Robust control of time-delay chaotic systems

Physics Letters A 314 (2003) 72–80 www.elsevier.com/locate/pla

Robust control of time-delay chaotic systems Changchun Hua ∗ , Xinping Guan Institute of Electrical Engineering, Yanshan University, Qinghuangdao 066004, China Received 13 September 2002; received in revised form 23 April 2003; accepted 14 May 2003 Communicated by A.P. Fordy

Abstract Robust control problem of nonlinear time-delay chaotic systems is investigated. For such uncertain systems, we propose adaptive feedback controller and novel nonlinear feedback controller. They are both independent of the time delay and can render the corresponding closed-loop systems globally uniformly ultimately bounded stable. The simulations on controlling logistic system are made and the results show the controllers are feasible.  2003 Elsevier B.V. All rights reserved. PACS: 05.45.+b Keywords: Robust control; Time-delay chaotic systems; Adaptive control

1. Introduction Chaos has been found in many engineering systems. A fundamental characteristic of a chaotic system is its extreme sensitivity to initial conditions. In many practical applications, chaos is required to be removed to obtain improved performance and avoidance of fatigue failure. Therefore, within the research area of nonlinear dynamics, the controlling or ordering of chaos is receiving increasing attention. Chaos control was firstly investigated by Ott et al. [1]. Since 1990, many papers appeared and many methods are proposed such as [1–11] and the references therein. Adaptive control method [3–5], time-delayed feedback control method [6], sliding control method [7], observer based control method [8] and other control methods [9–12] have been developed to control chaos. However, the existing papers are mainly focused on controlling chaotic systems without time delay. As we know, many systems have been found to contain time delay, and the behavior of timedelay systems are more complex, a simple nonlinear systems with time delay in state may produce chaos such as logistic system and MG system [17]. Therefore, it is important to investigate the control problem of time-delay chaotic systems. Tian and Gao [4] investigated the adaptive control problem of continuous chaotic systems with time delay. But there are no disturbances and uncertainties in the considered system, and the controller is dependent of the time delay. Guan et al. [13] investigated time-delayed feedback control for time-delay chaotic systems, while * Corresponding author.

E-mail address: [email protected] (C. Hua). 0375-9601/03/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/S0375-9601(03)00817-X

C. Hua, X. Guan / Physics Letters A 314 (2003) 72–80

73

this method needed to linearize the chaotic system at the equilibrium point, so when there are disturbances in system, this method may fail. In this Letter, robust control problem of nonlinear time-delay chaotic systems is investigated. We propose two classes of continuous state feedback controllers which are adaptive controllers and novel nonlinear feedback controllers. They are both independent of the time delays and can render the corresponding closed-loop systems globally uniformly ultimately bounded stable.

2. Problem formulation Consider the following class of time-delay chaotic systems with additional control input
 x˙ = Ax + Bu + BE x(t − τ ), t ,

(1)

where x ∈ R n , u ∈ R m and E(x(t − τ ), t) is an unknown nonlinear function with time delay which may contain uncertain sections. A and B are known matrices with proper dimensions. Tian and Gao [4] pointed out that if system (1) is simple first-order and satisfies eight conditions (C1–C8 in [4]), it may be chaotic. Many time-delay chaotic systems investigated are in the form of system (1). We assume that system (1) satisfies the following assumptions. Assumption 1. There exist positive matrices P , Q and positive scalar µ satisfying the following Riccati equation AT P + P A − µP BB T P = −Q.

(2)

Assumption 2. Nonlinear time-delay section E(x(t − τ ), t) satisfies the following inequality s   i   E(x(t − τ ), t)
αi x(t − τ ) ,

(3)

i=1

where s and αi are known scalar and unknown positive scalar, respectively. Remark 1. Assumption 1 is standard and denotes the internally stability of the nominal system. Different from the assumptions in existing literatures investigating robust control for time-delay systems, we assume that the uncertain section is not bounded by a linear function (for (3) s = 1), but bounded by a nonlinear function (3). Refs. [14] and [15] propose different control strategy to investigate the robust stabilization for system (1) with uncertainties bounded by a linear function and the bounds are not known. In this Letter we will investigate the control problem of this class of time delay chaotic systems with unknown scalars αi .

3. Adaptive control In this section, we will construct an adaptive state feedback controller to render the closed-loop system uniformly ultimately bounded stable. Theorem 1. For system (1) satisfying Assumption 1 and Assumption 2, the following state feedback controller can render the closed-loop system stable in the sense of uniform ultimate boundedness: ∂V T 1 , u = − µB T P x − θ¯ B T 2 ∂x

(4)

74

C. Hua, X. Guan / Physics Letters A 314 (2003) 72–80

where µ and P satisfy (2), and V=

s 

1
T i x Px , i

Vi =

Vi ,

i=1

(5)

θ¯ is the adaptive parameter with adaptive law 2   ˙θ¯ = k   ∂V B  − kl θ¯ ,  ∂x 

(6)

where k and l are adjustable positive scalars. Proof. We first define a Lyapunov–Krasovskii functional candidate for closed-loop system as follows W (x, θ¯ ) =

s 

s t 

Vi +

i=1

i=1 t −τ

2i  1
i x(z) dz + θ˜ 2 , 2k

(7)

¯ θ is defined as where
i is positive scalar, θ˜ = θ − θ, θ=

s  αi2 . 4
i

(8)

i=1

Then, by taking the time derivative of W (·) along the trajectories of closed-loop system, it is obtained that for any t  t0   s  ∂V 2
 1 d θ˜  ∂V dW (x, θ¯ ) 
T i−1 T  T  = BE x(t − τ ), t + θ˜ − θ¯  B x Px x A P + P A − µP BB T P x +  ∂x  dt ∂x k dt i=1

+

s 

s 2i  2i  
i x(t) −
i x(t − τ ) .

i=1

(9)

i=1

From (3) we know

   s   s i  2i  
  ∂V   αi2  ∂V 2 ∂V   BE x(t − τ ), t
 B  + Fi x(t − τ ) αi x(t − τ )
 ∂x B  ∂x 4
i  ∂x  i=1

(10)

i=1

substituting Eqs. (2), (6),(10) into (9), we have   s s   2i 1  ∂V 2 
T i−1 T dW (x, θ¯ ) 
− B x Px x Qx + θ˜  + Fi x(t) + θ˜ θ˙˜  dt ∂x k i=1


−

s  

i=1

 2i ¯ ¯ λi−1 min (P )λmin (Q) − Fi x + l θ (θ − θ ).

i=1

As we know, there always exists Fi satisfying the following inequality: λi−1 min (P )λmin (Q) − Fi = βi > 0. Further, the following inequality is obtained: s  dW (x, θ¯ ) 1 1
− βi x 2i − l θ¯ 2 + lθ 2 . dt 2 2 i=1

(11)

C. Hua, X. Guan / Physics Letters A 314 (2003) 72–80

75

As we know, θ is a constant and l is an adjustable parameter, it is easy to obtain that closed-loop system is robust uniformly ultimately bounded stable in light of Lyapunov stability theory. ✷ Remark 2. From Eq. (11), we know that one can obtain the upper bound on the steady-state as small as desired by decreasing the parameter l. So, the system designer can tune the size of the residual set by adjusting properly this parameter l in the adaptive law.

4. Novel nonlinear feedback control In this section we will propose a class of state feedback controllers to stabilize system (1). Theorem 2. For system (1) satisfying Assumption 1 and Assumption 2, the following novel feedback controller u = −ρ(x)

B T P xρ(x) 1 − (µ + v)B T P x T

B P x ρ(x) + f (t) 2

(12)

will render the closed-loop system uniformly ultimately bounded stable, where µ satisfies (2) and v is an adjusted positive scalar, ρ(x) and f (t) are as follows: ρ(x) =

s  

 ξi x i+1 + δi x i ,

f (t) = ϑe−rt ,

(13)

i=1

where ξi , δi , ϑ and r are adjustable parameters. Proof. Substituting Eqs. (12) and (13) into (1), we have


 B T P xρ(x) 1 T x˙ = Ax + BE x(t − τ ), t + B −ρ(x) T − (µ + v)B P x .

B P x ρ(x) + f (t) 2

(14)

Define the following Lyapunov function for system (1): W (x) = x T P x. Then by taking the time derivative of W (·) along the trajectories of closed-loop system (14), we obtain that for any t  t0

 W˙ = x T P A + AT P − µP BB T P x − vx T P BB T P x + 2x T P BE x(t − τ ), t −

2 B T P x 2 ρ 2 (x) .

B T P x ρ(x) + f (t)

(15)

We know that the following inequality holds:   2B T P x ρ(x) − 2f (t)


2 B T P x 2 ρ 2 (x) .

B T P x ρ(x) + f (t)

(16)

Substituting Eqs. (2) and (16) into (15), the following relation can be obtained: s  i   αi x(t − τ ) W˙
−x T Qx − vx T P BB T P x + 2B T P x  i=1 s  T      − 2 B Px ξi x i+1 + δi x i + 2ϑe−rt . i=1

(17)

76

C. Hua, X. Guan / Physics Letters A 314 (2003) 72–80

Now let us apply the well-known Razumikhin lemma [16]:
 W x(t − τ )
q 2 W (x), where q > 1 is a positive scalar. For W = x T P x, then we can obtain     x(t − τ )
qφ x(t), √ where φ = max(P )/ min(P ). Substituting (18) into (17), we have

(18)

s  i   αi (qφ)i x(t) W˙
−x T Qx − vx T P BB T P x + 2ϑe−rt + 2B T P x  i=1

  − 2B T P x 

s 



ξi x i+1 + δi x i



i=1 s     = −x T Qx − vx T P BB T P x + 2ϑe−rt + 2B T P x  αi (qφ)i − δi − ξi x x i . i=1

In the case δi  αi

(qφ)i ,

we can obtain

W˙
−x T Qx + 2ϑe−rt . It is easy to obtain that the closed-loop system (1) and (12) is robustly asymptotically stable. If δi < αi (qφ)i , we assume ηi = αi (qφ)i − δi , the following relation can be obtained: s     W˙
−x T Qx − vx T P BB T P x + 2ϑe−rt + 2B T P x  ηi x i − ξi x i+1 .

(19)

i=1

It is well known that for positive scalars ηi and ξi , the following inequality holds:

i+1 ξi ηi i i i+1 ηi x − ξi x

< , i ξi (i + 1) the proof is given in Appendix A. Substituting (20) into (19), we can get

i+1 s   T  T 2  ηi i ξi T −rt x P B .   ˙ W
−x Qx − v x P B + 2ϑe + i ξi (i + 1)

(20)

(21)

i=1

Let h=

s  ξi i=1

i

ηi i ξi (i + 1)

i+1 ,

then the following inequality can be obtained:  2   W˙
−x T Qx − v B T P x  + hB T P x  + 2ϑe−rt

√  T  h 2 h2 h2
−x T Qx + + 2ϑe−rt . = −x T Qx − v B P x  + √ + 2ϑe−rt + 4v 4v 2 v

(22)

As we know, h and v are adjustable positive parameters, it is easy to obtain that the closed-loop system is uniformly ultimately bounded stable. ✷

C. Hua, X. Guan / Physics Letters A 314 (2003) 72–80

77

Fig. 1. The behaviour of logistic time-delay chaotic system.

Remark 3. If δi is selected sufficiently large, the closed-loop system will be asymptotically stable. If δi < αi (qφ)i , we can get (22), and from (22) one can obtain the upper bound on the steady-state x(t) as small as desired by increasing parameter υ. In practical systems the designer can choose proper parameters v, ξi and δi to render the controlled system satisfying special performance level required and also the consuming energy less.

5. Illustrative example In this section we will make computer simulations on controlling logistic time-delay chaotic system. Consider the following system x˙ = −26x + bx(t − d) − cx 2 (t − d) + u.

(23)

If the parameters b = c = 106 and d = 0.5, the system (23) without control is chaotic, the behavior is shown in Fig. 1. And we know that system (23) is in the form of (1). Therefore, basing on Theorem 1 and Theorem 2, we can construct the following controllers.  Adaptive feedback controller. Basing on Section 3, for controller (4)–(6) we let µ = 0, V = 50 2i=1 1i x 2i , then the feedback controller is
 u(t) = −100θ¯ x + x 3 (24) with corresponding adaptive law 2
¯ θ¯˙ = 100 x + x 3 − 0.01θ.

(25)

Novel feedback controller. From Section 4, we can construct the following controller: u = −ρ(x)

xρ(x) − 30x, |x|ρ(x) + f (t)

(26)

where ρ(x) = 100|x| + 150x 2 + |x|3,

f (t) = e−t .

(27)

For simulation the initial value is chosen as x(0) = 1 and θ¯ (0) = 0, sample time is T = 0.001 s. The simulation results are shown in Figs. 2–5. Figs. 2 and 3 are the state and the control response curves of the closed-loop system,

78

C. Hua, X. Guan / Physics Letters A 314 (2003) 72–80

Fig. 2. The state response of closed-loop system with adaptive controller.

Fig. 3. The control input curve of closed-loop system with adaptive controller.

Fig. 4. The state response of closed-loop system with nonlinear controller.

Fig. 5. The control input curve of the closed-loop system with nonlinear feedback controller.

respectively, with the adaptive controller. Figs. 4 and 5 are the curves under nonlinear feedback controller. From the figures, we can see that the closed-loop system is uniformly ultimately bounded stable. If there exist disturbances in system (23), for example, b = 106 +10 sin(t), c = −106 +10 cos(t), the simulation results are shown Figs. 6–9 by using adaptive feedback controller (24) and (25) and novel feedback controller (26) and (27), respectively. From the figures we can also see that the both controllers can render the corresponding uncertain closed-loop systems stable in the sense of uniform ultimate boundedness. When there exist disturbances in chaotic systems, there is no existing method to solve this class of systems. In this Letter we propose two kinds of controllers, and the simulation results illustrate the validity of the controllers. Remark 4. In this section the simulation on controlling logistic system are investigated. In fact, for MG time-delay chaotic system x˙ = −20x +

k1 x(t − τ ) + u, 1 + x k2 (t − τ )

C. Hua, X. Guan / Physics Letters A 314 (2003) 72–80

Fig. 6. The state trajectory of system with adaptive controller.

Fig. 7. The control input curve of system with adaptive controller.

Fig. 8. The state trajectory of system with novel nonlinear feedback controller.

Fig. 9. The control input curve of system with novel nonlinear feedback controller.

79

where k1 = 40, k2 = 10 and τ = 0.2, the above system is chaotic. We know there exists proper parameter α satisfying    40x(t − τ )         1 + x 10 (t − τ ) 
α x(t − τ ) , so Assumption 2 is satisfied. Therefore, we can construct the corresponding controllers based on Section 3 and Section 4 to stabilize MG system. Here the simulation is omitted.

6. Conclusion In this Letter, the robust control problem of time-delay chaotic systems has been investigated. We propose adaptive feedback controller and novel nonlinear feedback controller. They are both independent of the time delay and can render the closed-loop systems uniformly bounded stable. We make computer simulations on controlling logistic chaotic system, and the results show that the both control approaches are feasible.

80

C. Hua, X. Guan / Physics Letters A 314 (2003) 72–80

Acknowledgement This work is supported by the National Nature Science Foundation of PR China (No. 60274023).

Appendix A Proof of Eq. (20). For f (y) = ay c −by c+1, where a, b and c are positive scalars, y is a positive variable parameter. We want to obtain the maximum value of f (y). The differential of f (y) is df (y) = acy c−1 − b(c + 1)y c . dy ac = 0 (y > 0), then we can obtain that acy c−1 − b(c + 1)y c = 0, further, that y = b(c+1) = y ∗ , and we  2  ∗ <0: want to prove that f (y ∗ ) is the maximum value, so we need to examine that d f (y) 2

Let

df (y) dy

dy

y=y

   d 2 f (y)  c−2 c−1  = ac(c − 1)y − b(c + 1)cy  ∗  y=y dy 2 y=y ∗



c−2

c−2 ac ac ac = ac(c − 1) − b(c + 1)c = −ac < 0. b(c + 1) b(c + 1) b(c + 1) So, f (y ∗ ) is the maximum value of f (y), that is

c+1
 b ac ay c − by c+1
f y ∗ = . c b(c + 1) The proof is completed. ✷

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

E. Ott, C. Grebogi, J.A. Yoke, Phys. Rev. Lett. 64 (1990) 1196. G. Chen, X. Dong, From Chaos to Order—Methodologies, Perspectives, and Applications, World Scientific, Singapore, 1998. Y.J. Cao, Phys. Lett. A 270 (2000) 171. Y. Tian, F. Gao, Physica D 117 (1998) 1. L. Chen, G. Chen, Y. Lee, Inform. Sci. 121 (1999) 27. G. Chen, X. Yu, IEEE Trans. Circuits Systems I 46 (1999) 767. K. Konishi, M. Hirai, H. Kokame, Phys. Lett. A 245 (1998) 511. E. Solak, O. Morgul, U. Ersoy, Phys. Lett. A 279 (2001) 47. F.N. Koumboulis, B.G. Mertzios, Chaos Solitons Fractals 11 (2000) 351. H. Richter, Chaos Solitons Fractals 12 (2001) 2375. Z.P. Jiang, IEEE Trans. Circuits Systems I 49 (2002) 244. L. Chen, Y.Z. Liu, Phys. Lett. A 262 (1999) 350. X. Guan, C. Chen, Z. Fan, H. Peng, Int. J. Bifurc. Chaos (2003) 193. H. Wu, IEEE Trans. Automat. Control 45 (2000) 1697. C. Hua, C. Long, X. Guan, G. Duan, in: American Control Conference, 2002, p. 3365. J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1978. M.C. Mackey, L. Glass, Science 197 (1977) 287.