Physics Letters A 323 (2004) 251–259 www.elsevier.com/locate/pla
Control of chaotic n-dimensional continuous-time system with delay Yonglu Shu a,∗ , Bangding Tan b , Chuandong Li a a Department of Applied Mathematics, Chongqing University, Chongqing 400044, PR China b Department of Electrical Engineering, Chongqing University, Chongqing 400044, PR China
Received 8 October 2003; received in revised form 31 January 2004; accepted 3 February 2004 Communicated by A.P. Fordy
Abstract We study the control of a first-order continuous-time chaotic vector system with delay. With and without parameter uncertainties are discussed respectively. A system consisting of two cellular equations is taken as a simulation example. 2004 Elsevier B.V. All rights reserved. PACS: 02.30.Ks; 05.45.Gg; 05.45.Jn Keywords: Adaptive control; Chaos; Delay; Stability
1. Introduction Over the last decades there has been a great interest to harness the very peculiar chaotic behaviour in deterministic systems. While suppression of chaos is aimed in many cases (e.g., chaos in the brain, cardiac chaos), its irregular behaviour is solicited in several other applications (e.g., secure communication). After the pioneering work on controlling chaos introduced by Ott et al. [8], there have been many other attempts to control chaotic systems. Actually, we can classify the developed methods into two main streams: parameter perturbations of an accessible system parameter, and introduction of an additive control law to the original chaotic system. Recently, adaptive control of chaos received an increasing interest. With adaptive control, chaotic trajectories can be controlled to converge to the specific orbits with certain or uncertain system parameters [4–6]. An adaptive control of a chaotic system with unknown parameters was studied by Yangs [6]. It is well known that a first-order time delay dynamic system may result in complicated chaotic motion [7]. In a recent paper [1], the authors studied the general scalar form of first-order continuous-time chaotic system with
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[email protected] (Y. Shu). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.02.007
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delay
y(t) ˙ = −y(t) + f y(t), y(t − τ ), p ,
(1)
where y is a scalar p may be a scalar or a vector parameter. They “proved” two theorems one with parameter certainties and another with parameter uncertainties, but both of their proof are incorrect. We remained the proof of [1] and generalized (1) to the following general vector form y˙ (t) = −y(t) + f y(t), y(t − τ ), p , (2) ) m where y(t) ∈ Rn , y˙ (t) = dy(t dt , p ∈ R is a parameter. A new first-order continuous-time chaotic system with delay (14) is put forward to illustrate our results. The details of the dynamical properties of the system (14) will be studied in a forthcoming paper.
2. The control model By introducing an external control term u into the system (1), we get the general form of control system as y˙ (t) = −y(t) + f y(t), y(t − τ ), p + u. (3) The objective of our control is to find a law of control term u to meet a given specified goal represented by a performance index function. Suppose the desired system is described by y˜˙ = g y˜ (t), t . (4) So (4) serves for a reference model and (3) is referred to the process model. Let e(t) = y(t) − y˜ (t), the control goal is to force the state difference e(t) toward zero as t tends to infinite, that is e(t) → 0,
as t → ∞.
(5)
From (3) and (4), we obtain the equation for e(t) as of the form e˙ (t) = −y(t) + f y(t), y(t − τ ), p − g y˜ (t), t + u.
(6)
2.1. Adaptive control with parameter certainties Taking into account of the effects of time delay to the system behavior, the delay term should be included in the control action u. We introduce the control action u as a nonlinear feedback control function u = y˜ (t) − f y˜ (t), y˜ (t − τ ), p + g y˜ (t), t − k y(t) − y˜ (t) , (7) where k is a scalar parameters. Using the control action (7) for the control problem (3)–(5), we have the following theorem. Theorem 1. If function f : Rn × Rn × Rm → Rn , f : (x1, x2 , p) → f(x1 , x2 , p) ∈ Rn is Frechet differentiable with respect to x1 , x2 , and its Frechet derivative satisfy following conditions: ∂f ∂f (a) m1 , ∂x m2 (m1 , m2 ∈ R) ∂x 1
2
and the parameter k satisfy m22 − k < 0. 4 Then the closed-loop control system described by (3)–(7) is asymptotically stable with the control goal. (b) m1 +
Y. Shu et al. / Physics Letters A 323 (2004) 251–259
Proof. Considering of Lyapunov function define below, 1 V = e(t), e(t) + 2
0
e(t + θ ), e(t + θ ) dθ.
−τ
Then V˙ = e(t), e˙ (t) +
0
e(t + θ ), e˙ (t + θ ) dθ
−τ
= e(t), e˙ (t) + e(t), e(t) − eτ (t), eτ (t) = e(t), −g(˜y, t) − y(t) + f y(t), yτ (t), p + y˜ (t) − f y˜ (t), y˜ τ (t), p + g y˜ (t), t − ke(t) + e(t), e(t) − eτ (t), eτ (t) = − e(t), ke(t) − eτ (t), eτ (t) + e(t), f y(t), yτ (t), p − f y˜ (t), y˜ τ (t), p . And by the mean value theorem [2] we have e(t), f y(t), yτ (t), p − f y˜ (t), y˜ τ (t), p f y(t), yτ (t), p − f y˜ (t), y˜ τ (t), p e(t) = f y(t), yτ (t), p − f y˜ (t), yτ (t), p + f y˜ (t), yτ (t), p − f y˜ (t), y˜ τ (t), p e(t) f y(t), yτ (t), p − f y˜ (t), yτ (t), p + f y˜ (t), yτ (t), p − f y˜ (t), y˜ τ (t), p e(t) ∂f(y + λ(˜y − y), yτ (t), p) sup ˜y − y e(t) ∂y 0<λ<1 ∂f(˜y(t), yτ + λ(˜yτ − yτ ), p) + sup ˜yτ − yτ e(t) ∂yτ 0<λ<1 2 m1 e(t) + m2 eτ (t) e(t). Thus
2 2 2 V˙ −k e(t) − eτ (t) + m1 e(t) + m2 eτ (t)e(t) 2 2 m2 e(t) − eτ (t) + m1 + m2 − k e(t)2 < 0. =− 2 4
Furthermore, V˙ = 0 if and only if e(t) = 0. So e(t) → 0, as t → ∞.
✷
Note. This theorem corrected and generalized Theorem 1 of [1]. Corollary 1. Consider the simplified Hopfield neural network [3] y˙ = −y + Af y(t) + Bf y(t − τ ) + c, where A, B ∈ Rn×n , c ∈ Rn , f(y(t)) = (f1 (y1 (t)), . . . , fn (yn (t)))T . Let u = y˜ (t) − Af y˜ (t) − Bf y˜ (t − τ ) + g y˜ (t), t − k y(t) − y˜ (t) − c n n 2 2 and max1in {supx∈R |fi (x)|} = m, A = i,j =1 aij , B = i,j =1 bij .
If k > nm(A + 14 nBm), then y˙ = −y + Af(y(t)) + Bf(y(t − τ )) + c + u synchronizes to y˜˙ = g(˜y(t), t).
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2.2. Adaptive controls with parameter uncertainties In real applications, the parameters of a chaotic system are not always accessible. And the parameters may be time-varying. When some parameters of a chaotic system are unknown, adaptive control technology is a effective mean to control the chaotic system. If p is unknown in the system (2), we take the control action u as of following: u = y˜ (t) − f y˜ (t), y˜ (t − τ ), p˜ + g y˜ (t), t − k y(t) − y˜ (t) . (8) We assume the estimate values p˜ of p satisfies the following dynamic equations: ˜ T ∂f(˜y(t), y˜ (t − τ ), p) e(t). p˜˙ = ∂ p˜
(9)
For a practical system, even we do not know the exact value of the parameter p, but we know that p is always bounded, therefore the estimate parameter p˜ is also bounded. Theorem 2. If function f : Rn × Rn × Rm → Rn , f : (x1, x2 , p) → f(x1, x2 , p) ∈ Rn is linear in p and is Frechet differentiable with respect to x1 , x2 , and its Frechet derivative satisfy following conditions: ∂f ∂f m1 , (a) ∂x m2 (m1 , m2 ∈ R) ∂x1 2 and the parameter k satisfy m22 − k < 0. 4 Then the closed-loop control system described by (3)–(7) is asymptotically stable with the control goal. (b) m1 +
Proof. Considering of Lyapunov function define below, V=
1 1 e(t), e(t) + (ep , ep ) + 2 2
0
e(t + θ ), e(t + θ ) dθ,
−τ
˜ then where ep = p − p, V˙ = e˙ (t), e(t) + (˙ep , ep ) + 2
0
−τ
e(t + θ ), e˙ (t + θ ) dθ
= e˙ (t), e(t) + (˙ep , ep ) + e(t), e(t) − eτ (t), eτ (t) = −g(˜y, t) − y(t) + f y(t), yτ (t), p + y˜ (t) − f y˜ (t), y˜ τ (t), p˜ + g y˜ (t), t − ke(t) , e(t) + (˙ep , ep ) + e(t), e(t) − eτ (t), eτ (t) = − e(t), ke(t) − eτ (t), eτ (t) + (˙ep , ep ) + e(t), f y(t), yτ (t), p − f y˜ (t), y˜ τ (t), p˜ 2 2 = −k e(t) − eτ (t) + (ep , e˙ p ) + f y(t), yτ (t), p − f y˜ (t), y˜ τ (t), p˜ , e(t) 2 2 ˜ T ∂f(˜y(t), y˜ (t − τ ), p) = −k e(t) − eτ (t) + ep , − e(t) ∂ p˜ + f y(t), yτ (t), p − f y˜ (t), y˜ τ (t), p , e(t)
Y. Shu et al. / Physics Letters A 323 (2004) 251–259
2 2 ˜ ∂f(y(t), y(t − τ ), p) ep , e(t) = −k e(t) − eτ (t) − ∂ p˜ + f y(t), yτ (t), p − f y˜ (t), y˜ τ (t), p , e(t) 2 2 = −k e(t) − eτ (t) + f y(t), yτ (t), p − f y˜ (t), yτ (t), p , e(t) ˜ ∂f(˜y(t), y˜ (t − τ ), p) ep , e(t) + f y˜ (t), y˜ τ (t), p − f y˜ (t), y˜ τ (t), p˜ − ∂ p˜ + f y˜ (t), yτ (t), p − f y˜ (t), y˜ τ (t), p , e(t) . And by the mean value theorem [2] we have f y(t), yτ (t), p − f y˜ (t), yτ (t), p , e(t) f y(t), yτ (t), p − f y˜ (t), yτ (t), p e(t) ∂f(y + λ(˜y − y), yτ (t), p) ˜y − y e(t) m1 e(t)2 sup ∂y 0<λ<1
255
(10)
(11)
and f y˜ (t), yτ (t), p − f y˜ (t), y˜ τ (t), p , e(t) f y˜ (t), yτ (t), p − f y˜ (t), y˜ τ (t), p e(t) ∂f(˜y(t), yτ + λ(˜yτ − yτ ), p) sup ˜yτ − yτ e(t) m2 e(t)eτ (t) ∂yτ 0<λ<1
(12)
as f is linear in p, we have ˜ ∂f(˜y(t), y˜ (t − τ ), p) f y˜ (t), y˜ τ (t), p − f y˜ (t), y˜ τ (t), p˜ − ep = 0. ∂ p˜
(13)
Thus 2 2 2 V˙ −k e(t) − eτ (t) + m1 e(t) + m2 eτ (t)e(t) 2 2 m22 m2 e(t) − eτ (t) − k e(t) < 0. =− + m1 + 2 4 Furthermore, V˙ = 0 if and only if e(t) = 0. So e(t) → 0, as t → ∞.
✷
Note. This theorem corrected and generalized Theorem 2 of [1].
3. Numerical simulation Consider the following systems with delay. y p sin y2 (t) + q sin y2 (t − τ ) y˙1 =− 1 + . q sin y1 (t) + p sin y1 (t − τ ) y˙2 y2
(14)
Let p = 3, q = 6, τ = 1 and y1 (t) = 2 for −1 t 0, y2 (t) = 1 for −1 t 0. The system (14) is chaotic, this fact is illustrated by its waveform with time span [0, 1000] shown in Fig. 1 and phase graph with time span [0, 1000] shown in Fig. 2.
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Fig. 1. The wave form of the system (14).
Y. Shu et al. / Physics Letters A 323 (2004) 251–259
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Fig. 2. The phase graph of the system (14).
3.1. The case of parameter fixed Take the reference system as ˙ y˜ 1 −0.1t 0.1 sin y˜2 (t) + cos y˜2 (t) ˜ . = g y (t), t = e 0.1 sin y˜1 (t) + cos y˜1 (t) y˙˜ 2
(15)
As e˙ = −g(˜y, t) − y(t) + f y(t), yτ (t), p + y˜ (t) − f y˜ (t), y˜ τ (t), p + g y˜ (t), t − ke(t)
= −(k + 1)e(t) + f y(t), yτ (t), p − f y˜ (t), y˜ τ (t), p
= −(k + 1)e(t) + f y˜ (t) + e , y˜ τ (t) + eτ (t), p − f y˜ (t), y˜ τ (t), p .
(16)
Substitute (14), (15) into (16) we get e˙1 e1 = −(k + 1) e˙2 e2 p sin(y˜2 (t) + e2 ) + q sin(y˜2 (t − τ ) + e2 (t − τ )) − p sin y˜2 (t) − q sin y˜2 (t − τ ) + (17) . q sin(y˜1 (t) + e1 ) + p sin(y˜1 (t − τ ) + e1 (t − τ )) − q sin y˜1 (t) − p sin y˜1 (t − τ ) √ √ As p = 3, q = 6, it is easy to verify that m1 = 47, m2 = 45, and we choose k = 19 > m1 + m22 /4 = √ 47 + 45/5, let τ = 1, and for t ∈ [−1, 0], y˜1 (t) = 1, y˜2 (t) = 2, e1 (t) = 3, e2 (t) = 4. The numerical result with time span [0, 10] is given by Fig. 3.
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Fig. 3. Error with parameter certainty.
3.2. The case of parameter uncertainty Now if we consider the parameter unknown case for the system (12), we assume the parameter p, q is bounded, 2 2 2 2 say p + q < 6. It is easy to verify that m1 = 2 + p + q , m2 = p2 + q 2 . We must make an assumption on p, q, we assume that p2 + q 2 < 6, then we choose k = 19 > 12 + 2 + p2 + q 2 m1 + m22 /3. We also take (14) as a reference model, after some simple calculation we get the error system
e˙ = −(k + 1)e(t) + f y˜ (t) + e , y˜ τ (t) + eτ (t), p˜ + ep − f y˜ (t), y˜ τ (t), p˜ . (18) Substitute (14), (15) into (18) we get
e˙1 = −(k + 1)e1 + (p˜ + ep ) sin y˜2 (t) + e2 + (q˜ + eq ) sin y˜2 (t − τ ) + e2 (t − τ )
− p˜ sin y˜2 (t) − q˜ sin y˜2 (t − τ ), e˙2 = −(k + 1)e2 + (q˜ + eq ) sin y˜1 (t) + e1 + (p˜ + ep ) sin y˜1 (t − τ ) + e1 (t − τ ) − q˜ sin y˜1 (t) − p˜ sin y˜1 (t − τ ). From (9) we get ˙ sin y˜1 (t − τ ) p˜ sin y˜2 (t) e1 = . sin y˜2 (t − τ ) q˙˜ sin y˜1 (t) e2 ˜˙ then there is ˜ we have e˙ p = −p, As ep = p − p, e˙p sin y˜1 (t − τ ) sin y˜2 (t) e1 =− . e˙q sin y˜1 (t) e2 sin y˜2 (t − τ )
(19) (20)
(21)
(22)
Y. Shu et al. / Physics Letters A 323 (2004) 251–259
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Fig. 4. Error with parameter uncertainty.
Let k = 19, p(0) ˜ = 2, q(0) ˜ = 1, ep (0) = −1, eq (0) = 2, τ = 1, and for t ∈ [−1, 0], y˜1 (t) = 1, y˜2 (t) = 2, e1 (t) = 3, e2 (t) = 4. The numerical illustration of the stability of the error system with time span [0, 50] is given by Fig. 4.
4. Conclusions We remained the proof of [1] which consider a general simple delay scalar system governed by a first-order, autonomous, continuous time equation. We also generalized the result of [1] to vector form by using Lyapunov function based method. The numerical simulation results meet our analysis.
References [1] S. Zhou, J. Yu, X. Liao, Adaptive, Control of Chaotic Continuous-time System with Delay, Communications, Circuits and Systems and West Sino Expositions, IEEE, 2002. [2] E. Zeidler, Nonlinear Functional Analysis with Applications, in: Fixed Point Theory, vol. I, Springer-Verlag, Berlin, 1985. [3] S. Zhou, The study of Hopf bifurcation, chaos and control for time delay neural network, PhD thesis, 2002. [4] Y.-C. Tian, F. Gao, Physica D 117 (1998) 1. [5] G. Chen, X. Yu, IEEE Trans. CAS-I: Fund. Appl. 46 (1999) 767. [6] T. Yang, C.-M. Yang, L.-B. Yang, Dynamics Control 8 (1998) 255. [7] H. Lu, Y. He, Z. He, IEEE Trans. Circuits Systems I 45 (1998). [8] E. Ott, C. Grebogi, J.A. Yorke, Phys. Rev. Lett. 64 (1990) 1196.