Adaptive control for a class of chaotic systems with nonlinear inputs and disturbances

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Adaptive control for a class of chaotic systems with

nonlinear inputs and disturbances

Kuo-Ming Chang

Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, No. 415, Chien-Kung Road, Kaohsiung 807, Taiwan, ROC

Accepted 29 June 2006

Abstract

In this paper, an adaptive controller design method based on hyperstability theory is proposed for a class of uniﬁed chaotic systems with so-called sector-bounded inputs and disturbances. Under the norm-bounded assumption of dis-turbances, for the controlled system, system states can be regulated to zero levels asymptotically in the presence of this class of disturbances. To show the validity and feasibility of the proposed adaptive controller, some simulations on con-trolling diﬀerent chaotic systems with disturbances, are made and investigated.

1. Introduction

Chaotic systems exhibit unpredictable and irregular dynamics and have been found in many engineering systems, such as lasers[1], Colpitts oscillators[2], nonlinear circuits[3], communication[4], and so on. Since the chaos control problem was first considered by Ott et al. in 1990[5], it has been investigated extensively by many researchers. Recently, many valuable control methods have been developed to control chaotic systems, such as the sliding mode control method[6,7], adaptive control method[8–10], backstepping method[11], observer-based method[12], and the time-delay feedback control method[13,14]. Recalling all the proposed control methods, we found that the common char-acteristic of chaos control methods is that the control input of chaotic systems is linear in nature. Owing to physical limitations, there usually exist nonlinearities in the plant actuators of control systems. The presence of nonlinearities in control input may cause serious influence upon system performance. Besides, the control input nonlinearity may result in unpredictable results in chaotic systems. Since the chaotic system is very sensitive to any system parameters, the nonlinear effect in the control input cannot be ignored in both control design and realization for chaotic systems. Considering these points, in this paper we will investigate the control problem of a class of unified chaotic systems with nonlinear inputs.

In this paper, the control objective is to regulate the system states of chaotic systems to reach zeros, asymptotically. In order to achieve the control goal, an adaptive control method based on hyperstability theory[15,16]is developed for chaotic systems with sector-bounded nonlinear inputs, which are subjected to system parameter variations and distur-bances. Under the assumptions of sector-bounded nonlinear inputs and norm-bounded disturbances, the proposed

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Chaos, Solitons and Fractals 36 (2008) 460–468

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control scheme has been analyzed theoretically that it has robust asymptotic stability with respect to system uncertain-ties of parameters, modeling error, and external disturbances. Finally, some numerical simulations are given to validate the proposed adaptive regulating control approach.

Throughout this paper, it is noted that,jjVjj represents the Euclidean norm of vector V and kmin(M) denotes the

minimum eigenvalue of matrix M. This paper is organized as follows. In Section2we give a description for a class of chaotic systems. Section3introduces the design method of adaptive control for chaotic systems and also presents the stability proof of the proposed adaptive control scheme. Some numerical simulations are provided to illustrate the validity of the proposed control method in Section4. Finally, a conclusion ends the paper in Section5.

2. Description of the system

Consider the following chaotic system described by _

X¼ AX þ f ðX Þ þ dðX ; tÞ; ð1Þ

where X(t)2 Rn is a n-dimensional state vector of the system, A2 Rn·n is the matrix of the system parameter, f(X):Rn! Rn

is the nonlinear part of the system, and d(X, t):Rn· R+! Rnis a disturbance vector. As we know, many

chaotic systems investigated, are in a form(1)without disturbance, such as a class of uniﬁed chaotic systems in the work of Lu et al.[17], and the Arneodo chaotic system. The uniﬁed chaotic system is as follows:

_x1¼ ð25a þ 10Þðx2 x1Þ;

_x2¼ ð28 35aÞx1 x1x3þ ð29a 1Þx2;

_x3¼ x1x2

8þ a 3 x3;

where a2 [0, 1]. When a 2 [0, 0.8), it is the Lorenz chaotic system, a = 0.8 is the Lu chaotic system, and a 2 (0.8, 1] is the Chen’s chaotic system. The Arneodo chaotic system is as follows:

_x1¼ x2;

_x2¼ x3;

_x3¼ 5:5x1 3:5x2 x3 x31:

In this paper, we will investigate the control problem of system(1)with an additional feedback force; and the corre-sponding controller is designed so that it can render the closed-loop system asymptotically stable under disturbances. Owing to physical limitations, there usually exist nonlinearities in the plant actuators of a control system. The nonlinear eﬀect in control input may cause serious unfavorable inﬂuence on chaotic system performance or stability. In view of this reason, a sector-bounded nonlinear input will be considered. In this paper, the control is assumed to be in a form of nonlinear input function vector /(u). Thus the controlled chaotic system is represented by

_

X¼ AX þ f ðX Þ þ dðX ; tÞ þ /ðuÞ; ð2Þ

where /ðuÞ ¼ ½/1ðu1Þ /2ðu2Þ /nðunÞ

T _{2 R}n _{is a continuous nonlinear function vector with /(0) = 0 and}

u¼ ½u1 u2 un T _{2 R}n_.

The following assumptions specify the class of chaotic systems considered in this paper.

Assumption 1. As shown inFig. 1, nonlinear input functions /i(ui), i = 1, . . . , n, are sector bounded by ui, i = 1, . . . , n,

respectively. It yields the result that there exist positive constants ci1, i = 1, . . . , n, and ci2, i = 1, . . . , n, such that the

following conditions are satisﬁed. ci16/iuðuiiÞ

6ci2, for i = 1, . . . , n.

Assumption 2. Uncertain disturbance vector d(X, t) is norm-bounded. It means that there exists a positive time function j(t), such thatkd(X, t)k6j(t).

3. Adaptive control design

FromAssumption 1, it can be straightforwardly obtained that ci1u2iðtÞ 6 uiðtÞ/iðuiðtÞÞ 6 ci2u2iðtÞ; i ¼ 1; . . . ; n:

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Then, we have

c11u21ðtÞ þ þ cn1u2nðtÞ 6 u1ðtÞ/1ðu1ðtÞÞ þ þ unðtÞ/nðunðtÞÞ 6 c12u21ðtÞ þ þ cn2u2nðtÞ:

Here, there exist two positive constants: c1= min{ci1ji = 1, . . . , n} and c2= max{ci2ji = 1, . . . , n}. From the above

inequality, it yields that

c1uTðtÞuðtÞ 6 uTðtÞ/ðuðtÞÞ 6 c2uTðtÞuðtÞ: ð3Þ

In this paper, we will consider controlling the chaotic system(2)to reach equilibrium 0. It is easy to apply our results to controlling the system’s other ﬁxed points or tracking problems. To achieve the control objective, an adaptive control design based on hyperstability theory will be given in this paper. For designing the controller, the controlled chaotic system(2)is rewritten as

_

X ¼ A1Xþ ðA A1ÞX þ f ðX Þ þ dðX ; tÞ þ /ðuÞ ¼ A1X B1X; ð4Þ

X¼ B1

1 ½ðA A1ÞX þ f ðX Þ þ dðX ; tÞ þ /ðuÞ; ð5Þ

where A12 Rn·nis a design Hurwitz matrix, and B12 Rn·nis chosen to be a nonsingular constant matrix. Then, a linear

combination of system state is deﬁned as Y ¼ CX ;

where C2 Rn·nis also a designed constant matrix. Hence, it follows that a linear time invariant system with output Y is given by the following dynamic equations:

_

X ¼ A1X B1X; ð6Þ

Y ¼ CX : ð7Þ

According to hyperstability theory, the linear system expressed in Eqs.(6) and (7)is an asymptotically hyperstable sys-tem if it satisﬁes two following conditions:

(1) The transfer function GðsÞ ¼ CðsI A1Þ1B1;

must be strictly positive real (SPR);

(2) The so-called passivity condition (Popov integral inequality) Z t

0

YTðsÞXðsÞds P r2

0 is true for all t P 0;

where r0is an arbitrary ﬁnite constant. Then G(s) is strictly positive real if, and only if, there exists a symmetric, positive

deﬁnite matrix P such that

Slope ci2 ) ( i iu φ Slope ci1 i u

Fig. 1. The sector bounded function.

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dðX ; tÞ ¼ sin 15t 0:1 sin x2þ 1:5 sin 10t 0:3ex1_{þ sin 5t} 2 6 4 3 7 5:

In this case, matrices A1, B1, and C, and all initial values are the same as those used in the previous case. Adaptation

gain k = 0.5 is applied on the adaptation law. All the behaviors of the proposed adaptive control scheme are illustrated inFigs. 8–13. It can be seen that the system states are regulated to zeros asymptotically even when the nonlinear input chaotic systems are undergoing system parameter abrupt variations and disturbances with modeling errors.

5. Conclusions

In practical control systems, owing to actuator physical limitations, the nonlinearity eﬀect in control input cannot be ignored in the chaotic systems. In this paper, the control problem of chaotic systems with so-called sector-bounded non-linear input and disturbances is investigated via the adaptive control method. The adaptive controller based on hyper-stability theory is designed. Under the norm-bounded assumption of disturbances, the adaptive controller is constructed without the knowledge of norm-bounded value by using an adaptation law. Simulation results show that the proposed adaptive controller can regulate the states of the uniﬁed chaotic system to zeros asymptotically, even under the chaotic system, with respect to system parameter variations, modeling errors, and external disturbances.

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