultestimation of time-varying delay systems. The FAFEalgorithm cannot only enhance the performance offault estimation for both constant and time-varyingfaults, but also get relatively less conservative results.Furthermore, we give detailed design steps of theproposed strategy in terms of LMI, which is convenientto calculate the design parameters. Then simulation

0 1 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

t/ s

f 2(t

) an

d es

timat

e of

f 2(t

)

Figure 8. Time-varying fault f2(t) (dotted) and its estimatef2(t) (solid) using the CAFE algorithm.

results show that using the FAFE algorithm, the lessconservative conditions can be obtained and the accu-racy and rapidity of fault estimation can be improvedevidently compared with the CAFE one. Extensionof the proposed method to fault accommodation ofnonlinear time-delay systems is an interesting issueand will be investigated in our future research work.

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