Direct adaptive fuzzy control for nonlinear systems with time-varying delays

Information Sciences 180 (2010) 776–792

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Information Sciences

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

Direct adaptive fuzzy control for nonlinear systemswith time-varying delays q

Bing Chen a,*, Xiaoping Liu b, Kefu Liu b, Peng Shi c, Chong Lin a

a Institute of Complexity Science, Qingdao University, Qingdao, Shandong 266071, PR Chinab Faculty of Engineering of Lakehead University, Thunder Bay, On, Canadac School of Technology, University of Glamorgan, Pontypridd, CF37 1DL, UK

a r t i c l e i n f o

Article history:Received 24 November 2008Received in revised form 29 October 2009Accepted 3 November 2009

Keywords:Nonlinear systemsAdaptive fuzzy controlTime-delayRazumikhin functionalOutput tracking

0020-0255/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.ins.2009.11.004

q This work was supported in part by the nationa* Corresponding author.

E-mail address: [email protected] (B

a b s t r a c t

This paper focuses on the problem of direct adaptive fuzzy control for nonlinear strict-feed-back systems with time-varying delays. Based on the Razumikhin function approach, anovel adaptive fuzzy controller is designed. The proposed controller guarantees that thesystem output converges to a small neighborhood of the reference signal and all the signalsin the closed-loop system remain bounded. Different from the existing adaptive fuzzy con-trol methodology, the fuzzy logic systems are used to model the desired but unknown con-trol signals rather than the unknown nonlinear functions in the systems. As a result, theproposed adaptive controller has a simpler form and requires fewer adaptation parameters.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

In recent years, considerable attention has been paid to fuzzy logic control. Some fuzzy logic control techniques have beendeveloped, for instance, see [3,15,16,23,35,36] and the references therein. Approximation-based adaptive fuzzy control orneural control has emerged as a popular and convenient tool in analysis and synthesis of complex and ill-defined systems,to which the application of conventional control techniques is not straightforward or feasible. The main idea of such a controlmethodology is to utilize fuzzy logic systems or neural networks to approximate the unknown nonlinearities in dynamic sys-tems and the adaptive backstepping technique to construct the controllers. Following such an idea, some adaptive neuralcontrol or fuzzy control schemes were proposed for nonlinear delay-free systems with strict-feedback structure. In[1,4,6,37], the adaptive neural tracking control was addressed for nonlinear delay-free systems in strict-feedback form.The corresponding adaptive fuzzy control schemes were proposed in [11,20,31–33,38,25].

All the aforementioned studies do not deal with state time-delays, which often occur in many dynamic systems, such asrolling mill systems, biological systems, metallurgical processing systems, network systems, and so on. It is well known thatthe existence of delays usually becomes the source of instability and performance deterioration of systems. Stability analysisand control synthesis of Takagi–Sugeno (T–S) fuzzy delayed systems were proposed in [2,12,17,26] and [24], respectively.Especially, approximation-based adaptive control has also been addressed for nonlinear systems with time-delay. In [5],the problem of adaptive fuzzy tracking control was addressed for nonlinear strict-feedback systems with time-delays. Theproposed tracking controller guarantees that all the closed-loop signals remain bounded, while the output converges to aneighborhood of the reference signal. By using a wavelet fuzzy network online approximation model, the stabilization

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l science foundation of China (Nos. 60674055 and 60774047).

. Chen).

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B. Chen et al. / Information Sciences 180 (2010) 776–792 777

problem was discussed for nonlinear strict-feedback systems with time-delay in [7]. In [21], the adaptive H1 control wasinvestigated via backstepping and fuzzy networks technique. In [8], the observer-based adaptive neural controller was de-signed for a special class of nonlinear time-delay systems. By using fuzzy logic systems to approximate the unknown non-linear functions, the adaptive fuzzy control scheme was proposed in [28] for nonlinear delayed systems with lower-triangular form. The main advantage of the results proposed in [28] is that the controllers contain fewer adaptation laws.

Usually, there are two approaches for stability analysis of time-delay systems. The first one is based on the Lyapunov–Krasovskii functionals [5,7,28], and the second one on the Lyapunov-Razumikhin functionals [9,13,18,34,19] and [27]. TheLyapunov–Krasovskii functional methods usually make stability analysis and controller design more complex than the Raz-umikhin functional approach. All the existing results on the approximation-based adaptive control are obtained by using theLyapunov–Krasovskii functional approach. To the authors’ knowledge, there have not been any reported results on the use ofthe Razumikhin functional approach for this purpose. In addition, all the existing adaptive fuzzy or neural controllers wereproposed only for the nonlinear systems with constant delays. Therefore, in theory, the existing adaptive fuzzy controllersare not capable of dealing with systems with time-varying delays. Thus, it is of importance to develop a new adaptive fuzzycontrol approach for nonlinear systems with time-varying delays.

The above observations motivate the present research, in which we investigate adaptive fuzzy control for nonlinear sys-tems with time-varying delays. By utilizing the Razumikhin functional approach and fuzzy backstepping technique, a noveldirect adaptive fuzzy controller is proposed. The proposed adaptive fuzzy controller guarantees that all the signals of theclosed-loop system are bounded, while the system output converges to a small neighborhood of the reference signal. Threemain contributions are made in this paper. (1) A Razumikhin lemma-based adaptive fuzzy controller design procedure is firstproposed. Compared with the existing results, the proposed fuzzy controllers can handle systems with both constant andtime-varying delays. (2) A novel direct adaptive fuzzy control approach is presented. Unlike the existing approximation-based adaptive control methods, the direct adaptive fuzzy control technique employs fuzzy logic systems to approximatethe desired but unknown control input signals rather than the nonlinear functions in dynamic systems. As a result, the directadaptive fuzzy controllers have a simpler form and fewer tuning parameters. Furthermore, it is more convenient to imple-ment the controllers in practice. (3) Different from the existing adaptive fuzzy control methods, for an nth order system theproposed control scheme contains only n adaption laws, no matter how many fuzzy rule bases are used. Therefore, the com-putation burden is reduced, and the algorithm is easily realized in real-time. In addition, the prior knowledge of the upperbound of the virtual control coefficients is not required.

The rest of the paper is organized as follows. Section 2 provides preliminaries and the formulation of the problem. Sec-tion 3 develops an adaptive fuzzy controller design procedure based on the Razumikhin functional approach and backstep-ping technique. Section 4 presents four examples to illustrate the effectiveness of the proposed controllers. These arefollowed by conclusions in Section 5.

2. Problem formulation and preliminaries

Consider the nonlinear time-delay system described by

_xi ¼ fi �xið Þ þ gi �xið Þxiþ1ðtÞ þ qi �xi t � siðtÞð Þð Þ; 1 6 i 6 n� 1;_xn ¼ fn �xnð Þ þ gn �xnð ÞuðtÞ þ qn �xn t � snðtÞð Þð Þ;y ¼ x1; ð1Þ

where xi 2 R;u 2 R and y 2 R are the state variable, the control input, and the output of the system (1), respectively, siðtÞ de-notes the time-varying delay in the ith equation, �xiðtÞ ¼ ½x1ðtÞ; x2ðtÞ; . . . ; xiðtÞ�T and xðtÞ ¼ �xnðtÞ ¼ ½x1ðtÞ; x2ðtÞ; . . . ; xnðtÞ�T ;fið:Þ; gið:Þ and qið:Þ are unknown smooth nonlinear functions satisfying fið0Þ ¼ qið0Þ ¼ 0. For t 2 ½�si;0�; xiðtÞ ¼ UiðtÞ; i ¼ 1;2; . . . ; n, where the initial function, UiðtÞ, is smooth and bounded.

For a given reference signal ydðtÞ, the control objective is to design a direct adaptive controller such that the system’s out-put converges to a small neighborhood of the reference signal and all the variables of the resulting closed-loop system re-main bounded. To this end, the following assumptions are imposed on the system (1).

Assumption 1. The signs of gið:Þ, for 1 6 i 6 n, are known and there exist positive constants g0 and gM such that fori ¼ 1;2; . . . ; n,

0 < g0 6 jgið:Þj 6 gM <1:

Apparently, Assumption 1 means that functions gið:Þ, for 1 6 i 6 n, are either strictly positive or strictly negative. With-out loss of generality, it is further assumed that gið:Þ > g0 > 0. In addition, in Assumption 1 g0 and gM can be unknown.

Assumption 2. There exist class-k1 functions Qið:Þ such that for 1 6 i 6 n,

qi �xi t � siðtÞð Þð Þj j 6 Qi k�xi t � siðtÞð Þkð Þ:

Remark 1. Let Qð:Þ be a class-k1 function and ai P 0 for 1 6 i 6 n. Then QðPn

i¼1aiÞ 6Pn

i¼1QðnaiÞ holds. Moreover, if uð:Þis a class-k1 function, then Qðuð:ÞÞ is also a class-k1 function.

778 B. Chen et al. / Information Sciences 180 (2010) 776–792

Assumption 3. The reference signal ydðtÞ and its time derivatives up to the nth order are continuous and bounded. Further-more, it is assumed that there exists a constant d such that jydðtÞj 6 d. In this paper, we adopt the singleton fuzzifier, prod-uct inference, and the center-defuzzifier to deduce the following fuzzy rules:

Ri : IF x1 is Fi1 and . . . and xn is Fi

n THEN y is Biði ¼ 1;2; . . . ; NÞ;

where x ¼ ½x1; . . . ; xn�T 2 Rn and y 2 R are the input and output of the fuzzy system, respectively, Fji and Bi are fuzzy sets in R.

The fuzzy inference engine performs a mapping from fuzzy sets in Rn to a fuzzy set in R based on the IF–THEN rules in thefuzzy rule base and the compositional rule of inference. The fuzzifier maps a crisp point x ¼ ½x1; . . . ; xn�T 2 Rn into a fuzzy setAx in R. The defuzzifier maps a fuzzy set in R to a crisp point in R. Since the strategy of singleton fuzzification, center-averagedefuzzification and product inference is used, the output of the fuzzy system can be formulated as

y xð Þ ¼

PNj¼1Wj

Qni¼1lFj

ixið ÞPN

j¼1

Qni¼1lFj

ixið Þ

h i : ð2Þ

where Wj is the point at which fuzzy membership function lBj ðWjÞ achieves its maximum value. It is assumed that

lBj ðWjÞ ¼ 1. Let SjðxÞ ¼Qn

i¼1l

Fji

ðxiÞPN

j¼1½Qn

i¼1l

Fji

ðxiÞ�; SðxÞ ¼ ½S1ðxÞ; S2ðxÞ; . . . ; SNðxÞ�T and W ¼ ½W1; . . . ; WN�T . Then the fuzzy logic system

(2) can be rewritten as

yðxÞ ¼WT SðxÞ: ð3Þ

It has been proved in [29] and [30] that if Gaussian functions are used as membership functions, the following lemmaholds.

Lemma 1. For any continuous function f ðxÞ over a compact set Xx � Rq and any given constant e > 0, there exists a fuzzy logicsystem WT SðxÞ such that:

f ðxÞ ¼WT SðxÞ þ dðxÞ; dðxÞj j 6 e;

where dðxÞ is the approximation error.

Razumikhin Lemma [10]. Suppose f : R� C ! Rn maps R� B (B is a bounded subset of C) into a bounded set of Rn andconsider the retarded functional differential equation (RFDE) _x ¼ f ðt; xtÞ. Suppose that uðsÞ;vðsÞ and wðsÞ are continuous non-decreasing functions and uðsÞ ! 1as s!1. If there are a continuous function V : R� Rn ! R, a continuous nondecreasingfunction p : Rþ ! Rþ; pðsÞ > s for s > 0 and a constant b0 P 0 such that:

V t þ s; x t þ sð Þð Þ < p V t; x tð Þð Þð Þ; 8s 2 ½�d; 0�;u kxkð Þ 6 V t; xtð Þ 6 v kxkð Þ;_V t; xtð Þ 6 �w kxkð Þ þ b0;

then the solutions of RFDE (f) are uniformly ultimately bounded (UUB). In this case, it is said that the system is UUB stable. Ifb0 ¼ 0, the system is said to be asymptotically stable.

In [14], it has been pointed out that for simplicity, the function pð:Þ can be chosen as pðsÞ ¼ ps with p > 1. Furthermore, ifthe Lyapunov function is quadratic, i.e., VðxÞ ¼ xT Px with P being a positive definite matrix, then the conditionVðt þ s; xðt þ sÞÞ < pðVðt; xðtÞÞÞ;8s 2 ½�d; 0� can be replaced by the condition kxðt þ sÞk 6 qkxðtÞk;8s 2 ½�d;0�, whereq ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipkM=km

p> 1; km and kM are the minimum and maximum eigenvalues of P, respectively.

In general, a backstepping-based design procedure for an nth order nonlinear strict-feedback system contains n designsteps. At Step i, the fuzzy network WT

i SðZiÞ will be used to approximate the unknown function �f iðZiÞ for i ¼ 1;2; . . . ; n. Thus,the virtual control signals aiðZiÞ and control law u are constructed as follows:

ai Zið Þ ¼�12a2

i

ziðtÞhiðtÞST Zið ÞS Zið Þ;1 6 i 6 n� 1; ð4Þ

uðtÞ ¼ �12a2

nznðtÞhnðtÞST Znð ÞS Znð Þ; ð5Þ

where Sð:Þ is the basis function vector, hi is the estimation of the unknown constant hi which will be specified at Step i, ai andan are positive design parameters, and

ziðtÞ ¼ xiðtÞ � ai�1 Zið Þ; ZiðtÞ ¼ �xTi ; h

T ; �yTdi

h iT; h ¼ h1; . . . ; hn

h iT;

ZðtÞ ¼ z1; . . . ; zn; ~hT

h iT; h ¼ h1; . . . ; hn½ �T ; ~h ¼ h� h;

with a0 ¼ yd and �ydi ¼ ½yd; _yd; . . . ; yðiÞd �T . The ith adaption law is defined by


_hi ¼ri

2a2i

z2i ðtÞS

T Zið ÞS Zið Þ � rihiðtÞ; i ¼ 1; . . . ; n; ð6Þ

with ri and ri being positive design parameters.

Remark 2. From (6), it is easy to prove that for any given initial condition hðt0Þ P 0; hðtÞ P 0 holds for t P t0. Thus, in thisresearch, it is always assumed that hðt0Þ P 0.

To develop a simple backstepping-based design procedure, we introduce some useful Lemmas, which will be used in thecontroller design procedure.

Lemma 2 [22]. For any u 2 R and e > 0, the following holds.

0 6 juj � u tan hue

� �6 de; d ¼ 0:2785:

Lemma 3. Let �xi ¼ ½x1; . . . ; xi�T ;�ziðtÞ ¼ ½z1; . . . ; zi�T and �ai ¼ ½yd;a1; . . . ; ai�1�T . Then there exists a class-k1 function uð:Þ suchthat:

k�xiðtÞk 6 uðkZðtÞkÞ þ d; 1 6 i 6 n;

where d is a constant defined as in Assumption 3 and uðsÞ ¼ sða0 þ b0sÞ with a0 and b0 being positive constants.

Proof. According to the definition of aj, there are positive constants cj such that:

k�xik 6 k�zik þ k�aik 6 kZðtÞk þXi�1

j¼1

kajk þ jydj 6 kZðtÞk þXi�1

j¼1

cjjZðtÞkkhk� �

þ d

6 kZðtÞk þXi�1

j¼1

cjkZðtÞk k~hk þ khk� ��

þ d 6 kZðtÞk þXn�1

i¼1

kZðtÞk cjkZðtÞk þ cjkhk� �

þ d

6 kZðtÞk a0kZðtÞk þ b0ð Þ þ d ¼ u kZðtÞkð Þ þ d;

where uðsÞ ¼ sða0sþ b0Þ with a0 ¼Pn�1

j¼1 cj and b0 ¼ 1þPn�1

j¼1 cjkhk. Obviously, uð:Þ is a class-k1 function. h

3. Control design and stability analysis

In this section, a backstepping-based adaptive fuzzy controller design procedure will be developed. For the purpose ofsimplicity, the time variable t is omitted from the corresponding functions and xðt � sðtÞÞ is denoted as xðsÞ.

Step 1. Let z1 ¼ x1 � yd. It follows from the first differential equation of the system (1) that:
_z1 ¼ f1 �x1ð Þ þ g1 �x1ð Þx2 þ q1 �x1 s1ð Þð Þ � _yd:
Consider a Lyapunov function candidate as

V1 ¼12

z21 þ

g0

2r1

~h21;

where ~h1 ¼ h1 � h1. Differentiating V1 yields:

_V1 ¼ z1 f1 �x1ð Þ þ g1 �x1ð Þx2 � _ydð Þ þ z1q1 �x1 s1ð Þð Þ � g0

r1

~h1_h1: ð7Þ

By Assumption 2 and Lemma 3:

z1q1ð�x1ðs1ÞÞ 6 jz1jQ1ðuðkZðs1ÞkÞ þ dÞ: ð8Þ

Since Q1ð:Þ is a class-k1 function, one has:

Q 1ðuðkZðs1ÞkÞ þ dÞ 6 Q 1ð2uðkZðs1ÞkÞÞ þ Q 1ð2dÞ: ð9Þ

Note that uð:Þ is also a class-k1 function, therefore, Q 1ð:Þ ¼ Q1ð2uð:ÞÞ is still a class-k1 function. Furthermore, by usingthe Razumikhin Lemma, there exists a constant q > 1 such that kZðs1Þk 6 qkZðtÞk. Consequently, it follows from (9) that:

Q 1ðuðkZðs1ÞkÞ þ dÞ 6 Q 1ðqkZðtÞkÞ þ Q1ð2dÞ: ð10Þ

Due to kZðtÞk 6 kZ1ðtÞk þPn

i¼2jzij with Z1 ¼ ½z1; ~hT �T , (10) implies that:

Q 1ðuðkZðs1ÞkÞ þ dÞ 6Xn

i¼2

Q1ðl1jzijÞ þ Q 1ðl1kZ1ðtÞkÞ þ Q 1ð2dÞ;

with l1 ¼ qðn� 1þ 1Þ. Because Q1ð:Þ is a class-k1 function, it can be expressed in the form Q 1ðsÞ ¼ s/1ðsÞ with /1ð:Þ being afunction. As a result, the above inequality can be rewritten as


Q1ðuðkZðs1ÞkÞ þ dÞ 6Xn

i¼2

l1jzij/1ðl1jzijÞ þ Q1ðl1kZ1ðtÞkÞ þ Q1ð2dÞ: ð11Þ

Substituting (11) into (8) and using completion of squares gives:

z1q1ð�x1ðs1ÞÞ 6Xn

i¼2

12

l21z2

1 þXn

i¼2

12

z2i /

21ðl1jzijÞ þ jz1jðQ1ðl1kZ1ðtÞkÞ þ Q1ð2dÞÞ

6

Xn

i¼2

12

l21z2

1 þXn

i¼2

12

z2i /

21ðl1jzijÞ þ z1F1 tanh

z1F1

b1

� �þ db1; ð12Þ

where Lemma 2 is used to deal with the term jz1jðQ1ðl1kZ1ðtÞkÞ þ Q1ð2dÞÞ and F1 ¼ Q1ðl1kZ1ðtÞkÞ þ Q1ð2dÞ. Define a functionas

�f 1ðZ1Þ ¼ f1 � _yd þXn

i¼2

12

l21z1 þ F1 tanh

z1F1

b1

� �;

Then, the following inequality can be verified.

_V1 6 z1ð�f 1 þ g1ð�x1Þx2Þ �g0

r1

~h1_h1 þ

Xn

i¼2

12

z2i /

21ðl1jzijÞ þ db1: ð13Þ

Apparently, when x2 is viewed as the intermediate input, a desired control signal for the first subsystem is a1 ¼ �g�11 ð�f 1þ

ðk1 þ 0:5g21Þz1Þ. With a1, one has:

z1ð�f 1 þ g1ð�x1Þx2Þ ¼ �ðk1 þ 0:5g21Þz2

1 þ z1g1ðx2 � a1Þ þ z1g1ða1 � a1Þ: ð14Þ

However, since a1 consists of the unknown functions �f 1 and g�11 , it cannot be implemented in practice. By Lemma 1, a

fuzzy logic system is thus used to approximate a1 such that for any given positive constant e1:

a1 ¼WT1SðZ1Þ þ d1ðZ1Þ; kd1k2

6 e21;

with d1ðZ1Þ indicating the approximation error. Furthermore, it follows that:

�z1g1a1 ¼ �z1g1WT1SðZ1ÞkW1kkW1k�1 þ z1g1d1ðZ1Þ 6

g0

2a21

z21

gM

g0kW1k2STðZ1ÞSðZ1Þ þ

12

a21gM þ

12

g21z2

1 þ12e2

1

¼ g0

2a21

z21h1STðZ1ÞSðZ1Þ þ

12

a21gM þ

12

g21z2

1 þ12e2

1; ð15Þ

where h1 ¼ gMg0kW1k2. On the other hand, it follows from Remark 2 and (4) that:

z1g1a1 6 �1

2a21

z21g0h1STðZ1ÞSðZ1Þ: ð16Þ

Combining (14) with (15) and (16) gives:

z1ð�f 1 þ g1ð�x1Þa2Þ 6 �k1z21 þ

12ða2

1gM þ e21Þ þ z1g1z2 þ

g0

2a21

z21~h1STðZ1ÞSðZ1Þ; ð17Þ

with z2 ¼ x2 � a1. Substituting (17) into (13) yields:

_V1 6 �k1z21 þ

12

a21gM þ e2

1

� �þ db1 þ z1g1z2 þ

g0

r1

~h1r1

2a21

z21STðZ1ÞSðZ1Þ � _h1

� �þXn

i¼2

12

z2i /

21ðl1jzijÞ: ð18Þ

Remark 3. Unlike the existing indirect adaptive fuzzy control approaches in [28] and the references therein, the idealcontrol signal, i.e., a1, is first constructed by using the unknown nonlinear functions and constants. Then, the virtual controla1 is designed by using a fuzzy logic system to approximate the unknown control signal a1 directly rather than the unknownfunctions f1ð�x1Þ and g1ð�x1Þ. Therefore, this method is called direct adaptive fuzzy control. Such an idea will be implementedat each step henceforth. As shown in (4) and (5), by using the direct adaptive method, the virtual control signals ai and thereal control signal u have simpler form and contain fewer design parameters. In addition, differing from the existing adaptivefuzzy control methods for nonlinear delayed systems, the proposed control laws are suitable for the case of time-varyingdelays, because the Razumikhin functional is used for stability analysis and control synthesis.

Step k. ð2 6 k 6 n� 1Þ First, it is assumed that after Step k� 1 for the following Lyapunov function:

Vk�1 ¼ Vk�2 þ12

z2k�1 þ

g0

2rk�1

~h2k�1;


with V0 ¼ 0 and ~hk�1 ¼ hk�1 � hk�1, hk�1 being the estimation of hk�1 ¼ gMg0kWk�1k2, the following inequality holds:

_Vk�1 6 �Xk�1

i¼1

kiz2i þ

Xk�1

i¼1

12ða2

i gM þ e2i Þ þ

Xk�1

i¼1

dbi þXk�1

i¼1

g0

ri

~hiri

2a2i

z2i STðZiÞSðZiÞ � _hi

� �þXn

i¼k

12

z2i

Xk�1

j¼1

ðk� jÞ/2j ðljjzijÞ

þXk�1

i¼2

zi xi �Xn

j¼1

oai�1

ohi

_hi

!þ zk�1gk�1zk; ð19Þ

where zk ¼ xk � ak�1. Then, consider a Lyapunov function candidate as

Vk ¼ Vk�1 þ12

z2k þ

g0

2rk

~h2k :

Differentiating Vk gives:

_Vk ¼ _Vk�1 þ zkðfk � _ak�1 þ gkxkþ1Þ þ zkqkð�xkðskÞÞ �g0

rk

~hk_hk; ð20Þ

where

�zk _ak�1 ¼ �zk

Xk�1

j¼1

oak�1

oxjðfj þ gjxjþ1Þ � zk

Xk�1

j¼1

oak�1

oxjqjð�xjðsjÞÞ � zk

Xk�1

j¼0

oak�1

oyðjÞd

yðjþ1Þd � zk

Xn

j¼1

oak�1

ohj

_hj: ð21Þ

For the term zkqkð�xkðskÞÞ, repeating the procedure from (8)–(12) with Z1 being replaced by Zk ¼ ½z1; . . . ; zk; ~hT �T and l1 bylk ¼ qðn� kþ 1Þ produces:

zkqkð�xkðskÞÞ 6Xn

i¼kþ1

l2k

2z2

k þXn

i¼kþ1

12

z2i /

2kðlkjzijÞ þ jzkjðQ kðlkkZkðtÞkÞ þ Q kð2dÞÞ: ð22Þ

Similarly, one also has the following inequality:

�zk

Xk�1

j¼1

oak�1

oxjqjð�xjðsjÞÞ 6

Xk�1

j¼1

Xn

i¼kþ1

l2k

2z2

koak�1

oxj

2

þXk�1

j¼1

Xn

i¼kþ1

12

z2i /

2j ðlkjzijÞ þ jzkj

Xk�1

j¼1

oak�1

oxj

��ðQ j lkkZkðtÞk

� �þ Q jð2dÞÞ:

ð23Þ

Now, combining (22) and (23) gives:

zkqkð�xkðskÞÞ�zk

Xk�1

j¼1

oak�1

oxjqj �xjðsjÞ� �

6

Xn

i¼kþ1

l2k

2z2

k þXk�1

j¼1

Xn

i¼kþ1

l2k

2z2

koak�1

oxj

2

þXk

j¼1

Xn

i¼kþ1

12

z2i /

2j lk zij jð Þþ jzkjFk

6

Xn

i¼kþ1

l2k

2z2

k þXk�1

j¼1

Xn

i¼kþ1

l2k

2z2

koak�1

oxj

2

þXk

j¼1

Xn

i¼kþ1

12

z2i /

2j ðlkjzijÞþzkFk tanh

zkFk

bk

� �þdbk;

where Lemma 2 is used to handle the term jzkjFk with Fk ¼ Q kðlkkZkðtÞkÞ þ Qkð2dÞ þPk�1

j¼1 joak�1

oxjjðQjðlkkZkkÞ þ Q jð2dÞÞ. Define:

�f kðZkÞ ¼ fk þ zk�1gk�1 � zk

Xk�1

j¼1

oak�1

oxjðfj þ gjxjþ1Þ � zk

Xk�1

j¼0

oak�1

oyðjÞd

yðjþ1Þd þ Fk tanh

zkFk

bk

� �þ 1

2

Xn

i¼kþ1

l2kzk

þXk�1

j¼1

Xn

i¼kþ1

l2k

2zk

oak�1

oxj

2

þ 12

zk

Xk�1

j¼1

ðk� jÞ/2j ðljjzkjÞ �xkðZkÞ;

where yðjÞd denotes the jth order derivative of yd. Substituting (19), (22) and (23) and �f k into (20) produces:

_Vk 6 �Xk�1

i¼1

kiz2i þ

Xk�1

i¼1

12

a2i gM þ e2

i

� �þXk

i¼1

dbi þXk�1

i¼1

g0

ri

~hiri

2a2i


� �

þXk

i¼2

zi xi �Xn

j¼1

oai�1

ohj

_hj

!þXn

i¼kþ1

12

z2i

Xk

j¼1

ðk� jþ 1Þ/2j ðljjzijÞ þ zkð�f k þ gkxkþ1Þ: ð24Þ

Alternatively, viewing xkþ1 as the virtual control input and taking the virtual control input signal as ak ¼ �g�1k ð�f k þ ðkkþ

0:5g2kÞzkÞ, we have:

zkð�f k þ gkxkþ1Þ ¼ �ðkk þ 0:5g2kÞz2

k þ zkgkzkþ1 þ zkgkðak � akÞ:

By Lemma 1, a fuzzy logic system is again utilized to approximate the unknown ak such that for any given constant ek > 0:

ak ¼WTk SðZkÞ þ dkðZkÞ;


where dkðZkÞ is the approximate error and satisfies kdkk26 e2

k . Then, repeating the procedure from (14)–(17) shows that:

zkð�f k þ gkxkþ1Þ 6 �kkz2k þ zkgkzkþ1 þ

12ða2

kgM þ e2kÞ þ

g0

2a2k

z2k~hkSTðZkÞSðZkÞ; ð25Þ

where zkþ1 ¼ xkþ1 � ak; ~hk ¼ hk � hk with hk ¼ gMg0kW�

kk2, and hk is the estimation of hk. Substituting this inequality into (24)

produces:

_Vk 6 �Xk

i¼1

kiz2i þ

Xk

i¼1

12ða2

i gM þ e2i Þ þ

Xk

i¼1

dbi þXk

i¼1

g0

ri

~hiri

2a2i


� �

þXk

i¼2

zi

xi �

Xn

j¼1

oai�1

ohj

_hj

!þXn

i¼kþ1

12

z2i

Xk

j¼1

ðk� jþ 1Þ/2j ðljjzijÞ þ zkgkzkþ1: ð26Þ

Step n. Consider the following Lyapunov function candidate:

Vn ¼ Vn�1 þ12

z2n þ

g0

2rn

~h2n:

Differentiating Vn gives:

_Vn ¼ _Vn�1 þ znðfn � _an�1 þ gnuÞ þ znqnð�xnðsnÞÞ; ð27Þ

where

�zn _an�1 ¼ �zn

Xn�1

j¼1

oan�1

oxjðfj þ gjxjþ1Þ � zn

Xn�1

j¼0

oan�1

oyðjÞd

yðjþ1Þd � zn

Xn�1

j¼1

oan�1

oxjqjð�xjðsjÞÞ � zn

Xn

j¼1

oan�1

ohj

_hj: ð28Þ

For the delay terms in (27) and (28), by following a similar line used in the procedure from (8)–(10), we have:

znqnð�xnðsnÞÞ 6 jznjðQ nð2uðqkZðtÞkÞÞ þ Qnð2dÞÞ;�zn

Xn�1

j¼1

oan�1

oxjqjð�xjðsjÞÞ

6 jznjXn�1

j¼1

oan�1

oxj

��ðQ jð2uðqkZðtÞkÞÞ þ Q jð2dÞÞ: ð29Þ

Now, define:

�f n ¼ fn þ zn�1gn�1 �Xn�1

j¼1

oan�1

oxjfj þ gjxjþ1� �

�Xn�1

j¼0

oan�1

oyðjÞd

yðjþ1Þd þ Fn

znFn

bn

� �þ 1

2zn

Xn�1

j¼1

ðn� jÞ/2j lj znj j� �

�xn;

Fn ¼ Q nð2uðqkZðtÞkÞÞ þ Q nð2dÞ þXn�1

j¼1

oan�1

oxj

��ðQ jð2uðqkZðtÞkÞÞ þ Q jð2dÞÞ:

By taking (24) with k ¼ n� 1 into account and using (28) and (29) and �f , (27) can be rewritten as

_Vn 6 �Xn�1

i¼1

kiz2i þ

Xn�1

i¼1

12

a2i gM þ e2

i

� �þXn

i¼1

dbi þXn�1

i¼1

g0

ri

~hiri

2a2i

z2i ST Zið ÞS Zið Þ � _hi

� �þXn

i¼2

zi xi �Xn

j¼1

oai�1

ohj

_hj

!

þ zn�f n þ gnu� �

: ð30Þ

For the last term in the equation above, by taking u ¼ �g�1n ð�f n ¼ ðkn þ 0:5g2

nÞznÞ, we have:

zn�f n þ gnu� �

¼ � kn þ 0:5g2n

� �z2

n þ zngn u� uð Þ: ð31Þ

Again, a fuzzy logic system is used to model the unknown u such that for any given constant en > 0:

un ¼WTnS Znð Þ þ dn Znð Þ;

where dn indicates the approximation error and satisfies kdnk26 e2

n. Subsequently, similar to (15) and (16), the followinginequalities can be obtained:

� zngnu 61

2a2n

z2ng0hnSTðZnÞSðZnÞ þ

12

z2ng2

n þ12

a2ngM þ e2

n

� �;

zngnu 6�12a2

nz2

ng0hnSTðZnÞSðZnÞ: ð32Þ


Substituting (31) and (32) into (30) results in

_Vn 6 �Xn

i¼1

kiz2i þ

Xn

i¼1

12ða2

i gM þ e2i Þ þ

Xn

i¼1

dbi þXn

i¼1

g0

ri

~hiri

2a2i


� �þXn

i¼2

zi xi �Xn

j¼1

oai�1

ohj

_hj

!: ð33Þ

In what follows, the function xiðZiÞ will be determined so that:

Xn

i¼2

zi xi �Xn

j¼1

oai�1

ohj

_hj

!6 0: ð34Þ

From the definition of _hi, the following holds:

�Xn

i¼2

zi

Xn

j¼1

oai�1

ohj

_hj ¼Xn

i¼2

zi

Xn

j¼1

oai�1

ohj

rjhj �Xn

i¼2

zi

Xn

j¼1

oai�1

ohj

rj

2a2j

z2j ST Zj� �

S Zj� �

: ð35Þ

For the second term on the right hand side of (35), we have:

�Xn

i¼2

ziumnj¼1

oai�1

ohj

rj

2a2j

z2j ST Zj� �

S Zj� �¼ �

Xn

i¼2

zi

Xi�1

j¼1

oai�1

ohj

rj

2a2j

z2j ST Zj� �

S Zj� ��Xn

i¼2

zi

Xn

j¼i

oai�1

ohj

rj

2a2j

z2j ST Zj� �

S Zj� �

: ð36Þ

Now, note that kSðZiÞk 6 1. For the second term on the right hand side of (36), one has:

�Xn

i¼2

zi

Xn

j¼i

oai�1

ohj

rj

2a2j

z2j ST Zj� �

S Zj� �

6

Xn

i¼2

Xn

j¼i

zioai�1

ohj

�� rj

2a2j

z2j ¼

Xn

i¼2

ri

2a2i

z2i

Xi

j¼2

zjoaj�1

ohi

��: ð37Þ

Substituting (36) and (37) into (35) yields:

�Xn

i¼2

zi

Xn

j¼1

oai�1

ohj

_hj 6Xn

i¼2

zi

Xn

j¼1

oai�1

ohj

rjhj �Xi�1

j¼1

oai�1

ohj

rj

2a2j

z2j ST Zj� �

S Zj� �þ ri

2a2i

z2i

Xi

j¼2

zjoaj�1

ohi

��

( );

which implies that if:

xi Zið Þ ¼ �Xn

j¼1

oai�1

ohj

rjhj þXi�1

j¼1

oai�1

ohj

rj

2a2j

z2j ST Zj� �

S Zj� �� ri

2a2i

z2i

Xi

j¼2

zjoaj�1

ohi

��;

then (34) is true. Finally, it follows from the definition of _hi that:

Xn

i¼1

g0

ri

~hiri

2a2i


� �¼Xn

i¼1

g0ri

ri

~hihi 6Xn

i¼1

rig0

2rih2

i � ~h2i

� �: ð38Þ

Substituting (34) and (38) into (33) yields:

_Vn 6 �Xn

i¼1

kiz2i �

Xn

i¼1

rig0

2ri

~h2i þ

Xn

i¼1

12

a2i gM þ e2

i

� �þXn

i¼1

dbi þXn

i¼1

rig0

2rih2

i 6 �a0

Xn

i¼1

12

z2i þ

Xn

i¼1

g0

2ri

~h2i

!þ d0; ð39Þ

where a0 ¼minf2ki;ri : i ¼ 1;2; . . . ; ng and d0 ¼Pn

i¼112 ða2

i gM þ e2i Þ þ

Pni¼1dbi þ

Pni¼1

rig02ri

h2i .

Now take the Lyapunov function candidate as VðZðtÞÞ ¼ Vn. Then, with the assumption kZðt � sÞk 6 qkZðtÞk; q > 1, thefollowing holds:

_V ZðtÞð Þ 6 �a0V ZðtÞð Þ þ d0: ð40Þ

As a result, the uniform ultimate boundedness of all the signals in the closed-loop system follows from the RazumikhinLemma. Furthermore, (40) implies that:

V ZðtÞð Þ 6 d0

a0þ V t0ð Þe�a0 t�t0ð Þ

� �þ d0

a0e�a0ðt�t0Þ:

Especially, we have 12 z2

1 6d0a0þ Vðt0Þe�a0ðt�t0Þ

� �þ d0

a0e�a0ðt�t0Þ, which implies that:

limt!1

z21 6 2

d0

a0; ð41Þ

Remark 4. The previous analysis shows that the tracking error, i.e., z1; depends on a0 and d0. Because a0 and d0 are unknown,an explicit estimation of the tracking errors is impossible. However, it is clearly seen from the definitions of a0 and d0 thatafter for any given ri, reducing ai, ei and bi, meanwhile increasing ri, will theoretically lead to smaller tracking errors.

At the present stage, we are in the position to summarize our main result in the following theorem.


Theorem 1. Consider the system (1) with Assumptions 1–3. Suppose that for 1 6 i 6 n� 1, the packaged unknown functionsai and u can be approximated by the fuzzy logic systems in the sense that the approximating errors are bounded. Then under theaction of the adaptive fuzzy controller (5) together with the intermediate virtual control signals (4) and the adaptive laws (6), allthe signals of the closed-loop system remain bounded and the tracking error satisfies (41).

4. Simulation

In this section, four examples are used to illustrate the effectiveness of the method developed in this paper. In all the fourexamples, the reference signal is chosen as yd ¼ 0:5ðsinðtÞ þ sinð0:5tÞÞ. The proposed adaptive fuzzy controller will be used tocontrol the systems to track the given reference signal.

Example 1. Consider the following 2nd-order nonlinear system proposed in [7].

_x1 ¼ x1e�0:5x1 þ 1þ x21

� �x2 þ sin x1 s1ð Þð Þ;

_x2 ¼ x22x1 þ 3þ cos x1ð Þð Þuþ x1 s2ð Þx2 s2ð Þ:

In [7], stabilization was addressed for the case of s1 and s2 being constants. Here, we consider s1 and s2 as time-varyingdelays. For the tracking problem, the system’s output is set to y ¼ x1. Because the existing adaptive fuzzy control approachesdeal with only the case of constant delays, they are not suitable to control this system. On the other hand, according to The-orem 1, the control laws (5) and adaption laws (6) can be used to control this system. To this end, the design parameters arechosen as a1 ¼ 0:1; a2 ¼ 0:075; r1 ¼ r2 ¼ 5 and r1 ¼ r2 ¼ 0:005. In the simulation studies, seven fuzzy sets:

F1i ¼ ðNLÞ; F2

i ¼ ðNMÞ; F3i ¼ ðNSÞ; F4

i ¼ ðZEÞ; F5i ¼ ðPSÞ; F6

i ¼ ðPMÞ; F7i ¼ ðPLÞ

are defined over interval [�1.5 1.5] for each variable, where NL, NM, NS, ZE, PS, PM, and PL denote negative large, negativemedium, negative small, zero, positive small, positive medium, and positive large, respectively. By choosing the center pointsas �1.5, �1, �0.5, 0, 0.5, 1, 1.5, respectively, the fuzzy membership functions are given as follows for i ¼ 1;2:

lF1i¼ exp

�ðxi þ 1:5Þ2

2

" #; lF2

i¼ exp

�ðxi þ 1Þ2

2

" #;

lF3i¼ exp

�ðxi þ 0:5Þ2

2

" #; lF4

i¼ exp

�ðxi þ 0Þ2

2

" #;

lF5i¼ exp

�ðxi � 0:5Þ2

2

" #; lF6

i¼ exp

�ðxi � 1Þ2

2

" #;

lF7i¼ exp

�ðxi � 1:5Þ2

2

" #: ð42Þ

They are shown in Fig. 1.

Remark 5. The reason of choosing the Gaussian-shaped membership functions is due to the universal approximationtheorem, i.e., Lemma 1. According to Lemma 1, with a sufficient number of the Gaussian-shaped membership functions, any

−1.5 −1 −0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

NL NM NS ZE PS PM PL

Fig. 1. The fuzzy rule bases over ½�1:5;1:5�.

0 10 20 30 40 50 60−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time(Sec)

Fig. 2. y (solid line) and yd (dash-dotted line) for Example 1.

0 10 20 30 40 50 60−1

−0.5

0

0.5

1

1.5

2

Time(Sec)

Fig. 3. Control input curve for Example 1.

0 10 20 30 40 50 60−1.5

−1

−0.5

0

0.5

1

1.5

2

Time(Sec)

Fig. 4. x2 for Example 1.


0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time(Sec)

Fig. 5. h1 (solid line) and h2 (dash-dotted line) for Example 1.


nonlinear function can be well approximated by a fuzzy logic system WT SðxÞ. The interval for the state variable x1 isconsidered to be �1.5, 1.5 since �1:5 6 yd 6 1:5. To simplify the simulation, this interval is also used for the other statevariables. As shown in Fig. 1, with seven membership functions, each universe of discourse of the variables can be wellcovered.

The simulation is run under the initial conditions xð#Þ ¼ ½0:250�T ;�3:5 6 # 6 0; h1ð#Þ ¼ h2ð#Þ ¼ ½00�T ;�3:5 6 # 6 0and for time-varying delays s1ðtÞ ¼ 2þ 1:5 sinðtÞ and s2ðtÞ ¼ 1:5� 1:5 cosð2tÞ. The simulation results are shown in Figs.2–5.

Example 2. Consider the following 3rd order nonlinear system.

_x1 ¼ x1e�0:5x1 þ 1þ x21

� �x2 þ 2x1 t � s1ð Þ2;

_x2 ¼ x22x1 þ 3þ cos x1x2ð Þð Þx3 þ 0:5x2 t � s2ð Þ sin x1 t � s2ð Þð Þ;

_x3 ¼ x1x2x3 þ 2þ sin x1x2x3ð Þð Þuþ x2 t � s3ð Þx3 t � s3ð Þ;y ¼ x1;

According to Theorem 1, the corresponding design parameters are chosen as a1 ¼ 0:15; a2 ¼ 0:05; a3 ¼ 0:015; r1 ¼5; r2 ¼ 8; r3 ¼ 10;r1 ¼ r2 ¼ r3 ¼ 0:05. In the simulation studies, we consider the fuzzy sets defined in Example 1. The delayterms are taken as s1ðtÞ ¼ 1� sinðtÞ; s2ðtÞ ¼ 1:5þ 1:5 sinðtÞ and s3ðtÞ ¼ 1:5� 1:5 cosð2tÞ. The simulation is run under the ini-tial conditions xð#Þ ¼ ½0;0;0�T ; h1ð#Þ ¼ h2ð#Þ ¼ h3 ¼ ½000�T for �3 6 # 6 0. Figs. 6–9 display the simulation results.

Example 3. Consider the following Brusselator model [6,31], which is a simplified model describing a certain set of chemicalreactions.

_ 2
x1 ¼ A� Bþ 1ð Þx1 þ x1x2 þ D1 x1; tð Þ;_x2 ¼ Bx1 � x2
1x2 þ 2þ cos x1ð Þuþ D2 x1; x2; tð Þ; ð43Þy ¼ x1;

where x1 and x2 stand for the concentrations of the reaction intermediates, A;B > 0 are parameters describing the constantsupply of ‘‘reservoir” chemicals, D1 and D2 denote the modeling errors, and u is control input. Since in the practical chemicalreactions, the existence of time-delays is inevitable, therefore, the Brusselator model is assumed to have the following form:

_x1 ¼ A� Bþ 1ð Þx1 þ x21x2 þ D1 x1; tð Þ þ h1 x1 t � s1ð Þð Þ;

_x2 ¼ Bx1 � x21x2 þ 2þ cos x1ð Þuþ D2 x1; x2; tð Þ þ h1 x1 t � s1ð Þ; x2 t � s2ð Þð Þ; ð44Þ

y ¼ x1;

where h1 and h2 denote the unknown delayed terms in the practical chemical reactions. As done in [6],f1ðx1Þ ¼ A� Bðþ1Þx1; g1ðx1Þ ¼ x2

1; f2ðx1; x2Þ ¼ Bx1 � x21x2, and g2 ¼ 2þ cosðx1Þ are assumed to be unknown, and x1–0. In the

simulation, choose D1 ¼ 0:7x21costð1:5tÞ;D2 ¼ 0:5ðx2

1 þ x22Þ sin3 t;h1 ¼ 2x2

1ðt � s1Þ and h2 þ 0:2x2ðt � s2Þ sinðx2ðt � s2ÞÞ withs1 ¼ 1s and s2 ¼ 2s. The reference signal is yd ¼ 3þ sinð0:5tÞ þ 0:5 sinð1:5tÞ. The simulation is carried out with the systemparameters A ¼ 1;B ¼ 3, the initial condition xð#Þ ¼ ½2:5;1;0;0�T and the design parameters a1 ¼ 0:15; a2 ¼ 0:05; r1 ¼r2 ¼ 8 and r1 ¼ r2 ¼ 0:15. Figs. 10–13 show the simulation results.

0 10 20 30 40 50 60−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time(sec)


0 10 20 30 40 50 60−50

−40

−30

−20

−10

0

10

20

30

40

50

Time(sec)


0 10 20 30 40 50 60−2

−1

0

1

2

3

4

Time(sec)

Fig. 8. x2 (solid line) and x3 (dash-dotted line) for Example 2.


0 10 20 30 40 50 600

0.5

1

1.5

Time(sec)

Fig. 9. h1 (solid line), h2 (dash-dotted line) and h3 (dotted line) for Example 2.


Example 4. A one-link manipulator with motor dynamics is described by

D€qþ B _ðqÞ þ N sinðqÞ ¼ h;

M _hþ Hh ¼ u� Km _qþ sd; ð45Þ

where q; _q; €q stand for the link position, velocity and acceleration, respectively. h and _h are the motor shaft angle and velocity.sd refers to the delay term of h. u is the control input representing the motor torque. By setting:

x1 ¼ q; x2 ¼ _q; x3 ¼ h;

the system (45) can be rewritten in the following strict-feedback form:

_x1 ¼ x2;

_x2 ¼ �BD

x2 �ND

sinðx1Þ þ1D

x3;

_x3 ¼ �HM

x3 �Km

Mx2 þ

uMþ sd

M;

with the appropriate units, the parameter values are given by D ¼ 1;M ¼ 0:05;B ¼ 1;Km ¼ 10;H ¼ 0:5 and N ¼ 10. In thesimulation, the delay term is chosen as sd ¼ x3ðt � sðtÞÞ sinðx3ðt � sðtÞÞÞ. In practice, the selection of fuzzy sets may affectthe performance of the designed controller considerably. According to [32], the fuzzy sets for each variable defined by(42) are used for this system. Noting that the first subsystem is a linear differential equation, the adaptive fuzzy controllerand adaptive laws can be constructed as follows:

0 10 20 30 40 50 601.5

2

2.5

3

3.5

4

4.5

Time(sec)


0 10 20 30 40 50 60−70

−60

−50

−40

−30

−20

−10

0

10

20

Time(sec)


0 10 20 30 40 50 60−2

−1.5

−1

−0.5

0

0.5

1

1.5

Time(sec)

Fig. 12. x2 for Example 3.

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time(sec)

Fig. 13. h2 (solid line), h3 (dash-dotted line) for Example 3.


0 10 20 30 40 50 60−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time(Sec)


0 10 20 30 40 50 60−10

0

10

20

30

40

50

Time(Sec)


0 10 20 30 40 50 60−10

−5

0

5

10

15

20

Time(Sec)

Fig. 16. x2 (solid line) and x3 (dash-dotted line) for Example 4.


0 10 20 30 40 50 600

2

4

6

8

10

12

14

Time(Sec)

Fig. 17. h2 (solid line), h3 (dash–dotted line) for Example 4.


a1 ¼ �k x1 � ydð Þ þ _yd;

a2 ¼ �1

2a22

h2 x2 � a1ð ÞST2S2;

u ¼ � 12a2

3

h3 x3 � a2ð ÞST3S3;

_h2 ¼r2

2a22

x2 � a1ð Þ2ST2S2 � r2h2;

_h3 ¼r3

2a23

x3 � a2ð Þ2ST3S3 � r3h3;

where the design parameters are: k ¼ 6; a2 ¼ a3 ¼ 0:35; r2 ¼ r3 ¼ 15;r2 ¼ r3 ¼ 0:05. In the simulation studies, the delayterm is taken as sðtÞ ¼ 1:5� 1:5 cosð2tÞ. The simulation is run under the initial conditions xð#Þ ¼ ½0;0;0�T ; h2ð#Þ ¼ h3 ¼½0;0�T for �3 6 # 6 0. Figs. 14–17 display the simulation results.

Note that the time-varying delays are considered in the above examples. The existing adaptive fuzzy controllers andadaptive neural controllers are not suitable for controlling these systems. From Figs. 2, 6 and 10, it can be seen that the pro-posed controller results in a satisfactory transient tracking performance after about a adaption period of 5 seconds. The otherfigures display the boundedness of the system state, the control input and the estimates of the parameters in the controlloop. By extensive simulations, it is found that the tracking performance can be further improved by appropriately increasingri and diminishing ai, at a cost of larger control gains. Theoretically, ai and ri can be chosen arbitrarily small and ri arbitrarilylarge. However, this is not the case in real implementation due to the limited actuator tolerance and computational capacity.

5. Conclusion

In this paper, the problem of adaptive fuzzy tracking control has been addressed for a class of nonlinear strict-feedbacksystems with time-varying delays. Based on the Razumikhin lemma, an adaptive fuzzy controller has been constructed. Theproposed controller ensures that all the signals of the resulting closed-loop system are bounded and the tracking error con-verges to a small neighborhood of the origin. A main feature of the proposed fuzzy control design scheme is that the pro-posed controller contains fewer adaption parameters and can handle systems with time-varying delays.

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Direct adaptive fuzzy control for nonlinear systems with time-varying delays

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