Observer-based fuzzy adaptive robust control of nonlinear systems with time delays and unmodeled dynamics

Neurocomputing 74 (2010) 369–378

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

Observer-based fuzzy adaptive robust control of nonlinear systems with time delays and unmodeled dynamics$ Shaocheng Tong n, Yongming Li Department of Mathematics, Liaoning University of Technology, Jinzhou 121001, China

a r t i c l e in fo

abstract

Article history: Received 28 October 2009 Received in revised form 12 January 2010 Accepted 24 March 2010 Communicated by H. Zhang Available online 4 May 2010

In this paper, an adaptive fuzzy output feedback backstepping control approach is developed for a class of nonlinear time-delay systems with unmeasured states and unmodeled dynamics. Using fuzzy logic systems to approximate unknown nonlinear functions, a fuzzy state observer is designed for estimating the unmeasured states. By combining adaptive backstepping technique with adaptive fuzzy control design and using changing supplying function idea, an observer-based adaptive fuzzy backstepping control approach is developed. It is proved that the proposed adaptive fuzzy control approach is able to guarantee that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded (SUUB) and the output of the controlled system converges to a small neighborhood of the origin. Simulation results are provided to show the effectiveness of the proposed approach. & 2010 Elsevier B.V. All rights reserved.

Keywords: Nonlinear time-delay systems Unmodeled dynamics Fuzzy logic systems State observer Adaptive output feedback control Backstepping design Stability analysis

1. Introduction In recent years, with the development of backstepping technique, adaptive fuzzy control theory and methods [1–8], many approximator-based adaptive fuzzy backstepping controllers have been developed for nonlinear systems in strict-feedback form [9–16]. Adaptive fuzzy backstepping controllers can provide a systematic methodology of solving the tracking or regulation problems, in which fuzzy logic systems are used to approximate the nonlinear uncertainties, and adaptive fuzzy controllers are constructed recursively. The main features of these adaptive approaches are that they can deal with some nonlinear systems without satisfying the matching conditions, and do not require that the nonlinear uncertainties must be linearly parameterized [9,10,12–15]. Therefore, approximator-based adaptive fuzzy backstepping control has become one of the most popular design approaches in fuzzy control community. In view of time delay frequently occurring in real engineering systems, the stability analysis and robust control for nonlinear time-delay systems have attracted considerable attention in recent years. It is well known that time delays may destroy the

$ This work was supported by the National Natural Science Foundation of China (No. 60674056). Outstanding Youth Funds of Liaoning Province (No. 2005219001) and Educational Department of Liaoning Province (No. 2006R29 and No. 2007T80). n Corresponding author. Tel.: + 86 416 4199101; fax: +86 416 4199415. E-mail address: [email protected] (S. Tong).

0925-2312/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2010.03.018

stability or degrade the performance of the controlled systems. Therefore, the stability analysis and controller synthesis of nonlinear time-delay systems are important both in theory and applications. By using the Lyapunov–Razumikhin functions or the Lyapunov–Krasovskii functions, many adaptive backstepping stabilizing control schemes are developed in [17–20] for nonlinear time-delay systems with parametric uncertainties. To deal with completely unknown nonlinear systems with time delays, several approximation-based adaptive fuzzy controllers have been reported in [21–24]. The proposed adaptive control approaches can guarantee that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded (SUUB), and the tracking error converges to a neighborhood of the origin. However, the adaptive fuzzy methods mentioned above are all based on the assumption that the states of the controlled systems are measured directly. As what authors stated in [5,7,8,25], in practice, the state variables are often unmeasured for many nonlinear systems. Therefore, the existing approaches cannot be implemented for the strict-feedback nonlinear systems with time delays and states unmeasured. Recently, observer-based adaptive fuzzy backstepping output feedback controllers have been developed in our works [26,27] for SISO uncertain strict-feedback uncertain nonlinear systems, but the proposed adaptive control approaches did not consider the problems of the nonlinear systems with time delays. To the best of the authors’ knowledge, few results are available for the strict-feedback nonlinear systems with unknown time delays, unmeasured states and the unmodeled dynamics.

370

S. Tong, Y. Li / Neurocomputing 74 (2010) 369–378

Motivated by the above observations, in this paper, an observer-based adaptive fuzzy backstepping output feedback control approach is proposed for a class of SISO strict- feedback nonlinear systems with time-delay, the unmodeled dynamics and without the measurements of the states. Fuzzy logic systems are used to model the unknown nonlinear system, and a state observer is designed for estimating the unmeasured states on the basis of the fuzzy modeling. Combining backstepping technique with adaptive fuzzy control designs and using the changing supply function idea, a stable observer-based adaptive fuzzy output feedback controller is developed. It is proved that the proposed adaptive fuzzy control approach is able to guarantee that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded (SUUB) and the output of the controlled system converges to a small neighborhood of the origin. Therefore, this paper has expanded the existing results of [21–24] and our previous works of [26,27].

2. Preliminaries 2.1. Problem formulations

1

where xi(t)AR, u(t)AR, and y(t)AR are the state variable, control input, and output of system. z(t)ARa represents the unmodeled dynamics. di(t) is the time-varying delay satisfying d_ i ðtÞ r di o1; here, di is positive scalars. xi ðtÞ ¼ ½x1 ðtÞ, x2 ðtÞ,    , xi ðtÞT , xðtÞ ¼ ½x1 ðtÞ, x2 ðtÞ,    , xn ðtÞT . F i ðUÞ and Hi(U) are unknown smooth functions. Some assumptions similar to references [2,3,20,28,29] are imposed on system (1) as follows: Assumption 1. [2,3,28]: There exists an input-to-state practically stable (ISpS) Lyapunov function U0(t,z) for the z-subsystem in (1) with y(t) as input. Namely, there are class KN functions a , a, a0 and g satisfying the following conditions:

gðtÞ t2

o þ 1 and limþ sup t-0

ki ðtÞ o þ1: a0 ðtÞ

2.2. Fuzzy logic systems A fuzzy logic system (FLS) consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on fuzzy rules, and the defuzzifier. The knowledge base for FLS comprises a collection of fuzzy If-then rules of the following form: R1 : if x1 is F1l and x2 is F2l and . . . xn is Fnl , Then y is Gl ,

l ¼ 1,2,. . .,N

where x¼(x1,y,xn)T and y are the fuzzy logic system input and output, respectively. Fil and Gl are fuzzy sets associating with the fuzzy functions mF l ðxi Þand mGl(y), respectively. N is the rule i number. Through singleton function, center average defuzzification and product inference [4], a fuzzy logic system can be expressed as n Q yl mFil ðxi Þ # yðxÞ ¼ l ¼ 1 " i ¼ 1 N n P Q mF l ðxi Þ i

i¼1

l¼1

ð5Þ

where yl ¼ maxmGl ðyÞ. yAR

Define fuzzy basis functions as n Q

jl ¼

i¼1

mF l ðxi Þ

N P

n Q

l¼1

i¼1

i

!

ð6Þ

mF l ðxi Þ i

T

Denoting y ¼ ½y1 , y2 , . . ., yN  ¼ ½y1 , y2 , . . ., yN  and S(x)¼[j1(x), y,jN(x)]T, then fuzzy logic system (5) can be rewritten as T

yðxÞ ¼ y SðxÞ

ð7Þ

Lemma 1. [4]: Let f(x) be a continuous function defined on a compact set O. Then for any e 40, there exists a fuzzy logic system like (7) such as T

sup9f ðxÞy SðxÞ9 r e

ð8Þ

xAO

2.3. Fuzzy state observer design ð2Þ

where t is a positive scalar. Assumption 2. [28]: There exists a known constant ri, 1rirn, such that 9Fi ðxi ÞFi ðx^ i Þ9 r ri Jxi x^ i J

lim sup

t-0 þ

N P

The class of uncertain nonlinear systems with time-delay and unmodeled dynamics to be studied in this paper is described by the following differential equations 8 z_ ðtÞ ¼ qðzðtÞ,xðtÞÞ > > > > _ > > < x 1 ðtÞ ¼ x2 ðtÞ þ F1 ðx1 ðtÞÞ þ H1 ðzðtÞ, xðtd1 ðtÞÞÞ x_ i ðtÞ ¼ xi þ 1 ðtÞ þ Fi ðxi ðtÞÞ þHi ðzðtÞ, xðtdi ðtÞÞÞ, i ¼ 2,. . .,n1 ð1Þ > > > x_ n ðtÞ ¼ uðtÞ þ Fn ðxn ðtÞÞ þ Hn ðzðtÞ, xðtdn ðtÞÞÞ > > > : yðtÞ ¼ x ðtÞ

a ðJzðtÞJÞ rU0 ðt,zðtÞÞ r aðJzðtÞJÞ   @U0 ðt,zðtÞÞ @U0 ðt,zðtÞÞ þ q ra0 ðJzðtÞJÞ þ g 9yðtÞ9 þ t @t @z

Assumption 4. [3,28,29]: The functions a0 in (2) , ki and g in (4) satisfy the following local property

ð3Þ

where x^ i is the estimation of xi and Jxi x^ i J is the 2- norm of vector xi x^ i .

In this section, assume that Fi ðx^ i ðtÞÞ in system (1) is an unknown smooth function, and the states are unmeasured. We T utilize fuzzy logic system F^ i x^ i ðtÞ ¼ yi Si ðx^ i Þ to approximate   continuous function Fi x^ i ðtÞ , and design a state observer to estimate the unmeasured states. Based on Lemma 1, assume that unknown function Fi ðx^ i ðtÞÞ can be expressed as   T Fi x^ i ðtÞ ¼ yi Si ðx^ i Þ þ ei , 8x^ i A Ox^  Rni ð9Þ i



Assumption 3. [20,28]: Time-varying nonlinear functions Hi(z(t),x(t di(t))) satisfy the following inequalities for i¼1,2,y,n, 2

9Hi ðzðtÞ, xðtdi ðtÞÞÞ9 r ki ðzðtÞÞ þ yðtdi ðtÞÞHi1 ðyðtdi ðtÞÞÞ þ ei

ð4Þ

where ei is a positive scalar, ki(U) and Hi1(U) are known functions with ki(0) ¼0 and Hi1(0) ¼0.

where yi is an ideal constant parameter and ei is the minimum approximation error [4,5,8], which are defined as 9 = T ^ ^ sup 9Fi ðxi Þyi Si ðxi Þ9 ; yi A R i : ^ x AO

 ^ ei ¼ Fi ðx^ i ðtÞÞyT i Si ðxi ðtÞÞ yi ¼ arg minn

8 <

i

i

S. Tong, Y. Li / Neurocomputing 74 (2010) 369–378

    T Letdi ¼ Fi x^ i ðtÞ yi Si x^ i ðtÞ , wi ¼ ei di , and we can have the following assumption: Assumption 5. There exist known constants ei0 and di0, i¼1,2,yn, such that 9ei9 r ei0 and 9di9r di0. From Assumption 5, we have 9wi 9 r9ei 9 þ 9di 9 r ei0 þ di0 ¼ wi0 . Using fuzzy logic system (9), the state observer is designed as   8   > < x_^ i ðtÞ ¼ x^ i þ 1 ðtÞ þ F^ i x^ i ðtÞ þ ki x1 ðtÞx^ 1 ðtÞ , i ¼ 1,. . .,n1   ð10Þ   > : x^_ n ðtÞ ¼ uðtÞ þ F^ n x^ n ðtÞ þ kn x1 ðtÞx^ 1 ðtÞ T where F^ i ðx^ i ðtÞÞ ¼ yi Si ðx^ i ðtÞÞ(i¼1,y,n) , ki is a parameter ,which is chosen to satisfy the following matrix inequality: ! n X 1 r2i þ n$ þ W þ d1 I oQ AT P þ PAþ 2WPP T þ W ð11Þ i¼1

where P and Q are positive-definite matrices, W, $ and d1 are positive design scalars, I is a unit matrix, and matrix A is defined as " # Kn1 In1 A¼ kn 0

371

(SUUB) and the output y(t) converges to a small neighborhood of the origin. For system (13), we introduce the following coordinate transformation: v1 ðtÞ ¼ yðtÞ vi ðtÞ ¼ x^ i ðtÞai1 ðt,yðtÞ, y1 ðtÞ, . . ., yi1 ðtÞ, x^ 1 ðtÞ, . . ., x^ i1 ðtÞÞ,

i ¼ 2, . . ., n

ð15Þ where functions ai  1(U)are the virtual control inputs, which will be defined later. Based on the backstepping design procedures, an adaptive fuzzy backstepping control scheme is given. The detailed design procedures are described as follows: Step 1: From system (13), we have _ ¼ v_ 1 ðtÞ ¼ x2 ðtÞ þF1 ðx1 Þ þ H1 ðzðtÞ,xðtd1 ðtÞÞÞ yðtÞ ¼ e2 ðtÞ þ x^ 2 ðtÞ þF1 ðx1 Þ þ H1 ðzðtÞ,xðtd1 ðtÞÞÞ Taking x^ 2 as a virtual control, and define v2 ¼ x^ 2 a1

ð17Þ

Substituting (17) into (16) yields v_ 1 ðtÞ ¼ e2 ðtÞ þv2 þ a1 þ F1 ðx1 Þ þH1 ðzðtÞ, xðtd1 ðtÞÞÞ

T

where Kn  1 ¼[k1,y,kn  1] . Remark 1. It should be pointed out that inequality (11) is a sufficient condition to guarantee the stability of the closed-loop system, which will be used for the adaptive fuzzy controller design and stability analysis later. To solve (11), we decompose A ¼ A þkD with     T 0 In1 A¼ , k ¼ k1    kn , and D ¼ 1 0    0 :: 0 0 Then (11) can be rewritten as the following LMI [30]: 3 E P 4 1 5 o0 I P  2W

ð12Þ

where T 1 E ¼ A P þ PA þDT NT þ ND þ ðW Sni¼ 1 r2i þ n$ þ W þ d1 ÞI þQ , N ¼ Pk. By solving LMI (12), one can obtain P, N and k. Furthermore, the matrix k can be computed as k¼P  1N. Combining (1) and (10), we obtain a composite system 8_ z ðtÞ ¼ qðzðtÞ,xðtÞÞ > >   > > ~ ðzðtÞ, xðtd1 ðtÞÞÞ, . . ., xðtdn ðtÞÞ > _ ¼ AeðtÞ þ F ðxn ðtÞÞF^ x^ n ðtÞ þ H > eðtÞ > > > < _ ¼ x2 ðtÞ þF1 ðx1 Þ þ H1 ðzðtÞ,xðtd1 ðtÞÞÞ yðtÞ     > _ > ^ > xi ðtÞ ¼ x^ i þ 1 ðtÞ þ F^ i x^ i ðtÞ þ ki x1 ðtÞx^ 1 ðtÞ , i ¼ 2,. . .,n1 > > >   >   > > : x^ n ðtÞ ¼ uðtÞ þ F^ n x^ n ðtÞ þ kn x1 ðtÞx^ 1 ðtÞ

ð18Þ

Consider the following Lyapunov function: T

V1 ¼ eT Pe þ v21 þ m1 y~ 1 y~ 1 þ W1 þ W0 Rt  d2 where y~ 1 ¼ y1 y1 ,W0 ¼ 1d v1 ðxÞH11 ðv1 ðxÞÞdx,  1

Rt

2W1 1d1

ð19Þ

td1 ðtÞ

W1 ¼ v1 ðxÞH11 ðv1 ðxÞÞdx, m 40 and d2 4 0z are design constants. td1 ðtÞ From (13), we have _ ¼ AeðtÞ þ eðtÞ

2

ð16Þ

n h      i X T ~ Bi Fi ðxi ðtÞÞFi x^ i ðtÞ þ Fi x^ i ðtÞ yi Si x^ i ðtÞ þ H i¼1

~ þd ¼ AeðtÞ þ F~ þ H

ð20Þ

h  iT   where F~ ¼ F1 ðx1 ðtÞÞF1 x^ 1 ðtÞ , . . ., Fn ðxn ðtÞÞFi x^ n ðtÞ , d ¼ ½d1 , h       T   T T ^ ^ . . ., dn  ¼ F1 x^ 1 ðtÞ y1 S1 x^ 1 ðtÞ , . . ., Fn xn ðtÞ yn Sn xn ðtÞ T The time derivative of V1 along (13), (18) and (20) is T _ 1 þW _ 0 _ T PeðtÞ þeðtÞT P eðtÞ _ þ2v1 ðtÞv_ 1 ðtÞ2m1 y~ 1 ðtÞy_ 1 ðtÞ þ W V_ 1 ¼ eðtÞ

~ þ dÞ þ 2v1 ðtÞðv2 ðtÞ þ a1 ¼ eðtÞT ðAT P þPAÞeðtÞ þ 2eðtÞT PðF~ þ H     þF1 ðx1 ðtÞÞF1 x^ 1 ðtÞ þF1 x^ 1 ðtÞ þ 2v1 ðtÞðe2 ðtÞ þ H1 Þ T _ 1 þW _ 0 2m1 y~ 1 ðtÞy_ 1 ðtÞ þ W

ð21Þ

ð13Þ Where eðtÞ ¼ xðtÞx^ ðtÞ,Fðx1 ðtÞ, x2 ðtÞ, . . ., xn ðtÞÞ ¼ ½F1 ðx1 ðtÞÞ, . . ., F1 ðxn ^ x^ ðtÞ, x^ ðtÞ, . . ., x^ ðtÞÞ ¼ ½F^ ðx^ ðtÞÞ, . . ., F^ ðx^ ðtÞÞT and HðUÞ ~ ðtÞÞT , Fð ¼ n n n 1 2 1 1 ½H1 ðUÞ, . . ., Hn ðUÞT . From (4), we have ~ 2r JHJ

n X i¼1

ki ðzðtÞÞ þ

n X

Hi1 ðyðtdi ðtÞÞÞyðtdi ðtÞÞ þ

i¼1

n X

ei

By using Young’s inequality and Assumption 2, we have 2v1 ðtÞe2 ðtÞ r

n

d1

v21 ðtÞ þ

d1 n

e22 ðtÞ

ð22Þ

ð14Þ

i¼1

2v1 ðF1 ðx1 ÞF1 ðx^ 1 ÞÞ r2Jv1 J9F1 ðx1 ÞF1 ðx^ 1 Þ9 r2r1 Jv1 JJx1 x^ 1 J r

3. Fuzzy adaptive control design In this section, by utilizing the output y(t) and the states estimates x^ i ðtÞ to determine an adaptive fuzzy controller and parameters adaptive laws, such that all the signals in the closedloop system are semiglobally uniformly ultimately bounded

2v1 ðtÞH1 r

n

d2

v21 ðtÞ þ

d2 n

1

$

r21 v21 ðtÞ þ $JeJ2

H12

where $, d1 and d2 are positive design scalars.

ð23Þ

ð24Þ

372

S. Tong, Y. Li / Neurocomputing 74 (2010) 369–378

Define

Wi ¼

W1

1di

Rt

v1 ðxÞHi1 ðv1 ðxÞÞdx

ði ¼ 1, . . ., nÞ,

it

is

tdi ðtÞ

noticed that 1

1 _ ir W W v1 ðtÞHi1 ðv1 ðtÞÞW v1 ðtdi ðtÞÞHi1 ðv1 ðtdi ðtÞÞÞ 1di

ð25Þ

Substituting (22)–(24) into (21) yields

ð28Þ

ð29Þ

where b40 is a small constant. and substituting (27)–(29) into (26) results in ~ þ dÞ V_ 1 reðtÞT ðAT P þ PAÞeðtÞ þ2eðtÞT PðF~ þ H 2 n 1 1 1X W1 Hj1 ðv1 ðtÞÞ þ 2v1 ðtÞ4v2 ðtÞ rðv1 ðtÞÞ l1 v1 ðtÞ 2 2 2j¼1

# 1 1 d1 2 d2 2 þ e ðtÞ þ H H ðv ðtÞÞ  d2 11 1 2 1d1 n 2 n 1 ð30Þ

@a1 @a1 _ @a1 _ _ yðtÞ x^ 1 ðtÞ y 1 ðtÞ @yðtÞ @y1 ðtÞ @x^ 1 ðtÞ ð31Þ

m

The time derivative of V2 along (30) and (31) is T_ _ 2 V_ 2 ¼ V_ 1 þ 2v2 v_ 2 þ2m1 y~ 2 y~ 2 þ W

~ þ dÞ r eðtÞT ðAT P þ PAÞeðtÞ þ 2eðtÞT PðF~ þ H

Using Assumption 2, we have @a1  F1 ðx1 ðtÞÞF1 ðx^ 1 ðtÞÞ @y

 r2 @a1 @a1 2 r 2r1 v2 ðtÞ þ $JeJ2 JeJ r 1 v2 ðtÞ @y $ @y

2v2 ðtÞ

ð35Þ

Substituting (35) into (34) yields @a1 _ yðtÞ @y

  r2 @a1 @a1 T x^ 2 ðtÞ 1 v2 ðtÞ þ y1 Sðx^ 1 ðtÞÞ @y 2$ @y 



n n @a1 2 d1 2 d2 2 e ðtÞ þ H þ $JeJ2 þ þ þ v2 ðtÞ @y n 2 n 1 d1 d2

 @a1 2 2 þ v2 ðtÞ þ d10 @y

r 2v2 ðtÞ

ð36Þ

~ þ dÞ V_ 2 reðtÞT ðAT P þPAÞeðtÞ þ 2eðtÞT PðF~ þ H 2 n 1 1 1X þ2v1 ðtÞ4v2 ðtÞ rðv1 ðtÞÞ l1 v1 ðtÞ W1 Hj1 ðv1 ðtÞÞ 2 2 2j¼1

# T 1 1 d1 2 d2 2  d2 e ðtÞ þ H þ $JeJ2 þ2ly~ 1 y1 ðtÞ H ðv ðtÞÞ þ 11 1 2 1d1 n 2 n 1

_ j þ 2b0 þ 2v2 v3 þ a2  @a1 y_ 1 ðtÞ @a1 x_^ 1 ðtÞ @a1 W @y1 ðtÞ @t @x^ 1 ðtÞ j¼0   2 r @a1 @a1 x^ 2 ðtÞ 1 v2 ðtÞ þ F^ 2 ðx^ 2 ðtÞÞ þk2 ðx1 ðtÞx^ 1 ðtÞÞ 2v2 ðtÞ @y 2$ @y þ

2 X

#

T

þ y1 S1 ðx^ 1 ðtÞÞ þ 29v2 w2 9þ



n

d1

þ

n

d2



 @a1 2 d1 2 d2 2 þ e2 ðtÞ þ v2 ðtÞ H @y n n 1

 h i T @ a1 2 2 þ $JeJ2 þ v2 ðtÞ þ d10 þ 2y~ 2 v2 ðtÞS2 ðx^ 2 ðtÞÞm1 y_ 2 ðtÞ @y

Consider the following Lyapunov function: 1 ~ T ~ y 2 y 2 þW2

ð34Þ

Substituting (36) into (33) results in

Step 2: From (13) and (15), differentiating v2 ¼ x^ 2 a1 yields v_ 2 ðtÞ ¼ x_^ 2 a_ 1

V2 ¼ V1 þ v22 þ

ð33Þ

@a1 _ yðtÞ @y

@a1  @a1 ½e2 ðtÞ þ H1  x^ 2 ðtÞ þ F1 ðx1 ðtÞÞ 2v2 ðtÞ @y @y @a1  x^ 2 ðtÞ þ F1 ðx1 ðtÞÞF1 ðx^ 1 ðtÞÞ ¼ 2v2 ðtÞ @y @a1 T ½e2 ðtÞ þ H1  þ y1 S1 ðx^ 1 ðtÞÞ þ d1 2v2 ðtÞ @y

2v2 ðtÞ

where l1 and l are positive design scalars and r(v1(t)) will be chosen later. Using the property of functiontanhðUÞ, i.e.,

@a1 ^ ^ þ F 2 ðx2 ðtÞÞ þ k2 ðx1 ðtÞx^ 1 ðtÞÞ @t

2

¼ 2v2 ðtÞ

The intermediate control a1 and the adaptation function y1 are chosen as 1 1 1 2 a1 ¼  rðv1 ðtÞÞ l1 v1 ðtÞ r v1 ðtÞyT1 S1 ðx^ 1 ðtÞÞ 2 2 2$ 1



 1 n n e10 v1 ðtÞ þ v1 ðtÞe10 tanh  2 d1 d2 b

 n 1X 1 1  W1 Hj1 ðv1 ðtÞÞ d2 H11 ðv1 ðtÞÞ ð27Þ  2j¼1 2 1d1



@a1 @a1 _ @a1 _ @a1 _ yðtÞ x^ 1 ðtÞ y 1 ðtÞ @yðtÞ @y1 ðtÞ @t @x^ 1 ðtÞ T_ 1 ~ ~ ^ ^ þ F 2 ðx2 ðtÞÞ þk2 ðx1 ðtÞx^ 1 ðtÞÞ þ 2m y y 2



2v2 ðtÞ

ð26Þ

¼ v3 ðtÞ þ a2 

_ j þ 2b0 þ 2v2 ½v3 þ a2 W

In addition, we have

~ þ dÞ r eðtÞT ðAT P þ PAÞeðtÞ þ 2eðtÞT PðF~ þ H   1 2 r1 v1 ðtÞ þ yT1 S1 ðx^ 1 ðtÞÞ þ 2v1 ðtÞ v2 ðtÞ þ a1 þ 2$

 n n d1 2 e ðtÞ v21 ðtÞ þ þ þ 29v1 ðtÞe1 9 þ n 2 d1 d2 h i T d2 2 _ 1 þW _ 0 H1 þ $JeJ2 þ 2y~ 1 v1 ðtÞS1 ðx^ 1 ðtÞÞm1 y_ 1 ðtÞ þ W þ n

T _ 1 þW _ 0 þ 2b0 þ $JeJ2 þ 2ly~ 1 y1 ðtÞ þ W

2 X j¼0

~ þ dÞ þ 2v1 ðtÞ½v2 ðtÞ þ a1 þ F1 ðx1 ðtÞÞF1 ðx^ 1 ðtÞÞ þ 2eðtÞT PðF~ þ H T þ y1 S1 ðx^ 1 ðtÞÞ þ e1 

 T n n d1 2 d2 2 _ 1 þW _ 0 þ e2 ðtÞ þ H1 2m1 y~ 1 ðtÞy_ 1 ðtÞ þ W þ v21 ðtÞ þ n n d1 d2

9v1 ðtÞe1 9v1 ðtÞe10 tanhðe10 v1 ðtÞ=bÞ r 0:2785b ¼ b0

n 1 1 1X þ2v1 ðtÞ4v2 ðtÞ rðv1 ðtÞÞ l1 v1 ðtÞ W1 Hj1 ðv1 ðtÞÞ 2 2 2j¼1

# 1 1 d1 2 d2 2 þ  d2 e ðtÞ þ H H ðv ðtÞÞ 11 1 2 1d1 n 2 n 1 T þ $JeJ2 þ2ly~ 1 y1 ðtÞ þ

V_ 1 ¼ eðtÞT ðAT P þ PAÞeðtÞ

y_ 1 ðtÞ ¼ mv1 ðtÞS1 ðx^ 1 ðtÞÞmly1 ðtÞ

2

ð37Þ

ð32Þ The intermediate control function a2 and the adaptation function y2 are chosen as 1 2

a2 ¼  l2 v2 ðtÞ þ

@a1 @a1 _ @a1 _ v1 ðtÞ þ x^ 1 ðtÞ y 1 ðtÞ þ @t @y1 ðtÞ @x^ 1 ðtÞ

S. Tong, Y. Li / Neurocomputing 74 (2010) 369–378

 w20 v2 ðtÞ T y2 S2 ðx^ 2 ðtÞÞk2 ðx1 ðtÞx^ 1 ðtÞÞw20 tanh b   r21 @a1 @a1 T þ x^ 2 ðtÞ þ y1 S1 ðx^ 1 ðtÞÞ v2 ðtÞ @y 2$ @y

2



 1 n n @a1 1 @a1 2  þ  v2 ðtÞ v2 ðtÞ 2 d1 d2 2 @y @y

373

y_ i ðtÞ ¼ mvi ðtÞSi ðx^ i ðtÞÞmlyi ðtÞ

ð44Þ

From (43)–(44), (42) becomes

ð38Þ

~ þ dÞ V_ i reðtÞT ðAT P þ PAÞeðtÞ þ2eðtÞT PðF~ þ H 2 n X þv1 ðtÞ4rðv1 ðtÞÞ W1 Hj1 ðv1 ðtÞÞ j¼1

y_ 2 ðtÞ ¼ mv2 ðtÞS2 ðx^ 2 ðtÞÞmly2 ðtÞ

ð39Þ

Using9v2 w2 9v2 w20 tanhðv2 w20 =bÞ r 0:2785b ¼ b0 , (37)–(39), we have

and

from



~ þ dÞ V_ 2 r eðtÞT ðAT P þ PAÞeðtÞ þ 2eðtÞT PðF~ þ H 2 n X W1 Hj1 ðv1 ðtÞÞ þ v1 ðtÞ4rðv1 ðtÞÞ d2 

2 X

j¼1

2 X

T

y~ j yj ðtÞ þ

j¼1

2 X

i X

T Vi ¼ Vi1 þ v2i þ m1 y~ i y~ i þWi

ð41Þ

~ þ dÞ r eðtÞT ðAT P þ PAÞeðtÞ þ 2eðtÞT PðF~ þ H 2 n X W1 Hj1 ðv1 ðtÞÞ þv1 ðtÞ4rðv1 ðtÞÞ j¼1

d2

~ þ dÞ r eðtÞT ðAT P þ PAÞeðtÞ þ 2eðtÞT PðF~ þ H 2 n X 1 W Hj1 ðv1 ðtÞÞ þ v1 ðtÞ4rðv1 ðtÞÞ

þðn1Þ$JeJ2 þ 2l

T

j¼1

_ j þ 2ði1Þb0 W

j¼0

i1 X @ai1

i

i

i

i

i

y_ l ðtÞ

i1 X @ai1

i

The intermediate control function ai and adaptation function yi are chosen as 1 2

ai ¼  li vi ðtÞ þ þ

i1 X @ai1 @ai1 _ vi1 ðtÞ þ y ðtÞ @t @ yl ðtÞ l l¼1

 wi0 vi ðtÞ T x_^ l ðtÞyi Si ðx^ i ðtÞÞki ðx1 ðtÞx^ 1 ðtÞÞwi0 tanh b @x^ l ðtÞ l¼1 i1 X @ai1

  r2 @a @a T þ i1 x^ 2 ðtÞ 1 vi ðtÞ i1 þ y1 S1 ðx^ 1 ðtÞÞ @y 2$ @y





 1 n n @ai1 2 1 @ai1 2  þ  vi ðtÞ vi ðtÞ 2 d1 d2 2 @y @y

n1 X @ai1

@yl ðtÞ l¼1

@a x_^ ðtÞ i1 ^ l ðtÞ l @ y ðtÞ @t @ x l l¼1 l¼1   2 r @ a @a1 þ F^ i ðx^ i ðtÞÞ þ ki ðx1 ðtÞx^ 1 ðtÞÞ 2vi ðtÞ i1 x^ 2 ðtÞ 1 vi ðtÞ @y 2$ @y # 2



n n @ a d T 1 2 þ y1 S1 ðx^ 1 ðtÞÞ þ 29vi wi 9þ e ðtÞ þ vi ðtÞ i1 þ @y n 2 d1 d2

2 d2 2 @a 2 H þ $JeJ2 þ vi ðtÞ i1 þ ði1Þd10 þ n 1 @y T _ þ 2y~ ½v ðtÞS ðx^ ðtÞÞmy_ ðtÞþ W ð42Þ þ 2vi vi þ 1 þ ai 

T

y~ j yj ðtÞ þ

ð43Þ

n1 X

_ j þ 2ðn1Þb0 W

j¼0 n1 X @ai1

@an1 @t  r2 @ a @a1 n1 x^ 2 ðtÞ 1 vn ðtÞ þ F^ n ðx^ n ðtÞÞ þkn ðx1 ðtÞx^ 1 ðtÞÞÞ2vn ðtÞ @y 2$ @y 2



i n n @an1 T þ y1 S1 ðx^ 1 ðtÞÞ þ29vn wn 9 þ þ vn ðtÞ @y d1 d2

2 d1 2 d2 2 @ a n1 e ðtÞ þ H þ $JeJ2 þ vn ðtÞ þ n 2 n 1 @y T 2 _ n þðn1Þd þ 2y~ ½vn ðtÞSn ðx^ n ðtÞÞm1 y_ n ðtÞþ W ð47Þ þ2vn u

ði1Þd1 2 ði1Þd2 2 e2 ðtÞ þ H1 n n i1 X

n1 X

j¼1

j¼1

y~ j yj ðtÞ þ

3  1 ðn1Þd2 2 7 ðn1Þd1 2 H11 ðv1 ðtÞÞ 5 þ e2 ðtÞ þ H1 1d1 n n



T_ _ i V_ i ¼ V_ i1 þ 2vi v_ i þ2m1 y~ i y~ i þ W

i1 X

ð46Þ

T_ _ n V_ n ¼ V_ n1 þ 2vn v_ n þ 2m1 y~ n y~ n þ W

Similar to step 2, the time derivative of Vi is

þ ði1Þ$JeJ2 þ2l

ð45Þ

The time derivative of Vn can be expressed as ð40Þ

þ

_ j þ 2vi vi þ 1 W

j¼0

Vn ¼ Vn1 þ v2n þ m1 y~ n y~ n þWn

j¼0

1 H11 ðv1 ðtÞÞ 1d1

j¼1

i X

Step n: In the final design step, the actual control input u appears. We consider the Lyapunov function as

Step i: (3rirn 1) a similar procedure is employed recursively at the i-step. Consider the following Lyapunov function:



T

y~ j yj ðtÞ þ

T

2

d2

i X

2

_ j þ2v2 v3 W

2$JeJ2 þ d10 þ4b0

#

lj v2j ðtÞ þ 2l

þi$JeJ2 þ ði1Þd10 þ 2ib0

# 1 2d1 2 2d2 2 e ðtÞ þ H H11 ðv1 ðtÞÞ þ 1d1 n 2 n 1

lj v2j ðtÞ þ2l

# 1 id1 2 id2 2 þ e ðtÞ þ H H ðv ðtÞÞ 11 1 1d1 n 2 n 1

j¼1

j¼1



d2



10

y_ l ðtÞ

@x^ l ðtÞ l¼1

x_^ l ðtÞ

n

Control u and adaptation functions yn are chosen as n1 n1 X X 1 @an1 @an1 _ @an1 _ u ¼  ln vn ðtÞ þ vn1 ðtÞ þ x^ ðtÞ  y l ðtÞ þ 2 @t @ y ðtÞ @x^ l ðtÞ l l l¼1 l¼1

 wn0 vn ðtÞ T yn Sn ðx^ n ðtÞÞkn ðx1 ðtÞx^ 1 ðtÞÞwn0 tanh b   r21 @an1 @an1 T þ x^ 2 ðtÞ þ y1 S1 ðx^ 1 ðtÞÞ vn ðtÞ @y 2$ @y

2



 1 n n @an1 1 @an1 2  þ  vn ðtÞ ð48Þ vn ðtÞ 2 d1 d2 2 @y @y

y_ n ðtÞ ¼ mvn ðtÞSn ðx^ n ðtÞÞmlyn ðtÞ Substituting (48) and (49) into (47) results in ~ þ dÞ V_ n reðtÞT ðAT P þPAÞeðtÞ þ2eðtÞT PðF~ þ H 2 n X W1 Hj1 ðv1 ðtÞÞ þv1 ðtÞ4rðv1 ðtÞÞ j¼1

ð49Þ

374

S. Tong, Y. Li / Neurocomputing 74 (2010) 369–378

d2 

# 1 H11 ðv1 ðtÞÞ þ d1 e22 ðtÞ þ d2 H12 1d1



n X

lj v2j ðtÞ þ2l

j¼1

n X

T

y~ j yj ðtÞ þ

j¼1

n X

Substituting (55) and (56) into (54) yields V_ n reðtÞT ðAT P þPAÞeðtÞ þ 2WeðtÞT PP T eðtÞ

_ j W

þW

j¼0

2

ð50Þ þW

r

2

T

eðtÞ PP eðtÞ þ

n W1 X

2

r2i JeJ2 þ WJeJ2 þ W1 JPdJ2 þ W1

1

n X

n X

ki ðzðtÞÞ þ d2 k1 ðzðtÞÞ

i¼1

ei þ v1 ðtÞ½rðv1 ðtÞÞ þ d1 e22 ðtÞ þ d2 e1 

i¼1 n X T þ2l y~ j yj ðtÞ þ n j¼1

Using Young-inequality and Assumptions 2–3, we have ~ W W1 ~T ~ F F eðtÞT P F r eðtÞT PP T eðtÞ þ 2 2 1 X n W W r2 Jx x^ J2 r eðtÞT PP T eðtÞ þ 2 2 i¼1 i i i T

n X i¼1

þ n$JeJ2 þðn1Þd10 þ 2nb0

W

1

n X

lj v2j ðtÞ

j¼1

$JeJ2 þ ðn1Þd210 þ 2nb0

ð57Þ

From (11), and using the inequality

2 2 i JeJ

r

ð51Þ

T  2ly~ i yi ðtÞ rlJy~ i J2 þ lJyi J2

ð58Þ

Eq. (57) can be rewritten as

i¼1

V_ n reðtÞT QeðtÞv1 ðtÞrðv1 ðtÞÞ n n n X X X  l Jy~ j J2 þ l Jyj J2  lj v2j ðtÞ þ cðzðtÞÞþ d

1 1 W 1 1 ~ 2 eðtÞT PðH~ þ dÞ r JeJ2 þ W JPdJ2 þ eðtÞT PP T eðtÞ þ W JHJ 2 2 2 2 W 2 1 1 W 2 T T r JeJ þ W JPdJ þ eðtÞ PP eðtÞ 2 2 2 n n 1 1 X 1 1 X þ W ki ðzðtÞÞ þ W ei 2 2 i¼1 i¼1

W

n 1 1 X þ W Hi1 ðv1 ðtdi ðtÞÞÞv1 ðtdi ðtÞÞ 2 i¼1

j¼1

j¼1

ð59Þ

j¼1

where

d ¼ W1

n X

ei þ d2 e1 þ W1 JPd0 J2 þ ðn1Þd210

i¼1

ð52Þ þ2nb0 cðzðtÞÞ ¼

n X

W1 ki ðzðtÞÞ þ d2 k1 ðzðtÞÞ

i¼1

If there is no unmodeled dynamics in the considered system, by choosing r(v1(t))¼ 0 and c(z(t)) ¼0, (59) becomes

By (51) and (52), we have eðtÞT ðAT P þPAÞeðtÞ þ2eðtÞT PðF~ þ H~ þ dÞ r eðtÞT ðAT P þPAÞeðtÞ þ 2WeðtÞT PPT eðtÞ n n X X 1 þW r2i JeJ2 þ WJeJ2 þ W1 JPdJ2 þ W1 ki ðzðtÞÞ 1

þW

i¼1 n X

V_ n r eðtÞT QeðtÞl

i¼1

ð53Þ

Substituting (53) into (50) results in V_ n reðtÞT ðAT P þ PAÞeðtÞ þ2WeðtÞT PP T eðtÞ n n n X X X 1 þW r2i JeJ2 þ WJeJ2 þ W1 JPdJ2 þ W1 ki ðzðtÞÞ þ W1 ei 1

þW

n X

i¼1

2

þ v1 ðtÞ4rðv1 ðtÞÞ

n X

1

W Hj1 ðv1 ðtÞÞd2

j¼1

þ d1 e22 ðtÞ þ d2 H12 

n X

lj v2j ðtÞ þ2l

j¼1

þ

n X

i¼1

Hi1 ðv1 ðtdi ðtÞÞÞv1 ðtdi ðtÞÞ

i¼1

n X

Jy~ j J2 þl

j¼1

n X 1 ei þ W1 Hi1 ðv1 ðtdi ðtÞÞÞv1 ðtdi ðtÞÞ 2 i¼1 i¼1

i¼1

n X

3  1 H11 ðv1 ðtÞÞ 5 1d1



T

y~ j yj ðtÞ

j¼1

_ j þ n$JeJ2 þ ðn1Þd2 þ 2nb0 W 10

ð54Þ

j¼0

n X

Jyj J2  

j¼1

n X

lj v2j ðtÞ þ d

ð60Þ

j¼1

It is clear that the closed-loop system is stable in the sense of semiglobal boundedness. In the sequel, we will show how to choose proper function r(v1(t)) such that the constructed output feedback controller can render the closed-loop system UUB stable. With the help of (2), one has U_ 0 ra0 ðJzðtÞJÞ þ gð9yðtÞ9Þ þ t

ð61Þ

By employing the changing supplying function idea [3,28], we define a new function as Z U0 ðt,zðtÞÞ U0 ¼ jðxÞdx ð62Þ 0

where j3aðXÞ ¼ jðaðXÞÞ, j:R + -R + is a smooth non-decreasing function such that j(x)40 and j(0) ¼0. It is clear that U 0 is a positive-definite function. Furthermore, by (62), we can obtain the time derivative of U 0 that satisfies _ r jðU ðt,zðtÞÞÞða ðJzðtÞJÞ þ gð9yðtÞ9Þ þ tÞ U 0 0 0

ð63Þ

With Assumption 1, it follows that

jðU0 ðt,zðtÞÞÞða0 ð:zðtÞ:Þ þ gð yðtÞ Þ þ tÞ Form (4) and (25), we have 2

d2 9H1 ðzðtÞ, xðtd1 ðtÞÞÞ9 r d2 k1 ðzðtÞÞ þ d2 yðtd1 ðtÞÞH1 ðyðtd1 ðtÞÞÞ þ d2 e1 ð55Þ n X j¼0

_ jr W

n n X X W1 W1 v1 ðtdj ðtÞÞHj1 ðv1 ðtdj ðtÞÞÞ  v1 ðtÞHj1 ðv1 ðtÞÞ 1d j j¼1 j¼1

þ

d2 1d1

1 ðJzðtÞJÞÞa0 ðJzðtÞJÞ r  ðj3a  2

 1 þ jðU0 ðt,zðtÞÞÞ  a0 ðJzðtÞJÞ þ gð9yðtÞ9Þ þ t 2

ð64Þ

Let us consider the following two cases. The first case is that

gð9yðtÞ9Þ r 14 a0 ðJzðtÞJÞ and t r 14 a0 ðJzðtÞJÞ. Under this case, we obtain jðU0 ðt,zðtÞÞÞða0 ðJzðtÞJÞ þ gð9yðtÞ9Þ þ tÞ

v1 ðtÞH11 ðv1 ðtÞÞd2 v1 ðtd1 ðtÞÞH11 ðv1 ðtd1 ðtÞÞÞ ð56Þ

1 r  ðj3a ðJzðtÞJÞÞa0 ðJzðtÞJÞ  2

ð65Þ

S. Tong, Y. Li / Neurocomputing 74 (2010) 369–378

The second case is that gð9yðtÞ9Þ 4 14 a0 ðJzðtÞJÞ t 4 14 a0 ðJzðtÞJÞ. Under this case, it is obtained that

and

jðU0 ðt,zðtÞÞÞða0 ðJzðtÞJÞ þ gð9yðtÞ9Þ þ tÞ 1 ðJzðtÞJÞÞa0 ðJzðtÞJÞ r ðj3a  2 þ ðj3a3a1 0 3ð4gð9yðtÞ9ÞÞÞgð9yðtÞ9Þ

þ ðj3a3a1 0 34tÞt

ð66Þ

Based on (65) and (66), (66) always holds for the two cases. If the functions do not satisfies the previous two cases, it is easy to check that (66) still holds. Furthermore, one can obtain 1 _ ðJzðtÞJÞÞa0 ðJzðtÞJÞ U 0 r ðj3a  2 1 ð67Þ þ ðj3a3a0 3ð4gð9yðtÞ9ÞÞÞgð9yðtÞ9Þ þ ðj3a3a1 0 34tÞ4t Based on Assumption 4, there exists a desired function

j(U)such that 1 ðj3aðJzðtÞJÞÞa0 ðJzðtÞJÞ Z cðJzðtÞJÞ ð68Þ 4 Similarly, we can select a desired function rðv1 Þ such that v1 rðv1 Þ Z ðj3a3a1 0 3ð4gð9yðtÞ9ÞÞÞgð9yðtÞ9Þ

ð69Þ

For the whole system, choose the following Lyapunov function: V ¼ Vn þU 0

ð70Þ

375

decreasing d1 , $, b and l. It is noted that if l is too small, it may not be enough to prevent the parameter estimates from drifting. If m, d1 , $ and b are small, the control energy is big. Therefore, in practical applications, the design parameters should be adjusted carefully for achieving suitable transient performance and control action. From the previous synthesis, we can obtain the following controller design procedure. Step 1): From z-subsystem and Assumption 1, one obtains U0(z(t)),aðJzðtÞJÞ,aðJzðtÞJÞ, a0 ðJzðtÞJÞ, g(9y(t)9) and t. Step 2): Based on system (1) and Assumption 3, determine functions Hi1(U)and ki(U). Step 3): Define fuzzy IF-THEN rules and membership functions, and determine fuzzy basis functions jl, and establish the fuzzy logic systems F^ i ðx^ i ðtÞÞ, then, construct state observer (10). Step 4): Specify a positive-definite matrix Q select appropriately design parameters ri, W, $ and d1 and solve LMI (12), obtain observer gain matrix k. Step 5): Select appropriately design parameters l1 40, d2 4 0, b40, m 40 and l 40, according to (68) and (69), determine design functions j(U) and r(v1(t)), then obtain the intermediate control function a1 (27) and adaptation function y1 (28). Step 6): Select appropriately design parameters li 40 and compute the partial derivations of the design function ai  1, then obtain the intermediate control function ai(43) and adaptation function yi(44) (i¼2,y,n), and finally obtain u (48).

Then the time derivative of V along the composite system (15) is 4. Simulation example

_ V_ ¼ V_ n þ U 0 r eðtÞT QeðtÞ

n X

lJy~ i J2 

i¼1 n P

where d ¼ i¼1 that as t-N JeðtÞJ r

n X

1 4

li v2i ðtÞ ðj3a ðJzðtÞJÞÞa0 ðJzðtÞJÞ þ d 

i¼1

ð71Þ  lJyi J2 þ d þð

d lmin ðQ Þ

j3a3a

!1=2

tÞ4t. From (65), we obtain

1 0 34

!1=2

,

Jy~ i J r

,

x_ 1 ðtÞ ¼ x2 ðtÞ þ x1 sinðx21 ðtÞÞ þ x_ 2 ðtÞ ¼ uðtÞ þ

d l

where lmin(Q) denotes the minimum eigenvalue of matrix Q and !1=2

d 9vi ðtÞ9 r li

The nonlinear system with time-delay is considered z_ ðtÞ ¼ zðtÞ þx1 ðtÞsinx2 ðtÞ

ðj3a ðJzðtÞJÞÞa0 ðJzðtÞJÞ r4d 

Note the boundness of v1(t). We obtain that the output signal y(t) is bounded. With the boundness of e1(t), one obtains that x^ 1 ðtÞ is bounded. Consider that with the boundedness of virtual control input a1(U), we obtain that x^ 2 ðtÞ is bounded based on (15). Via the recursive method, one can obtain that x^ 2 ðtÞand x^ 3 ðtÞ, . . ., x^ n ðtÞare bounded, According to the boundedness of error e(t) and the observer statex^ ðtÞ, we obtain that the state x(t) is bounded. Based on the previous analysis, it is obtained that the resulting closed-loop system is stable in the sense of semiglobal boundedness. The above analysis is summarized in the following theorem: Theorem:. For system (1), under Assumptions 1–5, after the application of the above design procedures, the proposed adaptive output feedback control scheme can guarantee that all the signals in the resulting closed-loop system are semiglobally bounded, and the output y(t) can converge to a small neighborhood of the zero by appropriately choosing the design parameters. Remark 2:. From 9vi ðtÞ9 r ðd=li Þ1=2 , we know that 9yðtÞ9 ¼ 9v1 ðtÞ9 r ðd=l1 Þ1=2 depends on the design constants d and l1. It is clear that the decrease of’ ðd=l1 Þ1=2 can be achieved through increasing l1 and decreasing d, therefore, one may choose the parameters l1 big enough to render the output of the system 9y(t)9 sufficiently small. Accelerating the convergence rate of the varieties in the system (1) can be achieved through increasing m, li and

x1 ðtd1 ðtÞÞ 2 z ðtÞ 1þ x21 ðtd1 ðtÞÞ

1 x2 ðtÞ þ ðsinx1 ðtÞÞx2 ðtÞ x41 ðtÞ þ1

x21 ðtd2 ðtÞÞsinx2 ðtÞ 2 z ðtÞ 1 þ x21 ðtd2 ðtÞÞ yðtÞ ¼ x1 ðtÞ þ

ð72Þ

where d1(t) and d2(t) are the time-varying delays satisfying d_ 1 ðtÞ rd1 o1 and d_ 2 ðtÞ r d2 o1. It is easy to check that (72) is in the form of (1), where F1 ¼ x1 sinðx21 ðtÞÞ,

F2 ¼

1 x2 ðtÞ þ ðsinx1 Þx2 ðtÞ, x41 ðtÞ þ1

H1 ¼

x1 ðtd1 ðtÞÞ 2 z ðtÞ, 1 þx21 ðtd1 ðtÞÞ

H2 ¼

x21 ðtd2 ðtÞÞsinx2 ðtÞ 2 z ðtÞ, 1 þx21 ðtd2 ðtÞÞ

d1 ðtÞ ¼ d2 ðtÞ ¼ 0:5ð1 þ sintÞ:

For (72), the Lyapunov function U0 ¼ z2 for z-subsystem gives U0 ¼ 2zðtÞðzðtÞ þx1 ðtÞsinx2 ðtÞÞ r z2 ðtÞ þx21 ðtÞ Selecting aðJzðtÞJÞ ¼ 0:5z2 , gð9yðtÞ9Þ ¼ y2 andt ¼ 0.

aðJzðtÞJÞ ¼ 1:5z2 ,

a0 ðJzðtÞJÞ ¼ z2 ,

It follows that Assumption 1 holds. Furthermore, it is easy to check that (72) also satisfies Assumptions 2–4. Given Q¼I, W ¼0.1, r1 ¼1, r2 ¼2, $ ¼ 0.1, d1 ¼ 0:1, d2 ¼ 0:1, by 2 3 E P 1 5 o0, the control gains are solving the following LIM 4 I P  2W obtained as

k1 ¼ 301:10,

k2 ¼ 340:12

376

S. Tong, Y. Li / Neurocomputing 74 (2010) 369–378

Fuzzy membership functions are defined as " # ðx^ 1 þ62jÞ2 mF ðxÞ ¼ exp , j ¼ 1, . . ., 5: 2 Fuzzy basis functions are chosen as ! ðx^ 1 þ 62jÞ2 exp 2 !, j1j ðx^ 1 Þ ¼ 5 ^ P ðx1 þ 62lÞ2 exp 2 l¼1 ! 2 Q ðx^ i þ 62jÞ2 exp 2 i¼1 !: j2j ðx^ 1 , x^ 2 Þ ¼ 5 2 P Q ðx^ i þ62lÞ2 exp 2 l¼1i¼1 T

The FLSs can be expressed as y1 S1 ðx^ 1 Þ ¼ 5 P j¼1

5 P j¼1

yT1j j1j , yT2 j2 ðx^ 2 Þ ¼

yT2j j2j , and which are used to approximate the unknown Fig. 1. Response curves of variables x1 (solid line) andx^ 1 (dotted line).

nonlinear functions F1 ðx^ 1 Þ and F2 ðx^ 2 Þ, respectively. The adaptive scheme are given as 1 1 1 2 r v1 ðtÞyT1 S1 ðx^ 1 ðtÞÞ 2 2 2$ 1



 1 n n e10 v1 ðtÞ þ v1 ðtÞe10 tanh  2 d1 d2 b

 n 1X 1 1 1  W Hj1 ðv1 ðtÞÞ d2 H ðv ðtÞÞ 11 1 2j¼1 2 1d1

a1 ¼  rðv1 ðtÞÞ l1 v1 ðtÞ

1 T u ¼  l2 v2 ðtÞv1 ðtÞy2 S2 ðx^ 1 ðtÞ, x^ 2 ðtÞÞk2 ðx1 ðtÞx^ 1 ðtÞÞ 2

   r2 w20 v2 ðtÞ @a1 @a1 T w20 tanh x^ 2 ðtÞ þ 1 v2 ðtÞ þ y1 S1 ðx^ 1 ðtÞÞ þ @y 2$ @y b





 1 2 2 @a1 2 1 @a1 2 @a1 _ @a1 _ x^ 1 ðtÞ v2 ðtÞ þ  v2 ðtÞ þ y 1 ðtÞþ  @y @y @y1 ðtÞ 2 d1 d2 2 @x^ 1 ðtÞ

y_ 1 ðtÞ ¼ mv1 ðtÞS1 ðx^ 1 ðtÞÞmly1 ðtÞ y_ 2 ðtÞ ¼ mv2 ðtÞS2 ðx^ 2 ðtÞÞmly2 ðtÞ where rðv1 ðtÞÞ ¼ 540v31 ðtÞ, H11 ðv1 ðtÞÞ ¼ v1 ðtÞ, d1 ¼ 0:6. The design parameters are chosen as

H21 ðv1 ðtÞÞ ¼ v31 ðtÞ,

l1 ¼ 10, l2 ¼ 20, m ¼ 2, l ¼ 0:1, e10 ¼ 0:2, b ¼ 0:01, w20 ¼ 0:5:

Fig. 2. Response curves of variables x2 (solid line) andx^ 2 (dotted line).

The initial values are chosen as zð0Þ ¼ 0:1, x1 ð0Þ ¼ 0:5, x2 ð0Þ ¼ 0:2, x^ 1 ð0Þ ¼ x^ 2 ð0Þ ¼ 0,

yT1 ð0Þ ¼ yT2 ð0Þ ¼ ½0,0,0,0,0: The simulation results are shown in Figs. 1–3 with the horizontal axis as the time. The response curve of output y(t) and x^ 1 ðtÞ is shown in Fig.1, from which one can see that the output y(t) can converge to a small neighborhood of the origin. Fig. 2 shows the response curves of x2(t) and x^ 2 ðtÞ. Fig. 3 shows the response curve of input u, from which we can see that the state variables are bounded. Example 2:. In order to further illustrate the effectiveness of the proposed control approach, we use control scheme in [26] to control the nonlinear system (72) in example 1. Case 1:. Consider system (72) without the unmodeled dynamics and time delays. For this case, we use the same fuzzy logic systems as in example 1, and choose the design parameters as k1 ¼ 2, k2 ¼ 1, g1 ¼ g2 ¼ 0:1, s ¼ 0:2, e10 ¼ e20 ¼ 0:1, c1 ¼ 0:96, c2 ¼ 0:1:

Fig. 3. Response curve of input u.

S. Tong, Y. Li / Neurocomputing 74 (2010) 369–378

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For given the symmetric positive matrix Q¼diag[4,4], by solving Lyapunov equation, one obtains the symmetric positive matrix P   4 4 P¼ 4 12 If the initial conditions are chosen as x1 ð0Þ ¼ 0:5, x2 ð0Þ ¼ 0:2, x^ 1 ð0Þ ¼ 0, x^ 2 ð0Þ ¼ 0, yT1 ð0Þ ¼ yT2 ð0Þ ¼ ½0,0,0,0,0:

the simulation results are shown by Figs. 4–6, where Fig. 4 shows the trajectories of state x1 and its estimate x^ 1 . Fig. 5 shows the trajectories of state x2 and its estimate x^ 2 and Fig. 6 shows the trajectory of input u. From Figs. 4–6, it is concluded that the adaptive fuzzy control approach of [26] can guarantee the boundedness of the signals x1, x^ 1 , x2, x^ 2 and u. The output y(t) converges to a small neighbordhood of the origin. Case 2:. Consider system (72) with the unmodeled dynamics and time delays. For this case, use the same fuzzy logic systems, the design parameters and the initial conditions as Case 1, and we obtain the simulation results, which are shown by Figs. 7–9. From Figs. 7–9, it is

Fig. 6. Response curve of input u.

Fig. 7. Response curves of variables x1 (solid line) and x^ 1 (dotted line). Fig. 4. Response curves of variables x1 (solid line) and x^ 1 (dotted line).

Fig. 5. Response curves of variables x2 (solid line) and x^ 2 (dotted line).

Fig. 8. Response curves of variables x2 (solid line) and x^ 2 (dotted line).

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Fig. 9. Response curve of input u.

concluded that the adaptive fuzzy control approach of [26] cannot guarantee the boundedness of the signals x1,x^ 1 , x2, x^ 2 and u if the nonlinear system (72) contains the unmodeled dynamics and time delays. 5. Conclusion In this paper, an adaptive fuzzy output feedback backstepping control approach is proposed for a class of nonlinear systems with unknown time delays and unmodeled dynamics, in which the unknown functions are not linearly parameterized and the state variables are not measured directly. Fuzzy logic systems are used to approximate the unknown nonlinear functions, and a state observer is developed for estimating the unmeasured states. By combining backstepping technique, adaptive fuzzy control design with a changing supplying function idea, a stable adaptive fuzzy output feedback control has been developed. It is proved that the proposed approach is able to guarantee that all the signals of the closed-loop system are semiglobally uniformly ultimately bounded, and the output of the system converges to a neighborhood of the origin. References [1] M. Kristic, I. Kanellakopoulos, P.V. Kokotovic, in: Nonlinear and Adaptive Control Design, Wiley, New York, 1995. [2] Z.P. Jiang, L. Praly, Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties, Automatica 34 (7) (1998) 825–840. [3] Z.P. Jiang, A combined backstepping and small-gain approach to adaptive output feedback control, Automatica 35 (6) (1999) 1131–1139. [4] L.X. Wang, in: Adaptive Fuzzy Systems and Control, Prentice Hall, Englewood Cliffs, NJ, 1994. [5] Y.G. Leu, W.Y. Wang, Observer-based adaptive fuzzy-neural control for unknown nonlinear dynamical systems, IEEE Transactions on Systems, Man, and Cybernetics, Part B 29 (5) (1999) 583–591. [6] H.G. Zhang, L.L. Cai, Z. Bien, A fuzzy basis function vector-based multivariable adaptive controller for nonlinear systems, IEEE Transactions on Systems Man and Cybernetics Part B 30 (1) (2000) 210–217. [7] S.C. Tong, H.X. Li, W. Wang, Observer-based adaptive fuzzy control for SISO nonlinear systems, Fuzzy Sets and Systems 148 (3) (2004) 355–376. [8] A. Boulkroune, M. Tadjine, M.M. Saad, et al., How to design a fuzzy adaptive controller based on observers for uncertain affine nonlinear systems, Fuzzy Sets and Systems 159 (8) (2008) 926–948. [9] B. Chen, X.P. Liu, Fuzzy approximate disturbance decoupling of MIMO nonlinear systems by backstepping and application to chemical processes, IEEE Transactions on Fuzzy Systems 13 (6) (2005) 832–847.

[10] B. Chen, X.P. Liu, Adaptive fuzzy output tracking control of MIMO nonlinear uncertain systems, IEEE Transactions on Fuzzy Systems 15 (2) (2007) 287–300. [11] S.C. Tong, Y.M. Li, Direct adaptive fuzzy backstepping control for a class nonlinear systems, International Journal of Innovative Computing, Information and Control 3 (4) (2007) 887–896. [12] Y.S. Yang, G. Feng, J.S. Ren, A combined backstepping and small-gain approach to robust adaptive fuzzy control for strict-feedback nonlinear systems, IEEE Transactions on Systems, Man, and Cybernetics, Part A 34 (3) (2004) 406–420. [13] Y.S. Yang, C.J. Zhou, Robust adaptive fuzzy tracking control for a class of perturbed strict-feedback nonlinear systems via small-gain approach, Information Sciences 170 (2–4) (2005) 211–234. [14] A.M. Zou, Z.G. Hou, Adaptive control of a class of nonlinear pure-feedback systems using fuzzy backstepping approach, IEEE Transactions on Fuzzy Systems 16 (4) (2008) 886–897. [15] S.S. Zhou, G. Feng, C.B. Feng, Robust control for a class of uncertain nonlinear systems: adaptive fuzzy approach based on backstepping, Fuzzy Sets and Systems 151 (1) (2005) 1–20. [16] S.C. Tong, Y.M. Li, P. Shi, Fuzzy adaptive backstepping robust control for SISO nonlinear systems with dynamic uncertainties, Information Sciences 179 (9) (2009) 1319–1332. [17] M.S. Mahmoud, N.F. Al-Muthairi, Design of robust controllers for time-delay systems, IEEE Transactions on Automatic Control 39 (5) (1994) 995–999. [18] W. Chen, H.G. Zhang, C.W. Yin, An output tracking control for nonlinear systems with uncertainties and disturbances using time delay control, Cybernetica 40 (3) (1997) 229–237. [19] X.H. Jiao, T.L. Shen, Adaptive feedback control of nonlinear time-delay systems: the LaSalle–Razumikhin-based approach, IEEE Transactions on Automatic Control 50 (11) (2005) 1909–1913. [20] C.C. Hua, X.P. Guan, P. Shi, Robust backstepping control for a class of time delayed systems, IEEE Transactions on Automatic Control 50 (6) (2005) 894–899. [21] M. Wang, B. Chen, K.F. Liu, et al., Adaptive fuzzy tracking control of nonlinear time-delay systems with unknown virtual control coefficients, Information Sciences 178 (22) (2008) 4326–4340. [22] M. Wang, B. Chen, X.P. Liu, et al., Adaptive fuzzy tracking control for a class of perturbed strict-feedback nonlinear time-delay systems, Fuzzy Sets and Systems 159 (8) (2008) 949–967. [23] M. Wang, B. Chen, P. Shi, Adaptive fuzzy control for a class of perturbed strictfeedback nonlinear time-delay systems, IEEE Transactions on Systems, Man, and Cybernetics, Part-B 38 (3) (2008) 721–730. [24] H.G. Zhang, S.X Lun, D.R. Liu, Fuzzy HN filter design for a class of nonlinear discrete-time systems with multiple time delays, IEEE Transactions on Fuzzy Systems 15 (3) (2007) 453–469. [25] N. Golea, A. Golea, K. Barra, Observer-based adaptive control of robot manipulators: fuzzy systems approach, Applied Soft Computing 8 (1) (2008) 778–787. [26] S.C. Tong, Y.M. Li, Observer-based fuzzy adaptive control for strict-feedback nonlinear systems, Fuzzy Sets and Systems 160 (12) (2009) 1749–1764. [27] S.C. Tong, X.L. He, H.G. Zhang, A combined backstepping and small-gain approach to robust adaptive fuzzy output feedback control, IEEE Transactions on Fuzzy Systems 17 (5) (2009) 1059–1069. [28] C.C. Hua, X.P. Guan, X.P. Guan, Robust output tracking control for a class of time-delay nonlinear using neural network, IEEE Transactions on Neural Networks 18 (2) (2007) 495–505. [29] Z.J. Wu, X.J. Xie, P. Shi, Robust adaptive output-feedback control for nonlinear systems with output unmodeled dynamics, International Journal Robust Nonlinear Control 18 (11) (2008) 1162–1187. [30] C. Wen, H.G. Zhang, C.W. Yin, Input/output linearization for nonlinear systems with uncertainties and disturbances using TDC, Cybernetics and Systems 28 (7) (1997) 625–634. Shaocheng Tong, received the B.A. degree in mathematics from Jinzhou Normal College, Jinzhou, China, the M.A. degree in fuzzy mathematics from Dalian Marine University, PRC, and the Ph.D degree in fuzzy control from Northeastern University, PRC, in 1982, 1988, and 1997, respectively. Currently, he is a Professor in the Department of Basic Mathematics, Liaoning University of Technology, Jinzhou, PRC. His research interests include fuzzy control theory, nonlinear adaptive control, and intelligent control.

Yongming Li, graduated from Liaoning University of Technology, PRC, in 2004. He received the M. Sc. degree from Liaoning University of Technology, PRC, in 2007. He is currently an assistant in the Department of Basic Mathematics, Liaoning University of Technology. His research interests include fuzzy control theory, nonlinear adaptive control, and intelligent control.