Guaranteed cost control of time-delay chaotic systems

Chaos, Solitons and Fractals 27 (2006) 1011–1018 www.elsevier.com/locate/chaos

Guaranteed cost control of time-delay chaotic systems Ju H. Park

a,*

, O.M. Kwon

b

a

b

Robust Control and Nonlinear Dynamics Laboratory, Department of Electrical Engineering, Yeungnam University, Kyongsan 712-749, Republic of Korea Department of Intelligent Control Research, Mechatronics Center, Samsung Heavy Industries Co. Ltd., Daejeon, Republic of Korea Accepted 6 April 2005

Abstract This article studies a guaranteed cost control problem for a class of time-delay chaotic systems. Attention is focused on the design of memory state feedback controllers such that the resulting closed-loop system is asymptotically stable and an adequate level of performance is also guaranteed. Using the Lyapunov method and LMI (linear matrix inequality) framework, two criteria for the existence of the controller are derived in terms of LMIs. A numerical example is given to illustrate the proposed method. Ó 2005 Elsevier Ltd. All rights reserved.

1. Introduction Since Lorenz found the first chaotic attractor in a simple three-dimensional autonomous systems in 1963, tremendous efforts have been devoted to chaos control including stabilization of unstable equilibria, and more generally, unstable periodic orbits during three decades. Also, since Mackey and Glass [1] first found chaos in time-delay system in 1977, there has been increasing attention in time-delay chaotic systems. For instance, see the papers; Lu and He [2], Tian and Gao [3], Chen and Yu [4], and the references therein. In the literature, one of the most frequent objectives consists in the stabilization of chaotic behaviors to one of unstable fixed points or unstable periodic orbits embedded within a chaotic attractor. That is, to design a stabilizing controller that guarantees the closed-loop system dynamics converges to the fixed point or periodic orbit. In this endeavor, Guan et al. [5], Sun [6], and Park and Kwon [7] have investigated the controller design problem of a class of time-delay chaotic systems using the famous Ott–Grebogi– Yorke (OGY) method [8]. They proposed two kind of controller, i.e., standard feedback control (SFC) and delayed feedback control (DFC), and derived the stabilization criteria using the Lyapunov method. For further information of DFC controller, see the paper [5]. In this article, we propose a novel feedback control scheme of integral type for the system. On the other hand, when designing controllers for dynamic systems, it is desirable to ensure satisfactory system performance. One way to address this problem is so-called guaranteed cost control [9]. The approach has the advantage of providing an upper bound on a given performance index and thus the system performance degradation is guaranteed to

*

Corresponding author. Fax: +82 53 8104629. E-mail address: [email protected] (J.H. Park).

0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.076

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J.H. Park, O.M. Kwon / Chaos, Solitons and Fractals 27 (2006) 1011–1018

be less than this bound. Thus, the guaranteed cost control problem for various dynamic systems have been investigated in recent years. For example, see the papers [10,11], and references therein. To the best of author
s knowledge, the problem of guaranteed cost control for time-delayed chaotic systems has been overlooked to date. Thus we consider the guaranteed cost control problem for a class of time-delays chaotic systems in this paper. Using the Lyapunov stability analysis and LMI framework, two stabilization criteria for the system are derived. The solutions of the criteria can be easily obtained by various convex optimization algorithms. Notation: Rn and Rn
m denote n-dimensional Euclidean space and the set of real n by m matrices, respectively. w denotes the symmetric part. X > 0 (X P 0) means that X is a real symmetric positive definitive matrix (positive semidefinite). I denotes the identity matrix with appropriate dimensions. kÆk refers to the induced matrix 2-norm. diag{  } denotes the block diagonal matrix. Cn;h ¼ Cð½h; 0; Rn Þ denotes the Banach space of continuous functions mapping the interval [h, 0] into Rn , with the topology of uniform convergence.

2. Problem statement Consider the following time-delay chaotic systems x_ ðtÞ ¼ AxðtÞ þ Bxðt  hÞ þ f1 ðt; xðtÞÞ þ f2 ðt; xðt  hÞÞ þ uðtÞ;

ð1Þ

where xðtÞ 2 Rn is the state variable, uðtÞ 2 Rn is the control input, A; B 2 Rn
n are constant system matrices representing the linear parts of the system, f1 ðt; xðtÞÞ; f2 ðt; xðt  hÞÞ 2 Rn are the nonlinear parts of the system, and h > 0 is the constant time delay. Suppose that the chaotic system (1) has an unstable fixed point or an unstable periodic orbit xðtÞ, and is currently in a chaotic state. Then, the goal in this article is to control the system asymptotically converges towards xðtÞ via the following memory feedback:   Z t uðtÞ ¼ K ðxðtÞ  xðtÞÞ þ BðxðsÞ  xðsÞÞ ds ð2Þ th

in which K is a gain matrix of the controller and xðtÞ is the desired fixed point. Since the chaotic system (1) has an unstable fixed point xðtÞ ¼ x ¼ constant, we have x_ ðtÞ ¼ AxðtÞ þ Bxðt  hÞ þ f1 ðt; xðtÞÞ þ f2 ðt; xðt  hÞÞ.

ð3Þ

By applying the control input (2) to system (1), we obtain the following error dynamics: Z t e_ ðtÞ ¼ ðA  KÞeðtÞ þ Beðt  hÞ þ F 1 ðt; eðtÞÞ þ F 2 ðt; eðt  hÞÞ  K BeðsÞ ds;

ð4Þ

th

where eðtÞ ¼ xðtÞ  xðtÞ; eðt  hÞ ¼ xðt  hÞ  xðt  hÞ ¼ xðt  hÞ  xðtÞ, and F 1 ðÞ ¼ f1 ðt; ðeðtÞ þ xðtÞÞÞ  f1 ðt; xðtÞÞ; F 2 ðÞ ¼ f2 ðt; ðeðt  hÞ þ xðt  hÞÞÞ  f2 ðt; xðt  hÞÞ. Then, the control goal is to force ke(t)k ! 0 as t ! 1.

3. Controller design In this section, based on the Lyapunov stability theory and LMI framework, the design problem of guaranteed cost control for system (1) will be discussed. For error system (4), since zero is a fixed point of F1(t, e(t)) + F2(t, e(t  h)), we have a Taylor expansion F 1 ðt; eðtÞÞ þ F 2 ðt; eðt  hÞÞ ¼ b0 eðtÞ þ ½HOT1 þ b1 eðt  hÞ þ ½HOT2 ;

ð5Þ

where b0 ¼ F 01 ðt; eðtÞÞ; b1 ¼ F 02 ðt; eðt  hÞÞ, [HOT]1 and [HOT]2 are higher order term in e(t) and e(t  h), respectively, and F 0i denotes the time derivative of Fi, (i = 1, 2).

J.H. Park, O.M. Kwon / Chaos, Solitons and Fractals 27 (2006) 1011–1018

1013

From the OGY-method [8], the control goal is the zero fixed point, so we can only consider the linearized part near zero point. Rewrite the local error system as follows: e_ ðtÞ ¼ ðA  K þ b0 IÞeðtÞ þ ðB þ b1 IÞeðt  hÞ  K

Z

t

ð6Þ

BeðsÞ ds. th

Associated with the system (6) is the following quadratic cost function: Z 1  T  e ðtÞC1 eðtÞ þ uT ðtÞC2 uðtÞ dt; J¼

ð7Þ

0

where C1 2 Rn
n and C2 2 Rn
n are given positive-definite matrices. Here, the objective of this article is to develop a procedure to design a state feedback controller u(t) for the system (4) and cost function (7) such that the resulting closed-loop system is asymptotically stable and the closed-loop value of the cost function (7) satisfies J 6 J, where J is some specified constant. Definition 1. For the error system (6) and cost function (7), if there exist a control law u*(t) and a positive J such that the system (6) is asymptotically stable and the closed-loop value of the cost function (7) satisfies J 6 J, then J is said to be a guaranteed cost and u*(t) is said to be a guaranteed cost law of the system (6) and cost function (7). Here, we introduce a well-known fact and two lemmas which are necessary for deriving main results of this paper. Fact 1 (Schur Complement). Given constant symmetric matrices R1, R2, R3 where R1 ¼ RT1 and 0 < R2 ¼ RT2 , then R1 þ RT3 R1 2 R3 < 0 if and only if "

R1

RT3

R3

R2

#

 < 0;

or

R2 RT3

R3 R1

< 0.

Lemma 1 [12]. For any constant matrix R 2 Rn
n , R = RT > 0, scalar c > 0, vector function x : ½0; c ! Rn such that the integrations concerned are well defined, then Z

c

xðsÞ ds

T Z c  Z c R xðsÞ ds 6 c xT ðsÞRxðsÞ ds.

0

0

ð8Þ

0

R ^ t xðsÞ ds, where xðtÞ 2 Rn and B ^ 2 Rn
n . Lemma 2 [13]. Consider an operator DðÞ : Cn;h ! Rn with Dðxt Þ ¼ xðtÞ þ B th For a given scalar d, where 0 < d < 1, if a positive definite symmetric matrix M exists, such that "

dM b hM B

T

b M hB M

# ð9Þ

<0

holds, then the operator Dðxt Þ is stable. Now, define an operator Dðet Þ : Cn;h ! Rn as Z t BeðsÞ ds; Dðet Þ ¼ eðtÞ þ

ð10Þ

th

where et = e(t + s), s 2 [h, 0]. From the definition of Dðet Þ, the transformed error system is _ t Þ ¼ e_ ðtÞ þ BeðtÞ  Beðt  hÞ ¼ ðA  KÞDðet Þ þ b1 eðt  hÞ  A Dðe

Z

t

BeðsÞ ds;

ð11Þ

th

where A ¼ A þ B þ b0 I. This transformation is called parameterized neutral model transformation. Then, we have the following theorem. Theorem 1. For given C1 and C2, the system (1) under the control (2) is asymptotically stabilized to the given fixed point xðtÞ if there exist positive definite matrices X, Z1, Z2, and any matrix Y which satisfy the following LMIs:

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J.H. Park, O.M. Kwon / Chaos, Solitons and Fractals 27 (2006) 1011–1018

2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

T

AX  Y þ X A  Y T

ABZ 1

b1 Z 2

hX

X

X C1

Y T C2

H

Z 1

0

hZ 1 BT

Z 1 BT

Z 1 BT C1

0

H

H

Z 2

0

0

0

0

H

H

H

Z 1

0

0

0

H

H

H

H

Z 2

0

0

H

H

H

H

H

C1

0

H

H

H

H

H

C2

H "

X H

hXBT X

3 7 7 7 7 7 7 7 7 7 < 0; 7 7 7 7 7 7 5

ð12Þ

# ð13Þ

< 0;

such that when ke(t)k is small enough, the control goal by the controller uðtÞ ¼ YX 1 ðeðtÞ þ i.e.,

Rt th

BeðsÞ dsÞ is guaranteed,

keðtÞk ! 0 as t ! 1. Also, the guaranteed cost of error system is Z 0 Z J ¼ DT ð0ÞP Dð0Þ þ ðs þ hÞeT ðsÞR1 eðsÞ ds þ h

0

eT ðsÞR2 eðsÞ ds.

ð14Þ

h

Proof. For P > 0; R1 > 0, and R2 > 0, construct a Lyapunov functional of the form V ¼ V 1 þ V 2 þ V 3;

ð15Þ

V 1 ¼ Dðet ÞT P Dðet Þ; Z t ðs  t þ hÞeT ðsÞR1 eðsÞ ds; V2 ¼ th Z t V3 ¼ eT ðsÞR2 eðsÞ ds.

ð16Þ

where

ð17Þ ð18Þ

th

The differential of the Lyapunov functional along the trajectory of error system (11) is  Z t dV 1 ¼ 2Dðet ÞT P ðA  KÞDðet Þ  A BeðsÞ ds þ b1 eðt  hÞ ; dt th Z t T Z t  Z t dV 2 ¼ heT ðtÞR1 eðtÞ  eT ðsÞR1 eðsÞ ds; 6 heT ðtÞR1 eðtÞ  eðsÞ ds ðh1 R1 Þ eðsÞ ds ; dt th th th dV 3 ¼ eT ðtÞR2 eðtÞ  eT ðt  hÞR2 eðt  hÞ; dt

T

T

¼ D ðet ÞRDðet Þ  2D ðet ÞR

ð20Þ ð21Þ

where Lemma 1 are utilized in (20). Define R ¼ hR1 þ R2 and then  T  Z t Z T e ðtÞReðtÞ ¼ Dðet Þ  BeðsÞ ds R Dðet Þ  th

ð19Þ

Z



t

BeðsÞ ds

th t

th

BeðsÞ ds þ

Z

t th

T Z BeðsÞ ds R

t

 BeðsÞ ds .

ð22Þ

th

From (19)–(21) and utilizing the relation (22), a new bound of the time-derivative of V is as follows: 3 X dV dV i ¼ 6 vT ðtÞXvðtÞ; dt dt i¼1

ð23Þ

J.H. Park, O.M. Kwon / Chaos, Solitons and Fractals 27 (2006) 1011–1018

where

1015

3 Dðet Þ 7 6Rt vðtÞ ¼ 4 th eðsÞ ds 5 eðt  hÞ 2

and

ð24Þ

2

3 b1 P 7 0 5.

P ðA  KÞ þ ðA  KÞT P þ R P AB  RB 6 X¼4 H h1 R1 þ BT RB H

R2

H T

Here, applying the relation (22) to the terms e (t)C1e(t) and using uT ðtÞC2 uðtÞ ¼ DT ðet ÞKT C2 KDðet Þ;

ð25Þ

gives that   dV 6 vT ðtÞX1 vðtÞ  eT ðtÞC1 eðtÞ þ uT ðtÞC2 uðtÞ ; dt

ð26Þ

where 2 6 X1 ¼ 4

P ðA  KÞ þ ðA  KÞT P þ KT C2 K H H

P AB h1 R1 H

3 3 2 I b1 P 7 7 6 0 5 þ 4 BT 5ðR þ C1 Þ½ I 0 R2

B 0 .

ð27Þ

Therefore, if X1 < 0, there exists the positive scalar c such that dV 6 ckDðet Þk2 . dt By Fact 1 (Schur Complement), the inequality X1 < 0 is equivalent to 2 P ðA  KÞ þ ðA  KÞT P þ KT C2 K P AB b1 P hI 6 H h1 R1 0 hBT 6 6 6 H H R2 0 X2 ¼ 6 6 H H H hR1 6 1 6 4 H H H H H H H H

ð28Þ

I BT 0 0 R1 2 H

3 I 7 BT 7 7 0 7 7 < 0. 0 7 7 7 0 5 C1 1

ð29Þ

1 and pre- and post-multiplying the matrix X2 by diag Letting X ¼ P 1 ; Y ¼ KX ; Z 1 ¼ hR1 1 ; Z 2 ¼ R2 , {X, Z1, Z2, I, I, C1}, give that X2 < 0 is equivalent to the following inequality: 2 3 T AX  Y þ X A  Y T þ Y T C2 Y ABZ 1 b1 Z 2 hX X X C1 6 7 6 H Z 1 0 hZ 1 BT Z 1 BT Z 1 BT C1 7 6 7 6 7 H H Z 2 0 0 0 7 < 0. X3 ¼ 6 ð30Þ 6 7 H H H Z 1 0 0 6 7 6 7 4 5 H H H H Z 2 0 H H H H H C1

Again, by Fact 1, the fact that X3 < 0 is equivalent to the LMI (12). The inequality (13) is equivalent to " # P hBT P <0 H P

ð31Þ

by pre- and post-multiplying the inequality (13) by diag{X1, X1}. If the above inequality (31) holds, then we can prove that a positive scalar d which is less than one exists such that " # dP hBT P <0 ð32Þ H P

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J.H. Park, O.M. Kwon / Chaos, Solitons and Fractals 27 (2006) 1011–1018

according to matrix theory. Therefore, from Lemma 2, if the inequality (13) holds, then operator Dðet Þ is stable. This implies that both the original local error system (6) and transformed error system (11) with stable operator Dðet Þ are asymptotically stable by Theorem 9.8.1 in [16]. Furthermore, we have eT ðtÞC1 eðtÞ þ uT ðtÞC2 uðtÞ < 

dV . dt

Integrating both sides of the above inequality from 0 to Tf leads to Z Tf ðeT ðtÞC1 eðtÞ þ uT ðtÞC2 uðtÞÞ dt 0

  < V ð0Þ  V ðT f Þ ¼ DT ð0ÞP Dð0Þ  DT ðT f ÞP DðT f Þ ! Z 0 Z Tf T T þ ðs þ hÞe ðsÞR1 eðsÞ ds  ðs þ hÞe ðsÞR1 eðsÞ ds h

þ

Z

T f h

0 T

e ðsÞR2 eðsÞ ds 

Z

h

!

Tf T

e ðsÞR2 eðsÞ ds ;

T f h

where Dð0Þ denotes Dðet Þjt¼0 . As both the operator Dðet Þ and the system (6) are asymptotically stable, when Tf ! 1, Z Tf DT ðT f ÞP DðT f Þ ! 0; eT ðsÞR1 eðsÞ ds ! 0; Z

T f h Tf

eT ðsÞR2 eðsÞ ds ! 0.

T f h

Therefore we have Z 1 ðeT ðtÞC1 eðtÞ þ uT ðtÞC2 uðtÞÞ dt < V ð0Þ ¼ J.

ð33Þ

0

This completes our proof.

h

Theorem 1 presents a method of designing a memory guaranteed cost feedback controller. The following theorem presents a method of selecting a controller minimizing the upper bound of the guaranteed cost (14). Before stating our second result, let us define Z 0 Z ðs þ hÞeðsÞeT ðsÞ ds; N2 NT2 ¼ N1 NT1 ¼ h

0

eðsÞeT ðsÞ ds.

ð34Þ

h

Theorem 2. Consider the system (6) with cost function (7). If the following optimization problem: min

X ;P1 ;P2 ;Z 1 ;Z 2 ;Y ;a

fa þ TraceðP1 Þ þ TraceðP2 Þg

subject to ðiÞ LMIs(12) and (13); " # a DT ð0Þ ðiiÞ < 0; Dð0Þ X " # P1 hNT1 < 0; ðiiiÞ hN1 hZ 1 " # P2 NT2 <0 ðivÞ N2 Z 2

ð35Þ

has a positive solution set (X, P1, P2, Z1, Z2, Y, a), then the control law (2) is an optimal robust guaranteed cost control law which ensures the minimization of the guaranteed cost (14) for error system (6).

J.H. Park, O.M. Kwon / Chaos, Solitons and Fractals 27 (2006) 1011–1018

1017

Proof. By Theorem 1, (i) in (35) is clear. Also, it follows from Fact 1 that (ii)–(iv) in (35) are equivalent to T 1 DT ð0ÞX 1 Dð0Þ < a, hNT1 Z 1 1 N1 < P1 , and N2 Z 2 N2 < P2 , respectively. On the other hand, Z

0

ðs þ hÞeT ðsÞR1 eðsÞ ds ¼

h

Z

0

h

Traceððs þ hÞeT ðsÞR1 eðsÞÞ ds ¼ TraceðN1 NT1 R1 Þ

¼ TraceðNT1 hZ 1 N1 Þ < TraceðP1 Þ; Z

0

eT ðsÞR2 eðsÞ ds ¼

h

Z

0

h

TraceðeT ðsÞR2 eðsÞÞ ds ¼ TraceðN2 NT2 R2 Þ ¼ TraceðNT2 Z 1 2 N2 Þ < TraceðP2 Þ.

Hence, it follows from (14) that J < a þ TraceðP1 Þ þ TraceðP2 Þ. Thus, the minimization of a + Trace(P1) + Trace(P2) implies the minimization of the guaranteed cost for the system (6). Note that this convex optimization problem guarantees that a global optimum, when it exists, is reachable (see Remark 1). h Remark 1. The problems in Theorems 1 and 2 are to determine whether the problem is feasible or not. It is called the feasibility problem. Also, the solutions of the problem can be found by solving eigenvalue problem with respect to solution variables, which is a convex optimization problem. For details of this kind of optimization problem, see Boyd et al. [14]. Various efficient convex optimization algorithms can be used to check whether the LMIs in Theorems 1 and 2 is feasible. In this article, in order to solve the LMIs, we utilize Matlab
s LMI Control Toolbox [15], which implements state-of-the-art interior-point algorithms, which is significantly faster than classical convex optimization algorithms [14].

4. Numerical example Consider a chaotic system of the form [17,5–7]: f

dxðtÞ G ¼ xðtÞ þ ðxðt  hÞ þ U B Þ½1 þ l cosðpxðt  hÞ þ U 0 þ U M Þ; dt 1þl

where x(t) is the normalized output voltage variation, G P 0 is the feedback gain and h P 0 is the feedback delay, l P 0 is the fringe constant, f P 0 is the response time, U0 and UB are the constant phase shifts, and UM is the input phase shift induced by the fiber strain. Let us keep U0 = 2.0, UB = 1.0, UM = 2.5 and put l = 1.0 and G = 2.0. When f = 0.1 and h = 0.01, the system demonstrates chaotic behavior (for simulation results, see [5]). In this case, we know x ¼ 0.85 is t one of the fixed points. Assume that the initial condition of the system x(t)  Based on this fixed R t = e for t 2 [h, 0]. point, the feedback controller can be chosen as uðtÞ ¼ K ð0.85  xðtÞÞ þ th 10ð0.85  xðsÞÞ ds . The error system is obtained that Z t _eðtÞ ¼ ð10  KÞeðtÞ þ 10eðt  hÞ þ F 1 ðt; eðtÞÞ þ F 2 ðt; eðt  hÞÞ  10K eðsÞ ds; th

where F 1 ðÞ ¼0; b0 ¼ 0; F 2 ðÞ ¼10ðeðt  hÞ þ xðt  hÞÞ cosðpðeðt  hÞ þ xðt  hÞÞ  0.5Þ þ 10 cosðpðeðt  hÞ þ xðt  hÞÞ  0.5Þ  10xðt  hÞ cosðpxðt  hÞ  0.5Þ  10 cosðpxðt  hÞ  0.5Þ. Then, we have b1 ¼ 10½cosðpxðt  hÞ  0.5Þ  pðxðt  hÞ þ 1Þ sinðpxðt  hÞ  0.5ÞjxðthÞ¼0.85 ¼ 53.6254. From the relationship in Eqs. (34) and (10), we have Dð0Þ ¼ 1.1010;

N1 ¼ 0.1229;

N2 ¼ 0.1005.

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J.H. Park, O.M. Kwon / Chaos, Solitons and Fractals 27 (2006) 1011–1018

Now, for given C1 = 1 and C2 = 0.1, by applying the optimization problem given in Theorem 2 to the above system, we found that the LMI solutions of Theorem 2 are as follows: X ¼ 0.0571;

Z 1 ¼ 1.2922
105 ;

P1 ¼ 11.6853;

P2 ¼ 0.0764;

Z 2 ¼ 0.0025;

Y ¼ 10.0000;

a ¼ 21.2407.

Thus, the gain of a stabilizing guaranteed cost controller is K ¼ YX 1 ¼ 175.2219; and the upper bound of cost function is J ¼ a þ P1 þ P2 ¼ 37.0025.

5. Concluding remarks In this article, the design method of guaranteed control controller for stabilizing time-delay chaotic systems has been proposed. The delay-dependent stabilization criteria are derived in terms of LMIs by using the Lyapunov functional stability method. The criteria can be easily obtained by convex optimization algorithms. The design procedure is illustrated by a numerical example.

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