observer-based robust adaptive interval type-2 fuzzy tracking control of multivariable nonlinear...

ARTICLE IN PRESS

Engineering Applications of Artificial Intelligence 23 (2010) 386–399

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Engineering Applications of Artificial Intelligence

0952-19

doi:10.1

n Tel.:

E-m

journal homepage: www.elsevier.com/locate/engappai

Observer-based robust adaptive interval type-2 fuzzy tracking controlof multivariable nonlinear systems

Tsung-Chih Lin n

Department of Electronic Engineering, Feng-Chia University, Taichung, Taiwan

a r t i c l e i n f o

Article history:

Received 22 December 2008

Received in revised form

13 November 2009

Accepted 19 November 2009Available online 4 January 2010

Keywords:

Interval type-2 fuzzy set

Upper and lower membership functions

Indirect adaptive control

HN approach

MIMO

Observer

76/$ - see front matter & 2009 Elsevier Ltd. A

016/j.engappai.2009.11.007

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ail address: [email protected]

a b s t r a c t

In this paper, in order to deal with training data corrupted by noise or rule uncertainties, a new

observer-based indirect adaptive interval type-2 fuzzy controller is developed for nonlinear MIMO

systems involving external disturbances using fuzzy descriptions to model the plant. Based on the

universal approximation theorem, a fuzzy logic controller equipped with a training algorithm is

proposed such that the tracking error, because of the matching error and external disturbance, is

attenuated to an arbitrary desired level using the HN tracking design technique. Simulation results

show that the interval type-2 fuzzy logic system can handle unpredicted internal disturbances—data

uncertainties, very well, but the adaptive type-1 fuzzy controller must expend more control effort in

order to handle noisy training data. In the meantime, the adaptive fuzzy controller can perform

successful control and guarantee that the global stability of the resulting closed-loop system and the

tracking performance can be achieved.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

During the past decades, fuzzy sets and their associated fuzzylogic have supplanted conventional technologies in many scien-tific applications and engineering systems, especially in controlsystems and pattern recognition. We have also witnessed a rapidgrowth in the use of fuzzy logic in a wide variety of consumerproducts and industrial systems. Since 1985, there has been astrong growth in their use for dealing with the control of,especially nonlinear, time varying systems. For instance, fuzzycontrollers have generated a great deal of excitement in variousscientific and engineering areas, because they allow for ill-definedand complex systems rather than requiring exact mathematicalmodels (Castro, 1995; Chen et al., 1996; Rovithakis and Christo-doulou, 1994; Narendra and Parthasarathy, 1990). The mostimportant issue for fuzzy control systems is to deal with theguarantee of stability and control performance, and recently therehave been significant research efforts on the issue of stability infuzzy control systems (Wang, 1993, 1994; Spooner and Passino,1996).

An adaptive controller differs from an ordinary controller inthat the controller parameters are variable, and there is amechanism for adjusting these parameters online based onsignals in the system. Some adaptive control design techniques

ll rights reserved.

405.

for feedback linearizable nonlinear systems have already beenproposed (Sastry and Isidori, 1989; Marino and Tomei, 1993a,1993b; Slotine and Li, 1991). The central idea of the feedbacklinearization is to transform algebraically a nonlinear systemdynamics into a (fully or partial) linear one, so that linear controlmethodologies can be applied. Adaptive controllers possess theessential ability to cope with unavoidable challenges imposed byinternal uncertainties, as well as by external environmentaluncertainties. Therefore, it is an important subject to design arobust adaptive controller to deal with a nonlinear system withuncertainties. In order to deal with increasingly complex systems,to accomplish increasingly demanding design requirements andthe need to attain these requirements with less precise advancedknowledge of the system and its environment, many researcherswere compelled to look for more applicable methods.

Recently, based on the feedback linearization technique (Maand Sun, 2000), adaptive fuzzy control schemes have beenintroduced to deal with nonlinear systems. Based on the universalapproximation theorem, a globally stable adaptive fuzzy con-troller is synthesized via the combination of IF-THEN rules. TheLyapunov synthesis approach is used to tune free parameters ofthe adaptive fuzzy controller by output feedback control law andadaptive law. The adaptive fuzzy controllers are classified intotwo categories: direct and indirect adaptive fuzzy controllers(Park and Yu, 1994). Furthermore, direct adaptive fuzzy con-trollers use fuzzy logic systems (FLSs) as controllers (Wang et al.,2002a, 2002b); therefore, linguistic fuzzy control rules can bedirectly incorporated into the controllers. On the other hand,

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indirect adaptive fuzzy controllers use fuzzy logic systems tomodel the plant and construct the controllers assuming that thefuzzy logic systems represent the true plant (Leu et al., 1999a,1999b; Wang et al., 1995, 2002a, 2002b); therefore, fuzzy IF-THENrules describing the plant can be directly incorporated into theindirect adaptive fuzzy controller.

Quite often, the information that is used to construct the rulesin an FLS is uncertain. There are three possible ways of ruleuncertainty (Karnik et al., 1999; Liang and Mendel, 2000; Mendeland John (2000), Mendel et al., 2006): (i) the words that are usedin antecedents and consequents of rules can mean different thingsto different people; (ii) consequents obtained by polling a groupof experts will often be different for the same rule because theexperts will not necessarily be in agreement; and (iii) noisytraining data. Therefore, antecedent or consequent uncertaintiestranslate into uncertain antecedent or consequent membershipfunctions. Type-1 FLSs are unable to handle rule uncertaintiesdirectly, since their membership functions are type-1 fuzzy sets.On the other hand, type-2 FLSs involved in this paper whoseantecedent or consequent membership functions are type-2 fuzzysets can handle rule uncertainties. A type-2 FLS is characterizedby IF-THEN rules, but its antecedent or consequent sets are type-2. Hence, type-2 FLSs can be used when the circumstances are toouncertain to determine exact membership grades such as whentraining data are corrupted by noise.

The type-2 FLS has been successfully applied to fuzzy neuralnetworks (Wang et al., 2004), VLSI testing (Lin, 2009c), and fuzzycontroller designs (Wu and Tan, 2006; Wang et al., 2007; Castilloet al., 2006a, 2006b; Lin et al., 2005; Hsiao et al., 2008; Ho et al.,2008; Ross, 2004). An indirect adaptive interval type-2 fuzzycontrol is proposed in (Kheireddine et al. (2007). Moreover, directand indirect adaptive interval type-2 fuzzy control is developed in(Lin et al., 2009a, 2009b) for a multi-input/multi-output (MIMO)nonlinear system. In Kheireddine et al. (2007) and Lin et al.(2005), the full state must be assumed to be available formeasurement; this assumption may not hold in practice eitherbecause the state variables are not accessible for direct connec-tion or because sensing devices or transducers are not available.In this paper, our main objective is to create a technique fordesigning a state observer-based (Park and Yu, 1994) indirectadaptive interval type-2 fuzzy controller for a model-free non-linear MIMO system with external disturbances and noisytraining data such that the HN tracking performance can beachieved. The illustration example shows that the interval type-2fuzzy logic system can handle unpredicted internal disturban-ce—data uncertainties, very well. Moreover, the overall adaptivecontrol scheme not only guarantees global stability, but also thetracking error due to the matching error and external disturbanceis attenuated to an arbitrary desired level by using the HN

tracking design technique.This paper is organized as follows. First, the problem formula-

tion is presented in Section 2. A brief description of the intervaltype-2 fuzzy logic system is then introduced in Section 3. InSection 4, the observer-based indirect adaptive interval type-2fuzzy controller design for MIMO systems is given and Lyapunovstability theorem is adopted to testify the stability of thetype-2 controller system. A simulation example to demon-strate the performance of the proposed method is providedin Section 5. Section 6 gives the conclusions of the advocateddesign methodology.

2. Problem formulation

Consider a class of nonlinear multi-input/multi-output(MIMO) dynamic systems described by the following differential

equations (Tong et al., 2000, 2005):

xðniÞ

i ¼ fiðxÞþXp

j ¼ 1

gijðxÞujþdi; i¼ 1;2; . . .; p

yi ¼ xi ð1Þ

where u9½u1;u2; . . .;up�T and y9½y1; y2; . . .; yp�

T are the controlinput and output of the system, respectively, fiðxÞ and gijðxÞ fori, j=1, 2, y, p are unknown smooth but bounded nonlinearities, ni

is a positive integer, d9½d1; d2; . . .; dp�T is the external disturbance

vector, and x9½y1; y01; . . .; y

ðn1�1Þ1 ; . . .; yp; y0p. . .; y

ðnp�1Þp �T . The control

objective is to force the system output y to follow the givenreference signal yr9½yr1; yr2; . . .; yrp�

T under plant uncertaintiesand external disturbances. Let GðxÞ9½G1ðxÞ;G2ðxÞ; . . .;GpðxÞ� andGiðxÞ9½g1iðxÞ; g2iðxÞ; . . .; gpiðxÞ�

T . Hence, (1) can be rewritten as

yðn1Þ

1

yðn2Þ

2

^

yðnpÞ

p

2666664

3777775¼

f1ðxÞ

f2ðxÞ

^

fpðxÞ

266664

377775þGðxÞ

u1

u2

^

up

266664

377775þ

d1

d2

^

dp

266664

377775 ð2Þ

If fi(x) and gij(x) are known and free of external disturbance, i.e.di=0, since det GðxÞa0 and based on the equivalent approach, thecontrol law can be obtained as

u1

u2

^

up

266664

377775¼G�1ðxÞ �

f1ðxÞ

f2ðxÞ

^

fpðxÞ

266664

377775�

kT1c

e1

kT2c

e2

^

kTpc

ep

2666664

3777775þ

yðn1Þ

r1

yðn2Þ

r2

^

yðnpÞ

rp

2666664

3777775

0BBBBB@

1CCCCCA ð3Þ

where e1 ¼ y1�yr1; e2 ¼ y2�yr2

; . . .; ep ¼ yp�yrp , ei9½ei; _ei; . . .;

eðni�1Þi �T and feedback gain vector kT

ic¼ ½ki1; ki2; . . .; kini

�. Inserting

(3) into (2) and after simple manipulation, we have

eðn1Þ

1 þk1n1eðn1�1Þ

1 þ � � � þk11e1

eðn2Þ

2 þk2n2eðn2�1Þ

2 þ � � � þk21e2

^

eðnpÞ

p þkpnp eðnp�1Þp þ � � � þkp1ep

2666664

3777775¼

0

0

^

0

2666437775 ð4Þ

If all the coefficients kij are chosen such that all polynomials in(4) are Hurwitz, which implies that lim

t-1eiðtÞ ¼ 0, the main control

objective is achieved. However, fiðxÞ and gijðxÞ are unknown, theideal controller (3) cannot be implemented, and not all systemstates x can be measured. We have to design an observer toestimate the state vector x.

2.1. State observer scheme

Replacing the functions fiðxÞ, gijðxÞ and error vector ei

in (3) byestimation functions fiðxÞ; gijðxÞ, and error vector ei, the control law(3) is rewritten as

u1

u2

^

up

266664

377775¼ G�1ðxÞ �

f1ðxÞ

f2ðxÞ

^

fpðxÞ

266664

377775�

kT1c

e1

kT2c

e2

^

kTpc

ep

2666664

3777775þ

yðn1Þ

r1

yðn2Þ

r2

^

yðnpÞ

rp

2666664

3777775

0BBBBB@

1CCCCCA ð5Þ

Assumption 1. The system GðxÞ as defined above is nonsingular,i.e. G�1ðxÞ exists and is bounded for all xAUx, where Ux � Rn andn1þn2þ � � � þnp ¼ n. Also, external disturbance is bounded, i.e.di

�� rdim, where dim is the upper bound of noise di.

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T.-C. Lin / Engineering Applications of Artificial Intelligence 23 (2010) 386–399388

Applying (5) to (2) and after some simple manipulations, wecan obtain the error equation

_ei ¼ Aiei�Bik

Tic

eiþBi fiðxÞ�fiðxÞþXp

j ¼ 1

ðgijðxÞ�gijðxÞÞuj

24

35þdi

8<:

9=;

ei ¼ CTi e

ið6Þ

where ei=yri�yi,

Ai ¼

0 1 0 0 � � � 0 0

0 0 1 0 � � � 0 0

� � � � � � � � � � � � � � � � � � � � �

0 0 0 0 � � � 0 1

0 0 0 0 � � � 0 0

26666664

37777775; Bi ¼

0

0

^

0

1

26666664

37777775;

and Ci ¼

1

0

^

0

0

26666664

37777775

ð7Þ

From (6), the state vector ei

can be estimated by the followingobserver (Leu et al., 1999a, 1999b):

_ei¼ Aiei�Bik

Tic

eiþkioðei�eiÞ

ei ¼ CTi ei ð8Þ

where kio¼ ko

i1; koi2; . . .; k

oini

h iis the observer gain vector.

The observation errors are defined as

~ei ¼ ei�ei

and

~ei ¼ ei�ei

Subtracting (8) from (6), we can obtain the error dynamics

_~ei ¼ ðAi�kio

CTi Þ ~eiþBi fiðxÞ�f ðxÞþ

Xp

j ¼ 1


24

35þdi

8<:

9=;

¼Li ~eiþBi fiðxÞ�f ðxÞþXp

j ¼ 1


24

35þdi

8<:

9=;

~ei ¼ CTi~ei ð9Þ

where

Li ¼ Ai�kio

CTi ¼

�koi1 1 0 0 � � � 0 0

�koi2 0 1 0 � � � 0 0

^ ^ ^ ^ ^ ^ ^ ^

�koini

0 0 0 � � � 1 0

266664

377775

Since the (Ci, Ai) pair is observable, the observer gain vector ko

can be chosen such that the characteristic polynomial of Li isstrictly Hurwitz and we know that there exists a positive definitesymmetric n�n matrix Pi that satisfies the Lyapunov equation

LTi PiþPLi ¼�Qi ð10Þ

where Qi is an arbitrary n�n positive definite matrix.In order to treat the HN tracking problem, the approximation

of the interval type-2 fuzzy logic systems is expressed in Section3. First, we replace fiðxÞ and gijðxÞ in (10) by the interval type-2fuzzy logic systems fiðx9yfi

Þ and gijðx9ygijÞ, respectively. In the

meantime, to ensure stability of the closed-loop system, it isnecessary to add a robust compensator usi to deal with fuzzyapproximation error and external disturbances. Therefore, the

control in (5) is expressed as

u1

u2

^

up

266664

377775¼ G�1ðx9yÞ �

f1ðx9yf 1Þ

f2ðx9yf 2Þ

^

fpðx9yfpÞ

2666664

3777775�

kT1c

e1

kT2c

e2

^

kTpc

ep

2666664

3777775þ

yðn1Þ

r1

yðn2Þ

r2

^

yðnpÞ

rp

2666664

3777775�

us1

us2

^

usp

266664

377775

0BBBBB@

1CCCCCAð11Þ

where Gðx9yÞ is the fuzzy logic system model of GðxÞ. Hence,according to assumption 1, Gðx9yÞ is nonsingular too. Applying(11) to the MIMO system Eqs. (2) and after some simplemanipulations, the error dynamic Eq. (9) can be rewritten as

_~e1

_~e2

_~ep

266664

377775¼

L1e~1

L2e~2

^

Lpe~p

266664

377775þ

B1 f1ðxÞ�f1ðx9yf 1Þþ

Xp

j ¼ 1

ðgijðxÞ�gijðx9ygijÞÞuj

24

35�us1þd1

8<:

9=;

B2 f2ðxÞ�f2ðx9yf 2Þþ

Xp

j ¼ 1


24

35�us2þd2

8<:

9=;

^

Bp fpðxÞ�fpðx9y fpÞþ

Xp

j ¼ 1


24

35�uspþdp

8<:

9=;

266666666666666664

377777777777777775ð12Þ

Therefore, the error dynamic equation of the ith subsystem can bedescribed as

_~ei ¼Li ~eiþBi fiðxÞ�f ðxÞþXp

j ¼ 1


24

35�usiþdi

8<:

9=; ð13Þ

If oj denotes the sum of matching errors due to fuzzyapproximations of ujðxÞ and the external disturbance dj, thenour objective is to design the fuzzy control algorithm to achievethe following asymptotically stable tracking while oj=0.

Also, given a disturbance attenuation level r40, the followingHN tracking performance index is required (Chen et al., 1996):Z T

0

~eiT Qi ~ei dtr ~ei

Tð0ÞPi ~eið0Þþ

1

rið ~y

T

fið0Þ~yfið0ÞÞ

þXp

j ¼ 1

1

2rij

~yT

gijð0Þ~ygijð0Þþr2

Z T

0o2

i dt ð14Þ

where TA[0,N). Qi and Pi are positive definite matrices of proper

dimension, ~yfi ¼ y�fi�y

fiand ~ygij ¼ y�

gij�y

gijare the parameter

approximation error vectors, and ri and rij are design parameters.If the system starts with the initial conditions

eið0Þ ¼ 0; ~yfið0Þ ¼ 0; and ~ygijð0Þ ¼ 0, then the HN tracking per-

formance index can be written as

supo A ½0;T�

:~ei:Qi

:oi:rr ð15Þ

where : ~ei:2

Q ¼R T

0~eT

i Qi ~ei dt and :oi:2¼R T

0 o2i dt, i.e. the L2-gain

from oi to the tracking error ei must be equal to or less than r.

3. Interval type-2 fuzzy logic system

In this section, the interval type-2 fuzzy set and the inferenceof the type-2 fuzzy logic system will be presented. A type-2 fuzzyset in universal set X is denoted as ~A, which is characterized by atype-2 membership function u ~A ðxÞ in (16), which can be referredto as a secondary membership function or referred to as asecondary set, which is a type-1 fuzzy set in [0 ,1]. In (16), fx(u) is asecondary grade, which is the amplitude of a secondary member-ship function; i.e. 0r fx(u)r1. The domain of a secondarymembership function is called the primary membership of x. In(16), Jx is the primary membership of x, where uAJxD[0,1] 8xAX;

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u is a fuzzy set in [0, 1], rather than a crisp point in [0, 1].

~A ¼

ZxAX

u ~A ðxÞ=x¼

ZxAX

ZuA Jx

fxðuÞ=u

� �=x; JxD ½0;1� ð16Þ

When fx(u)=1, 8uAJxD[0,1], then the secondary MFs are intervalsets such that u ~A ðxÞ in (16) can be called an interval type-2 MF(Karnik et al., 1999; Liang and Mendel, 2000; Mendel and John,2000). Therefore, the type-2 fuzzy set can be rewritten as

~A ¼

ZxAX

u ~A ðxÞ=x¼

ZxAX

ZuA Jx

1=u

� �=x; JxD ½0;1� ð17Þ

Also, a Gaussian primary MF with uncertain mean and fixedstandard deviation having an interval type-2 secondary MF can becalled an interval type-2 Gaussian MF (16). A 2-D interval type-2Gaussian MF with an uncertain mean in [m1, m2] and a fixedstandard deviation s is shown in Fig. 1. It can be expressed as

u ~A ðxÞ ¼ exp �1

2

x�m

s

� �2� �

; mA m1;m2½ � ð18Þ

It is obvious that the type-2 fuzzy set is in a region called afootprint of uncertainty (FOU), and is bounded by an upper MFand a lower MF (Karnik et al., 1999; Liang and Mendel, 2000;Mendel and John (2000)), which are denoted as u ~A ðxÞ and u ~A

ðxÞ,

Fig. 1. Interval type-2 fuzzy set with uncertain mean.

Fig. 2. Structure of the type

respectively. Hence, (16) can be re-expressed as

~A ¼

ZxAX

ZuA ½u ~A

ðxÞ;u ~A ðxÞ�1=u

" #=x ð19Þ

A type-2 fuzzy logic system (FLS) is very similar to a type-1FLS, as shown in Fig. 2 (Mendel, 2007), the major structuredifference being that the defuzzifier block of a type-1 FLS isreplaced by the output processing block in a type-2 FLS, whichconsists of type reduction followed by defuzzification.

There are five main parts in a type-2 FLS: a fuzzifier, rule base,inference engine, type reducer and defuzzifier. A type-2 FLS is a

mapping f : Rp-R1. After fuzzification, fuzzy inference, typereduction, and defuzzification, a crisp output can be obtained.

Consider a type-2 FLS having p inputs x1AX1, y, xpAXp and oneoutput yAY. The type-2 fuzzy rule base consists of a collection ofIF-THEN rules, as in the type-1 case. We assume there are M rulesand the rule of a type-2 relation between the input spaceX1�X2�y�Xp and the output space Y can be expressed as

Rl : IF x1 is ~F l1 and � � � and xp is ~F l

p; THEN y is ~Gl l¼ 1;2; � � � ;M

ð20Þ

where ~F ljs are antecedent type-2 sets (j=1,2,y,p) and ~Gls are

consequent type-2 sets.The inference engine combines rules and gives a mapping from

input type-2 fuzzy sets to output type-2 fuzzy sets. To achieve thisprocess, we have to compute the unions and intersections of type-2 sets, as well as the compositions of type-2 relations. The outputof the inference engine block is a type-2 set. By using theextension principle of the type-1 defuzzification method, typereduction takes us from type-2 output sets of the FLS to a type-1set called the ‘‘type-reduced set’’. This set may then be defuzzifiedto obtain a single crisp value.

There are many kinds of type reduction, such as centroid,height, modified weight, and center-of-sets (Karnik et al., 1999;Liang and Mendel, 2000; Mendel and John, 2000; Mendel, 2007).The center-of-sets type reduction will be used in this paper andcan be expressed as

YcosðY1; . . .;YM ; F1; . . .; FMÞ ¼ yl; yr½ �

¼R

y1 � � �R

yM

Rf 1 � � �

Rf M 1=

PMi ¼ 1

f iyi

PMi ¼ 1

f i

ð21Þ

where Ycos is the interval set determined by two end points yl and

yr, and f iAFi ¼ f i; fi�

h. In the meantime, an interval type-2 FLS

with singleton fuzzification and meet under minimum or product

-2 fuzzy logic system.

ARTICLE IN PRESS


t-norm f i and fi

can be obtained as

f i¼ m

~F i1

ðx1Þ � � �m~F i

p

ðxpÞ ð22Þ

and

fi¼ m ~F i

1ðx1Þ � � �m ~F i

pðxpÞ ð23Þ

Also, yiAYi, Yi ¼ yil; y

ir

� �is the centroid of the type-2 interval

consequent set ~Gi, the centroid of a type-2 fuzzy set (Karnik et al.,1999; Liang and Mendel, 2000; Mendel and John, 2000). For anyvalue yAYcos, y can be expressed as

y¼

PMi ¼ 1

f iyi

PMi ¼ 1

f i

ð24Þ

where y is a monotonic increasing function with respect to yi.

Also, yl is the minimum associated only with yil , and yr the

maximum associated only with yir . Note that yl and yr depend only

on a mixture of f i or fivalues. Therefore, the left-most point yl and

the right-most point yr can be expressed as a fuzzy basis function(FBF) expansion, i.e.

yl ¼

PMi ¼ 1

f il yi

l

PMi ¼ 1

f il

¼XMi ¼ 1

yilx

il ð25Þ

and

yr ¼

PMi ¼ 1

f ir yi

r

PMi ¼ 1

f ir

¼XMi ¼ 1

yirx

ir ð26Þ

respectively, where xil ¼ f i

l =PM

i ¼ 1 f i and xir ¼ f i

r=PM

i ¼ 1 f i.

Let the FBF vectors be denoted as xl¼ x1

l ; x2l ; . . .;x

Ml

h iand

xr¼ x1

r ; x2r ; . . .; x

Mr

h i. Let yT

l¼ y1

l ; y2l ; . . .; y

Ml

� �and

yTr¼ y1

r ; y2r ; . . .; y

Mr

� �. Then (25) and (26) can be rewritten as

yl ¼

PMi ¼ 1

f il yi

l

PMi ¼ 1

f il

¼XMi ¼ 1

yilx

il ¼ yT

l

xl

ð27Þ

and

yr ¼

PMi ¼ 1

f ir yi

r

PMi ¼ 1

f ir

¼XMi ¼ 1

yirx

ir ¼ yT

r

xr

ð28Þ

For illustrative purposes, we briefly provide the computationprocedure for yr. Without loss of generality, assume the yi

r arearranged in ascending order, i.e. y1

r ry2r r � � �ryM

r .[Step 1]: Compute yr in (28) by initially setting f i

r ¼ ðfiþ f iÞ=2

for i=1, 2, y, M, where f i and fi

have been pre-computed by (22),(23) and let y0r9yr .

[Step 2]: Find Rð1rRrM�1Þ such that yRr ry0r ryRþ1

r .[Step 3]: Compute yr in (28) with f 1

r ¼ f i for irR and f 1r ¼ f

ifor

i4R and let y00r9yr .[Step 4]: If y00r ay0r , then go to step 5. If y00r ¼ y0r , then stop and

set yr9y00r .[Step 5]: Set y0r equal to y00r and return to step 2.

The point to separate two sides by number R can be decidedfrom the above algorithm, one side using lower firing strengths f i

and another side using upper firing strengths fi. Therefore, the yr

in (28) can be rewritten as

yr ¼

PRi ¼ 1

f iyirþ

PMi ¼ Rþ1

fiyi

r

PRi ¼ 1

f iþ

PMi ¼ Rþ1

fi¼XR

i ¼ 1

qi

r

yirþ

XMi ¼ Rþ1

q

i

r

yir ¼ ½Q

r

Q r�y

r

yr

" #¼ xT

r

Yr

ð29Þ

where qir¼ f i=Dr , qi

r ¼ fi=Dr

, and Dr ¼ ðPR

i ¼ 1 f iþPM

i ¼ Rþ1 fiÞ. In the

meantime, we have Qr¼ ½q1

r; q2

r; . . .; qR

r�, Q

r¼ ½q1

r ; q2r ; . . .;q

Rr �,

xT

r¼ ½Q

rQ

r�, and YT

r¼ ½y

ryr�.

The procedure to compute yl is similar to that for yr. In step 2, L

is to be determined (1rLrM�1), such that yLl ry0lryLþ1

l . In step

3, let f il ¼ f

ifor irL and f i

l ¼ f i for i4L; yl in (27) can also be

rewritten as

yl ¼

XL

i ¼ 1

f iyilþ

PMi ¼ Lþ1

fiyi

l

PLi ¼ 1

f iþ

PMi ¼ Lþ1

fi¼XL

i ¼ 1

qi

l

yilþ

XMi ¼ Lþ1

qily

il ¼ ½Q

l

Ql�

yl

yl

" #¼ xT

l

Yl

ð30Þ

where qil¼ f i=Dl, qi

l ¼ fi=Dl

, and Dl ¼ ðPL

i ¼ 1 f iþPM

i ¼ Lþ1 fiÞ. In the

meantime, we have Ql¼ ½q1

l; q2

l; . . .; qR

l�, Q

l¼ ½q1

l ; q2l ; . . .;q

Rl �,

xT

l¼ ½Q

lQ

l�, and YT

l¼ ½y

lyl�.

The defuzzified crisp value from an interval type-2 FLS isobtained as

yðxÞ ¼ylþyr

2¼

1

2ðxT

r

Yr

þxT

l

Yl

Þ ¼1

2xT

r

xT

l

" #Y

rY

l

� �¼ xTY ð31Þ

where ð1=2Þ½xT

rxT

l� ¼ xT and ½YT

rYT

l� ¼YT .

Remark 1. (Mendel et al., 2006) : The summation sign is simplyshorthand for lots of + signs. The + indicates the union betweenmembers of a set, whereas the union sign represents the union ofthe sets themselves. Hence, by using both the summation andunion signs, we are able to distinguish between the union setsversus the union of members within a set.

4. Adaptive controller design using interval type-2 fuzzy logicsystem

To begin with, we replace fiðx9yfiÞ and gijðx9ygij

Þ in specific fuzzylogic systems as (31), i.e.

fiðx9yfiÞ ¼ yT

fixðxÞ ð32Þ

and

gijðx9ygijÞ ¼ yT

gijxðxÞ ð33Þ

In order to adjust the parameters in the fuzzy logic systems,we have to derive adaptive laws. Hence, the optimal parameterestimations y�

fiand y�

gijare defined as

y�fi ¼ arg minyfi AOfisup

x AOx

9fiðxÞ�fiðx9yfiÞ9

24

35 ð34Þ

y�fi ¼ arg minygij AOgijsup

x AOx

9gijðxÞ�gijðx9ygijÞ9

24

35 ð35Þ

ARTICLE IN PRESS


where Ofi, Ogij, and Ox are compact sets of suitable bounds on

yfi, y

gij, and x, respectively, and they are defined as Ofi ¼

yfi9 y

fi

�� rMfi

n o, Ogij ¼ y

gij9 y

gij

�� rMgij

n oand Ox ¼ x9 x

�� rMx

n o,

where Mfi, Mgij, and Mx are positive constants.

Define the minimum approximation errors as

oi ¼ fiðxÞ�fiðx9y�

fÞþXp

j ¼ 1

ðgijðxÞ�gijðx9y�

gijÞÞuj

24

35þdi ð36Þ

The error dynamics (13) can be expressed as

_~ei ¼Li ~eiþBi ðfiðx9y�

fiÞ�fiðx9yfi

ÞÞþXp

j ¼ 1

ðgijðx9y�

gijÞ�gijðx9ygij

ÞÞuj

24

35

�BiusiþBioi ð37Þ

Following the preceding consideration, we have the followingtheorem.

Theorem 1. Consider the nonlinear MIMO system(2)with con-

trol(11). If the robust compensator usiand the fuzzy-based adaptive

law are chosen as

usi ¼1

liBT

i Pi ~ei ð38Þ

_yfi ¼ ri ~eTi PiBix

TðxÞ ð39Þ

and

_ygij ¼ rij ~eTi PiBix

TðxÞuj ð40Þ

where li40, ri40, rij40, andPi ¼ PiT 40are solutions of the

following Ricatti-like equation:

PiAiþATi PiþQi�PiBi

2

li�

1

r2

BT

i Pi ¼ 0 ð41Þ

whereQi ¼QTi 40 is a prescribed weighting matrix. Therefore, the

HNtracking performance can be achieved for a prescribed attenua-

tion level r and all the variables of the closed-loop system are

bounded.

The proof of theorem 1 is given in the Appendix.

Remark 2. If oi is not equal to zero, we can expect oi to be smallbased on the universal approximation theorem. From (34), (35), and

Fig. 3. Overall scheme of the adaptive in

the constraint sets Ofi and Ogij of the optimal parameters y�fi

and y�gij

,

respectively, if we can constrain yfi

and ygij

within the sets, then ui in

(6) and usi in (38) will be bounded due to the fact that, in this case,

fiðx9yfiÞ and gijðx9ygij

Þ are bounded. Obviously, the adaptive laws in

(39), (40) are unable to guarantee that yfiAOfi and y

gijAOgij.

Therefore, all the adaptive laws have to be modified by using theparameter projection algorithm (Castro, 1995; Wang, 1993; Parkand Yu, 1994), such that the parameter vectors will remain insidethe constraints. The modified adaptive laws are given as follows:

�

ter

Use the following adaptive law to adjust the parameter vector yfi:

_yfi ¼

r1xðxÞBTi Pi ~ei if ð9y

fi9oMfiÞ or ð9y

fi9¼Mfi

and ~eTi PiBix

TðxÞy

fiZ0Þ

Projfr1xðxÞBTi~eig if ð9y

fi9¼Mfi and ~eT

i PiBixðxÞyfio0Þ

8>><>>:

ð42Þ

where the projection operator Proj{n} is defined as

Proj r1xðxÞBTi P ~eig ¼ r1xðxÞBT

i Pi ~ei�r1 ~eTi PiBi

yfiyT

fixTðxÞ

9yfi92

8<: ð43Þ

Use the following adaptive law to adjust the parameter vector
� yg:
_ygij ¼

rijxðxÞBTi Pi ~eiuj if ð9y

gij9oMgijÞ or ð9y

gij9¼Mgij

and ~eTi PiBix

TðxÞy

gijujZ0Þ

ProjfrijxðxÞBTi Pi ~eiujg if ð9y

gij9¼Mgij and ~eT

i PiBixTðxÞy

gijujo0Þ

8>>><>>>:

ð44Þ

where the projection operator Proj{n} is defined as

Proj rijxðxÞBTi Pi ~eiuj

n o¼ rijxðxÞBi ~eiuj�rij ~e

Ti PiBi

ygijyT

gijxTðxÞuj

9ygij92

ð45Þ

The overall multivariable robust adaptive interval type-2 fuzzytracking control scheme is shown in Fig. 3.

To summarize the above analysis, the design algorithm formultivariable robust adaptive interval type-2 fuzzy trackingcontrol is proposed as follows:

[Step 1]: Specify the feedback gain vector and observer gainvector k

icand k

io, respectively, such that the characteristic

matrices Li are strictly Hurwitz matrices.

val type-2 fuzzy control system.

ARTICLE IN PRESS


[Step 2]: Solve the state Eq. (8) to obtain estimate state vectorxi ¼ y

ri�ei.

[Step 3]: Specify the parameters Mfi, Mgij, and Mx based on thepractical constraints. Also, specify the parameters li, ri, rij, r, and apositive definite matrix Qi to solve the Lyapunov Eq. (41).

[Step 4]: Define the membership function mFliðxÞ for i=1, 2, y,

M and compute the fuzzy basis functions xðxÞ. Using (31) and from(32) and (33), the fuzzy logic control systems for the plant can beconstructed.

[Step 5]: Using (42)–(45), the adaptive laws, adjust theparameter vectors y

fiand y

gij.

[Step 6]: Use (38) to compute the robust compensator usi. Thenthe control law can be obtained.

1

5. Simulation example

In this section, we will apply our indirect adaptive type-1 fuzzycontroller and interval type-2 fuzzy controller to the mass–spring–damper system described in Fig. 4. The diagram shows massesM1 and M2 being subjected to forces u1 and u2, respectively.If the springs are pulled right to further distances y1 and y2,the restoring forces will be the new tensions in the springs. K1

and K2 are the moduli of elasticity. It can be shown experi-mentally that in such cases the resistance to motion isdirectly proportional to the velocity of the mass. So there willbe damping forces fB1

and fB2; B1 and B2 are the damping factors.

The equations of motion for this mechanical system can beexpressed as

M1 €y1 ¼ u1�fK1ðxÞ�fB1

ðxÞþ fK2ðxÞþ fB2

ðxÞ�fC1ðxÞþ fC2

ðxÞþd1 ð46Þ

M2 €y2 ¼ u2�fK2ðxÞ�fB2

ðxÞ�fC2ðxÞþd2 ð47Þ

where x=[x1, x2, x3, x4]T, fK1ðxÞ and fK2

ðxÞ denote the springforces due to K1 and K2, respectively, fB1

ðxÞ and fB2ðxÞ are

the friction forces, and fC1ðxÞ and fC2

ðxÞ are the coulombfriction forces. Let x1 ¼ y1; x2 ¼ _x1 ¼ _y1; x3 ¼ y2; x4 ¼ _x3 ¼ _y2;

the state space representation of the system can be

fk1 fk2

K2K1

B1B2

fB1 fc1

fB1 fc2

M1 M2u1

u2

y1 y2

Fig. 4. Mass–spring–damper system.

Table 1Interval type-2 and type-1 fuzzy membership functions.

Variance (s) Mean (m)

m1 m2 m (type-1)

mF1iðxiÞ 1 �1.32 �1.08 �1.2

mF2iðxiÞ 1/8 �0.55 �0.45 �0.5

mF3iðxiÞ 1/8 �0.1 0.1 0

expressed as

_x1 ¼ x2

_x2 ¼1

M1ð�fK1

ðxÞþ fK2ðxÞ�fB1

ðxÞþ fB2ðxÞ�fC1

ðxÞþ fC2ðxÞÞþ

1

M1ðu1þd1Þ

ð48Þ_x3 ¼ x4

_x4 ¼1

M2ð�fK2

ðxÞ�fB2ðxÞ�fC2

ðxÞÞþ1

M2ðu2þd2Þ ð49Þ

This mass–spring–damper system suffers from plant uncer-tainties, unmodeled force, and external disturbances. Thenominal parameters of the system are given by M10=0.25,M20=0.2, K10=1, K20=2, B10=2, and B20=2.2. The perturbationsare given by DM1=0.05sin (y1), DM2=0.05sin (y1�y2), DK1=0.1,DK2=0.12, DB1=0.2, and DB2=0.15. Also, the nonlinear springforces and friction forces are assumed to be fK1

ðxÞ ¼ K10y1þDK1 y31,

fK2ðxÞ ¼ K20ðy2�y1ÞþDK2ðy2�y1Þ

3, fB1ðxÞ ¼ B10 _y1þDB1 _y1

2, andfB2ðxÞ ¼ B20ð _y2� _y1ÞþDB2ð _y2� _y1Þ

2. In addition, there are coulombfriction forces fC1

ðxÞ ¼ 0:02sgnð _y1Þ and fC2ðxÞ ¼ 0:02sgnð _y2� _y1Þ.

Therefore, the state space representation can be rewritten as

_x1

_x2

_x3

_x4

266664

377775¼

0 1 0 0

0 0 0 0

0 0 0 1

0 0 0 0

26664

37775

x1

x2

x3

x4

266664

377775þ

0 0

1 0

0 0

0 1

26664

37775

f1ðxÞþg1ðxÞu1þd�1

f2ðxÞþg2ðxÞu2þd�2

" #

ð50Þ

y1

y2

" #¼

1 0 0 0

0 0 1 0

� � x1

x2

x3

x4

266664

377775 ð51Þ

Variance (s) Variance (s) Mean (m)

m1 m2 m (type-1)

mF4iðxiÞ 1/8 0.45 0.55 0.5

mF5iðxiÞ 1 1.08 1.32 1.2

-5 -4 -3 -2 -1 0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

()x

µ

Fig. 5. Interval type-2 Gaussian membership functions for xii=1, 2, 3, 4.

ARTICLE IN PRESS


where

f1ðxÞ ¼1

M10þDM1ð�fK1

ðxÞþ fK2ðxÞ�fB1

ðxÞþ fB2ðxÞ�fC1

ðxÞþ fC2ðxÞÞ

f2ðxÞ ¼1

M20þDM2ð�fK2

ðxÞ�fB2ðxÞ�fC2

ðxÞÞ; g1ðxÞ ¼1

M10þDM1

g2ðxÞ ¼1

M20þDM2; d�

1 ¼1

M10þDM1d1; and d

�

2 ¼1

M20þDM2d2

Fig. 7. Estimated systems x1 and x2

Fig. 6. Estimated systems x1 and x2 of the type-1 controller.

Substituting all the parameters into equations (58) and (59),we have

f1ðxÞ ¼1

0:25þ0:05 sinx1�x1�0:1x3

1þ2ðx3�x1Þþ0:12ðx3�x1Þ3

h�2x2�0:2x2

2þ2:2ðx4�x2Þþ0:15ðx4�x2Þ2

�0:02 sgnðx2Þþ0:02 sgnðx4�x2Þ�

f2ðxÞ ¼1

0:2þ0:05 sinðx1�x3Þ�2ðx3�x1Þ�0:12ðx3�x1Þ

3�2:2ðx4�x2Þ

h

�0:15ðx4�x2Þ2�0:02 sgnðx4�x2Þ

ig1ðxÞ ¼

1

0:25þ0:05 sinx1and g2ðxÞ ¼

1

0:2þ0:05 sinðx1�x3Þ

The design objective is to control the states x1 and x3 of thesystem to track the reference trajectories yr1(t)=0.5sin(t) andyr2(t)=0.5cos(t), respectively, if only the system outputs y1 and y2

are measurable. Suppose the external disturbancesd1=0.2sin(3t)exp(�0.2t) and d2=0.2cos(3t)exp(�0.1t). Thechoices of g and h are to improve the convergence rate of theclosed-loop system controlled by our proposed controller.

Now we will apply our observer-based direct adaptive MIMOfuzzy controller to the mass–spring–damper system. According tothe design procedure, the design is given in the following steps:

[Step 1]: The observer and feedback gain matrices are chosenas

kTo¼ 89 184 0 0;0 0 89 184½ � and kT

c¼ 4 4 0 0;0 0 4 4½ �:

[Step 2]: Solve the state equation (8) to obtain estimate statevector xi ¼ y

ri�ei.

[Step 3]: Choose Q=4I4�4, R=r2I2�2, and r=0.8. Solving eq.(41) to get the positive definite symmetric 4�4 matrix

P¼

3:118 �1:5 0 0

�1:5 0:7425 0 0

0 0 3:118 �1:5

0 0 �1:5 0:7425

26664

37775

of the interval type-2 controller.

ARTICLE IN PRESS

Fig. 9. Output trajectories of the type-2 controller for noise free: (a) Output trajectories of y1 (dashed line) and yr1 (solid line), (b) Output trajectories of _y1(dashed line) and_yr1 (solid line), (c) Output trajectories of y2 (dashed line) and yr2 (solid line), and (d) Output trajectories of _y2(dashed line) and _yr2 (solid line).

Fig. 8. Output trajectories of the type-1 controller for noise free: (a) Output trajectories of y1 (dashed line) and yr1 (solid line), (b) Output trajectories of _y1 (dashed line) and_yr1 (solid line), (c) Output trajectories of y2 (dashed line) and yr2 (solid line) and (d) Output trajectories of _y2(dashed line) and _yr2 (solid line).


ARTICLE IN PRESS


Also, specify the design parameters Mx ¼ Mfi ¼ Mgij ¼ 200, i, j=1, 2;r1=10, r2=10, r11=r12=r21=r22=0.5r2, and l1=l2=0.02.

[Step 4]: The following interval type-2 and type-1 fuzzymembership functions for xi, i = 1, 2, 3, 4 are selected for Fj

i ,j=1, y, 5 as shown in Table 1.

Also, the footprints of uncertainty of the type-2 membershipfunction for xi, i=1, y, 4 are as shown in Fig. 5.

[Step 5]: Using (42)–(45), the adaptive laws, adjust theparameter vectors y

fiand y

gij.

[Step 6]: Use (37) to compute the robust compensator usi. Thenthe control law can be obtained.

The initial values are given as x(0)=[0.8 0 �0.5 0]T,xð0Þ ¼ �0:6 0 0:5 0

� �T, y

f 1ð0Þ ¼ y

f 2ð0Þ ¼ 1 � � � 1

� �1�25

, andy

gijð0Þ ¼ 1 � � � 1

� �1�25

; i; j¼ 1;2 to adjust the parameter vec-tors y

fiand y

gij. From Figs. 6 and 7, we see that the system states

of a type-1 controller and an interval type-2 controller can beestimated quickly.

For r=0.8, Figs. 8(a)–(d) and 9(a)–(d) show the responses ofthe mass–spring–damper system for a type-1 controller and aninterval type-2 controller, respectively.

2 4 6 8 10 12 14 160

50

100

150

200

250

300

350

X: 16Y: 196

t(sec)

Nor

m e

2(t)

Norm e2(t)

X: 16Y : 128.9

e12

e22

Fig. 11. Tracking performanceR 20

0 e2ðtÞdt for the interval type-2 controller (noise-

free).

2 4 6 8 10 12 14 160

50

100

150

200

250

300

350

X: 16Y : 217.5

t (sec)

Nor

m e

2(t)

Norm e2(t)

X: 16Y : 127.9

e12

e22


0 e2ðtÞdt for the type-1 controller (noise-free).

Both figures indicate the fact that tracking performances canbe guaranteed. The tracking performance is shown in Figs. 10 and11 and they show that the tracking error of the type-1 fuzzycontroller is larger than that of the interval type-2 fuzzycontroller. Figs. 12 and 13 show control inputs for type-1 andinterval type-2, respectively.

In order to show that the interval type-2 and type-1 FLS canhandle the measurement uncertainties, training data are cor-rupted by white Gaussian noise with signal-to-noise ratio (SNR)20 dB. Figs. 14 and 15 show the responses of the two-linkmanipulator for type-1 and interval type-2, respectively. Figs. 16

Fig. 12. Trajectory of the control input (type-1): (a) u1 for time 0–20 s and (b) u2

for time 0–20 s.


for time 0–20 s.

ARTICLE IN PRESS


and 17 show the tracking errors of the type-1 fuzzy controller andthe interval type-2 fuzzy controller.

The following figures, Figs. 18 and 19, show control inputs forthe type-1 controller and interval type-2 controller, respectively.

Fig. 15. Output trajectories of the type-2 controller with noise: (a) Output trajectories o_yr1 (solid line), (c) Output trajectories of y2 (dashed line) and yr2 (solid line) and (d) O

Fig. 14. Output trajectories of the type-1 controller with noise: (a) Output

trajectories of y1 (dashed line) and yr1 (solid line), (b) Output trajectories of _y1

(dashed line) and _yr1 (solid line), (c)Output trajectories of y2 (dashed line) and yr2

(solid line), and (d) Output trajectories of _y2 (dashed line) and _yr2 (solid line).

Tables 2 and 3 show that the proposed interval type-2 fuzzycontroller gives better performance compared with the type-1controller as system internal uncertainties (noisy training data)and external disturbance appear. Moreover, the adaptive type-1fuzzy controller must expend more control effort to deal withnoisy training data.

From the above simulation results, we can see that in order todeal with the noisy training data, there exists vibration inthe type-1 control input, i.e. more control effort is expended.

f y1 (dashed line) and yr1 (solid line), (b) Output trajectories of _y1 (dashed line) and

utput trajectories of _y2 (dashed line) and _yr2 (solid line).

2 4 6 8 10 12 14 160

50

100

150

200

250

300

350

X: 16Y: 237.5

t(sec)

Nor

m e

2 (t)

Norm e2(t)

X: 16Y: 141.2

e12

e22


0 e2ðtÞdt for the type-1 controller with noise.

ARTICLE IN PRESS


Nevertheless, the interval type-2 fuzzy logic system canhandle unpredicted internal disturbance—data uncertainties, verywell.


for time 0–20 s.

6. Conclusions

Due to rule uncertainties and the training data corrupted bynoise, the circumstances are too uncertain to determine exactmembership grades. Moreover, the state variables are notaccessible and only output variables can be measured. A newobserver-based indirect adaptive interval type-2 fuzzy controlleris developed to handle such uncertainties for a class of multi-variable nonlinear dynamical systems involving external distur-bances. The free parameters of the adaptive fuzzy controller canbe tuned online by an output feedback control law and adaptivelaws based on the Lyapunov synthesis approach. From thesimulation results, the interval type-2 fuzzy logic system canhandle unpredicted internal disturbance—data uncertainties verywell; however, the adaptive type-1 fuzzy controller must expendmore control effort in order to deal with noisy training data.Furthermore, it is obvious that the tracking performance obtainedfrom the interval type-2 fuzzy controller is better than thetracking performance obtained from the type-1 fuzzy controller.The overall adaptive interval type-2 control scheme guaranteesstability of the resulting closed-loop system in the sense that allthe states and signals are uniformly bounded and HN trackingperformance can be achieved.


for time 0–20 s.

Table 2Comparison of control effort of type-1 and interval type-2 controllers applied to the system.

PTn ¼ 1

ui

�� ; i¼ 1;2; T ¼ tf=h¼ 8000

r=0.8 r=0.6

u1 u2 u1 u2

Type-1 Type-2 Type-1 Type-2 Type-1 Type-2 Type-1 Type-2

Noise-free 10,239.54 10,169.84 10,344.01 10,235.77 10,312.61 10,224.69 10,201.87 10,371.42

SNR=20 dB 15,386.17 10,478.21 15,348.39 10,501.34 15,507.98 10,587.23 15,514.21 10,415.69

2 4 6 8 10 12 14 160

50

100

150

200

250

300

350

X: 16Y: 144.8

t(sec)

Nor

m e

2(t)

Norm e2 (t)

X: 16Y: 125

e12

e22


0 e2ðtÞdt for the interval type-2 controller with

noise.

ARTICLE IN PRESS

Table 3Comparison of tracking performance of type-1 and interval type-2 controllers.

R 160 e1

2ðtÞdt

ðr¼ 0:8Þ

R 160 e2

2ðtÞdt

ðr¼ 0:8Þ

R 160 e1

2ðtÞdt

ðr¼ 0:6Þ

R 160 e2

2ðtÞdt

ðr¼ 0:6Þ

Type-1 Type-2 Type-1 Type-2 Type-1 Type-2 Type-1 Type-2

Noise-free 217.5 196 127.9 128.9 170.8 195.4 144.8 129.5

SNR=20 dB 237.5 144.8 141.2 125 304.7 180.5 294.2 133.1


Appendix A. Proof of Theorem 1

Proof. Let us reconsider the error dynamic Eq. (37) and takeinto account the minimum approximation errors. Substi-tuting (32), (33) into (37), the error dynamic equation can berewritten as

_~ei ¼Li ~eiþBi~y

T

fixðxÞþXp

j ¼ 1

~yT

gijxðxÞuj�BiusiþBioi ð52Þ

where ~yfi ¼ y�fi�y

fiand ~ygij ¼ y�

gij�y

gij.

In order to analyze the closed-loop system stability, the

Lyapunov-like function candidate is chosen as

V ¼ V1þV2þ � � � þVp

where

Vi ¼1

2~eT

i Pi ~eiþ1

2ri

~yT

fi~yfiþ

1

2rij

Xp

j ¼ 1

~yT

gij~ygij ð53Þ

The time derivative of V along the error trajectory (37) is

_V i ¼1

2_~e

T

i Pi ~eiþ1

2~eiPi

_~eT

i þ1

2rið_~y

T

fi

~yfiþ~y

T

fi_~yfiÞþ

Xp

j ¼ 1

1

2rijð_~y

T

gij

~ygijþ~y

T

gij_~ygijÞ

ð54Þ

Substituting (52) into (54) and using (38) yields

_V i ¼1

2~eT

i LTi PiþPiLi�

2

liPiB

Ti BiPi

� �~eiþ

1

2oT

i BTi Pi ~eiþ ~e

Ti PiBioi�

�

þ1

riðri ~e

Ti PiBix

TðxÞþ

_~yT

fiÞ ~yfiþ

Xp

j ¼ 1

1

rijðrij ~e

Ti PiBix

TðxÞujþ

_~yT

gijÞ ~ygij

ð55Þ

From the adaptive laws (39) and (40) and using (41), we can

obtain

_V i ¼�1

2~eT

i Qi ~ei�1

2ð1

r~eT

i PiBi�roiÞ2þ

1

2r2o2

i ð56Þ

Since ð1=r ~eTi PiBi�roiÞ

2Z0, from (56) we have

_V ir�1

2~eT

i Qi ~eiþ1

2r2o2

i ð57Þ

Since the term ð1=2Þr2o2i is of the order of the approximation

error, this is the best we can hope to obtain. If oi=0, from (57) we

have

_V ir�1

2~eT

i Qi ~eir0

Integrating (57) from t=0 to t=T, we can obtain

ViðTÞ�Við0Þr�1

2

ZT

0

~eTi Qi ~ei dtþ

1

2r2

ZT

0

o2i dt ð58Þ

Since Vi(T)Z0 and using (53), (58) can be rewritten as follows

1

2

ZT

0

~eTi Qi ~ei dtrVið0Þþ

1

2r2

ZT

0

o2i dt

¼1

2~eT

i ð0ÞPi ~eið0Þþ1

2ri

~yT

fið0Þ~yfið0Þþ

1

2rij

Xp

j ¼ 1

~yT

gijð0Þ~ygijð0Þ

þ1

2r2

ZT

0

o2i dt ð59Þ

This demonstrates that all the states and signals involved in the

closed-loop system are bounded. Moreover, the HN performance

can be achieved. The proof is complete.

References

Castro, J.L., 1995. Fuzzy logical controllers are universal approximators. IEEETransactions on Systems, Man, and Cybernetics, Part B 25, 629–635.

Chen, B.S., Lee, C.H., Chang, Y.C., 1996. HN tracking design of uncertain nonlinearSISO systems: adaptive fuzzy approach. IEEE Transactions on Fuzzy Systems 4,32–43.

Castillo, O., Nohe’, C’azarez, Rico, D., 2006a. Intelligent control of dynamic systemsusing type-2 fuzzy logic and stability issues. International MathematicalForum 1 (28), 1317–1382.

Castillo O., Nohe’ C’azarez, Melin P., 2006b. Design of stable type-2 fuzzy logiccontroller based on a fuzzy Lyapunov approach. In: Proceedings of the IEEEInternational Conference on Fuzzy Systems, Canada, pp. 2331–2336.

Hsiao, M.Y., Li, T.H., Lee, J.Z., Chao, C.H., Tsai, S.H., 2008. Design of interval type-2fuzzy sliding-mode controller. Information Science 178, 1696–1716.

Ho, H.F., Wong, Y.K., Rad, A.B., 2008. Adaptive fuzzy approach for a class of uncertainnonlinear systems in strict-feedback form. ISA Transactions 47, 286–299.

Karnik, N.N., Mendel, J.M., Liang, Q., 1999. Type-2 fuzzy logic systems. IEEETransactions on Fuzzy Systems 7, 643–658.

Kheireddine, C., Lamir, S., Mouna, G., Khier, B., 2007. Indirect adaptive intervaltype-2 fuzzy control for nonlinear systems. International Journal of Modeling,Identification and Control 2, 106–119.

Leu, Y.G., Lee, T.T., Wang, W.Y., 1999a. Observer-based adaptive fuzzy-neuralcontrol for unknown nonlinear dynamical systems. IEEE Transactions onSystems, Man, and Cybernetics, Part B 29, 583–591.

Leu, Y.G., Lee, T.T., Wang, W.Y., 1999b. Robust adaptive fuzzy-neural controllers foruncertain nonlinear systems. IEEE Transactions on Robotics and Automation15, 805–817.

Liang, Q., Mendel, J.M., 2000. Interval type-2 fuzzy logic system: theory and design.IEEE Transactions on Fuzzy Systems 8, 535–550.

Lin, P.Z., Lin, C.M., Hsu, C.F., Lee, T.T., 2005. Type-2 fuzzy controller design using asliding-mode approach for application to DC–DC converters. IEEE Proceedings—

Electrical Power Application 152 (6), 1482–1488.Lin, T.C., Liu, H.L., Kuo, M.J., 2009a. Direct adaptive interval type-2 fuzzy control of

multivariable nonlinear systems. Engineering Applications of Artificial Intelli-gence 22, 420–430.

Lin T.C., Kuo M.J., Hsu C.H., 2009b. Robust adaptive tracking control ofmultivariable nonlinear systems based on interval type-2 fuzzy approach.International Journal of Innovative Computing Information and Control,accepted for publication.

Lin, T.C., 2009c. Analog circuit fault diagnosis under parameter variations based ontype-2 fuzzy logic systems. International Journal of Innovative Computing.Information and Control, accepted for publication.

Ma, X.J., Sun, Z.Q., 2000. Output tracking and regulation of nonlinear system basedon Takgi-Sugeno fuzzy model. IEEE Transactions on Systems, Man, andCybernetics, Part B 30, 47–59.

Mendel, J.M., John, R.I.B., 2000. Type-2 fuzzy sets made simple. IEEE Transactionson Fuzzy Systems 10, 117–127.

Mendel, J.M., John, R.I., Liu, F., 2006. Interval type-2 fuzzy logic systems madesimple. IEEE Transactions on Fuzzy Systems 14 (6), 808–821.

Mendel, J.M., 2007. Type-2 fuzzy sets and systems: an overview. IEEE Computa-tional Intelligence Magazine 99, 20–29.

Marino, R., Tomei, P., 1993a. Globally adaptive output-feedback control onnonlinear systems, part I: linear parameterization. IEEE Transactions onAutomatic Control 38, 17–32.

Marino, R., Tomei, P., 1993b. Globally adaptive output-feedback control onnonlinear systems, part II: nonlinear parameterization. IEEE Transactions onAutomatic Control 38, 33–48.

Narendra, K.S., Parthasarathy, K., 1990. Identification and control of dynamicalsystems using neural networks. IEEE Transactions on Neural Networks 1, 4–27.

Park, A.S., Yu, W., 1994. Sanchez EN, Perez JP. Nonlinear adaptive tracking usingdynamic neural networks, IEEE Transactions on Neural Networks 10,1402–1411.

ARTICLE IN PRESS


Rovithakis, G.A., Christodoulou, M.A., 1994. Adaptive control of unknown plantsusing dynamical neural networks. IEEE Transactions on Systems, Man, andCybernetics, Part B 24, 400–412.

Sastry, S.S., Isidori, A., 1989. Adaptive control of linearization systems. IEEETransactions on Automatic Control 34, 1123–1131.

Slotine, J.E., Li, W., 1991. In: Applied Nonlinear Control. Prentice-Hall, EnglewoodCliffs, NJ.

Spooner, J.T., Passino, K.M., 1996. Stable adaptive control using fuzzy systemsand neural networks. IEEE Transactions on Fuzzy Systems 4, 339–359.

Ross, Timothy J., 2004. In: Fuzzy Logic with Engineering Applications. Wiley.Tong, S., Tang, J., Wang, T., 2000. Fuzzy adaptive control of multivariable nonlinear

systems. Fuzzy Sets and Systems 111, 153–167.

Tong, S., Chen, B., Wang, T., 2005. Fuzzy adaptive output control feedback controlfor MIMO nonlinear systems. Fuzzy Sets and Systems l, 285–299.

Wang, L.X., 1993. Stable adaptive fuzzy control of nonlinear systems. IEEETransactions on Fuzzy Systems 1, 146–155.

Wang, L.X., 1994. In: Adaptive Fuzzy Systems and Control: Design and StabilityAnalysis. Prentice-Hall, Englewood Cliffs, NJ.

Wang, C.H., Wang, W.Y., Lee, T.T., Tseng, P.S., 1995. Fuzzy B-splinemembership function (BMF) and its applications in fuzzy-neuralcontrol. IEEE Transactions on Systems, Man, and Cybernetics, Part B 25,841–851.

Wang, C.H., Liu, H.L., Lin, T.C., 2002a. Direct adaptive fuzzy neural control withstate observer and supervisory controller for unknown nonlinear dynamicalsystems. IEEE Transactions on Fuzzy Systems 10, 39–49.

Wang, C.H., Lin, T.C., Lee, T.T., Liu, H.L., 2002b. Adaptive hybrid intelligent controlfor uncertain nonlinear dynamical systems. IEEE Transactions on Systems,Man, and Cybernetics, Part B 32, 583–597.

Wu, D., Tan, W.W., 2006. A simplified type-2 fuzzy logic controller for real-timecontrol. ISA Transactions 45 (4), 503–516.

Wang, M., Chen, B., Dai, S.L., 2007. Direct adaptive fuzzy tracking control for a classof perturbed strict-feedback nonlinear systems. Fuzzy Sets and Systems 158,2655–2670.

Wang, C.H., Cheng, C.S., Lee, T.T., 2004. Dynamical optimal training for type-2 fuzzyneural (T2FNN). IEEE Transactions on Systems, Man, and Cybernetics, Part B 34(3), 1462–1477.

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