Physics Letters A 310 (2003) 40–43 www.elsevier.com/locate/pla
Robust adaptive tracking control for a class of uncertain chaotic systems Zhi Li a,b,∗ , Guanrong Chen b , Songjiao Shi a , Chongzhao Han c a Department of Automation, Shanghai Jiaotong University, Shanghai 200030, China b Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong, China c School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China
Received 22 August 2002; received in revised form 23 December 2002; accepted 16 January 2003 Communicated by A.R. Bishop
Abstract This Letter considers the robust adaptive tracking control problem of a class of uncertain chaotic systems with time-varying unknown but bounded parameters. A novel controller is designed based on the Lyapunov stability theory. The proposed controller ensures that the states of the controlled chaotic systems globally asymptotically track the desired bounded trajectories. Simulation results verify the effectiveness of the proposed controller. 2003 Elsevier Science B.V. All rights reserved. PACS: 05.45.Gg Keywords: Robust adaptive control; Tracking control; Uncertain chaotic system; Rössler system
1. Introduction In recent years, there have been numerous achievements in the design of effective controllers for chaotic systems [1–4]. More recently, some adaptive control methods [5–9] are proposed for classes of chaotic systems with uncertain parameters based on the Lyapunov stability theory. The main advantage of the existing methods is that the controller is directly constructed by using analytic formulas without knowing in advance the values of the unknown parameters. However, these
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E-mail address:
[email protected] (Z. Li).
methods are valid only for a small class of chaotic systems with unknown constant parameters. When these unknown parameters are not constant, how do we deal with the control problems for such systems? In this Letter, a novel robust adaptive controller is designed based on the Lyapunov stability theory for a class of chaotic systems with time-varying unknown but bounded parameters, which can achieve globally asymptotically tracking of the desired bounded trajectories. There are some salient differences between the result obtained in this Letter and previously published works [7–9]. Firstly, the assumption that system’s unknown parameters are constant is no longer needed. These unknown parameters can vary in bounded intervals and the bounds of the intervals need not be known.
0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00115-4
Z. Li et al. / Physics Letters A 310 (2003) 40–43
In order to overcome the limitation of the previous control schemes, the key idea here is to set the unknown bounds of the uncertain parameter as a new update object. Moreover, the method of constructing the controller here is different from the previous works.
Consider an uncertain nonlinear chaotic system of the form x˙ = f (x) + F (x)θ (t) + u,
(1)
where x ∈ R n represents the state vector of the system, f ∈ R n is a vector of continuously differentiable nonlinear functions, F ∈ R n×p denotes a matrix of continuously differentiable nonlinear functions, u ∈ R n is the control input, and θ (t) ∈ R p is a timevarying uncertain parameter vector, which is assumed to vary in a bounded closed set and the bound of the set is unknown. p Assumption 1. θ (t) ∈ Θ ⊂ R p , Θ = i=1 [θ i , θ i ] is a bounded closed set and θ i , θ i (i = 1, 2, . . . , p) are unknown constant parameters, where θ (t) = [θ1 (t), θ2 (t), . . . , θp (t)]T . From Assumption 1, we have p 1/2 2 θ (t) = θi (t)
p
max |θ i |2 , |θ i |2
In order to achieve the goal of Eq. (4), there are many possible choices for the controller. We choose the controller in the following form:
1/2 := β.
(2)
Note that β ∈ R is an unknown constant parameter. Let xd be an arbitrarily given desired bounded smooth trajectory. The error dynamical system between Eq. (1) and the desired trajectory is easily derived as e˙ = f (e + xd ) + F (xd + e)θ (t) − x˙d + u.
(3)
The goal of control is to find a robust adaptive controller, u, such that the solutions of Eq. (3) are robust asymptotically stable at e = 0, that is lim e(t) = lim x(t) − xd (t) = 0 (4) t →∞
for all θ (t) that satisfy Assumption 1.
(5)
where βˆ ∈ R is an estimated value of the unknown constant parameter β ∈ R, which will be used as the updating object in the design of the controller below. Substituting Eq. (5) into Eq. (3), we have ˆ e˙ = f˜(e, xd ) + F (xd + e)θ (t) + α(e, xd , β),
(6)
where e = x − xd , f˜(e, xd ) = f (xd + e) − f (xd ), and ˆ is a compensator to be designed. α(e, xd , β) Assumption 2. For the arbitrarily given xd , e˙ = f˜(e, xd ) is asymptotically stable at e = 0, that is, ∀(e, t) ∈ R n × R + , there exists a positive number, γ > 0, and a scalar function with well-defined continuous first derivatives V : R n × R + → R + such that (1) λmin e2 V (e, t) λmax e2 , ∂V (e, t) dV = + grad V , f˜(e, xd ) (2) dt ∂t −γ e2 , where grad V =
i=1
t →∞
3. Main results
ˆ u = x˙d − f (xd ) + α(e, xd , β),
2. Problem formulation
i=1
41
(7)
(8)
∂V
∂V ∂V T ∂e1 , ∂e2 , . . . , ∂en .
Theorem 1. If Assumptions 1 and 2 hold, then there ˆ and exists a robust adaptive compensator α(e, xd , β) ˙ ˆ a parameter estimation update law β such that the solutions of Eq. (6) are robust asymptotically stable at e = 0, where ˆ =− α(e, xd , β)
F (e + xd )µ(e, xd )βˆ 2 , µ(e, xd )βˆ + εe2
µT (e, xd ) = (grad V )T F (e + xd ), ˆ 0 ) > 0, 0 < ε < γ . β˙ˆ = µ(e, xd ), β(t
(9) (10) (11)
Proof. For Eq. (6), choose the Lyapunov function ˜ t) = V (e, t) + 1 β˜ 2 . Since V (e, t) satisfies W (e, β, 2
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Z. Li et al. / Physics Letters A 310 (2003) 40–43
Assumption 2, we have
4. Simulation
∂V (e, t) dW = + grad V , f˜(e, xd ) dt ∂t ˆ + grad V , F (e + xd )θ + α(e, xd , β)
In this section, the Röossler chaotic system [10] is used as an example to demonstrate the effectiveness of the proposed controller. The Rössler chaotic system is described by −x2 − x3 x˙1 x˙2 = x1 x˙3 x1 x3 0 0 0 a + x2 0 (13) b + u, 0 c 0 1 −x3
˙ˆ ˜ β) + β(−
−γ e2 + F T (e + xd ) grad V β F (e + xd )µ(e, xd )βˆ 2 − β˜ β˙ˆ µ(e, xd )βˆ + εe2 = −γ e2 + µ(e, xd )β − (grad V )T
µ(e, xd )2 βˆ 2 − β˜ β˙ˆ µ(e, xd )βˆ + εe2 −γ e2 + µ(e, xd )β − µ(e, xd )βˆ + εe2 − β˜ β˙ˆ = −(γ − ε)e2 + β˜ µ(e, xd ) − β˙ˆ
where
−
= −(γ − ε)e2 .
−x2 − x3 f (x) = , x1 x1 x3
(12)
Therefore, the solutions of Eq. (6) in about equilibrium point e(t) = 0, and βˆ = β, are globally uniformly stable. It follows that e(t) and βˆ are globally bounded for all t 0. Vector e(t) is square-integrable by inequality (12). From Eq. (5), along with Eqs. (9), (10) and (11), and xd is bounded, we can show that the controller u is globally bounded for all t 0. Consequently, Eq. (3), along with Eq. (5) and Assumption 1, implies that the e(t) ˙ is bounded for all t 0. By Barblat’s lemma [12], we conclude that e(t) → 0, as t → ∞. That is, limt →∞ e(t) = 0. This completes the proof. ✷ From Eq. (3) with Eq. (5) and Theorem 1, we obtain the following result.
0 F (x) = x2 0
0 0 1
0 , 0 −x3
u is the control input, and θ (t) = [ a b c ]T is the unknown time-varying parameter vector of the system. It has been proved that the Rössler system presents chaos when a = b = 0.2, c = 5.7 and u = 0. Let the desired orbit, xd = [ 0 cos t sin t ]T , the initial state vector of the system, and the unknown parameter vector, be taken in the simulation as x(0) = [ 2 −4 −0.3 ] and
θ (t) = 0.2(1 − sin t)
0.2 cos t
5.7
T
,
respectively. Since 3 1/2 θ (t) = θi (t)2
i=1 3
1/2 2 2 max |θ i | , |θ i | = (32.69)1/2,
i=1
Theorem 2. If Assumptions 1 and 2 hold, then there exists a robust adaptive controller u and a parameter estimation update law β˙ˆ such that the states of Eq. (1) robust asymptotically track the arbitrarily given desired bounded and smooth orbit xd , where ˆ α(e, xd , β) ˆ u = x˙d − f (xd ) + α(e, xd , β),
the true value of the unknown parameter is taken as β = (32.69)1/2, and the initial value of the estimated ˆ = 0.2. unknown parameter β is taken as β(0) According to Eqs. (5), (9), (10) and (11), we construct the controller u and the update law for estimation of the unknown constant parameter β as follows:
and β˙ˆ are given by Eqs. (9), (10) and (11), respectively.
ˆ u = x˙d − f (xd ) + α(e, xd , β),
(14)
Z. Li et al. / Physics Letters A 310 (2003) 40–43
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λmin = 10, λmax = 15, γ = 20. Then, e˙ = f˜(e, xd ) + ur (e, xd ) satisfies Assumption 2. By Theorems 1 and 2, we conclude that the robust adaptive controller u along with the update law given by Eqs. (14)–(18) will make the states of the Rössler system robustly asymptotically track the desired orbit xd = [ 0 cos t sin t ]T , as verified by the simulation results shown in Fig. 1.
5. Conclusion
Fig. 1. The time response of the error states between the Rössler system and the desired orbit xd = [ 0 cos t sin t ]T , when the controller is deactivated (t < 15 s) and the controller with parameter update law given by Eqs. (14)–(18) is activated as Eq. (13) at t = 15 s.
ˆ = ur (e, xd ) − α(e, xd , β)
F (e + xd )µ(e, xd )βˆ 2 , (15) µ(e, xd )βˆ + εe2
µT (e, xd ) = (grad V )T F (e + xd ), ˆ = 0.2, β(0) β˙ˆ = µ(e, x ),
(16)
ε = 8,
(17)
In this Letter, by employing the Lyapunov stability theory and making unknown bounds of the uncertain parameters as a new update objective in the construction of the tracking controller, we have solved the problem of robust adaptive tracking control for a class of nonlinear continuous-time chaotic systems with unknown time-varying parameters. Applying the Lyapunov design technique to the robust adaptive tracking control of nonlinear discrete-time uncertain chaotic systems remains a topic for further research, which is quite different from the continuous-time case [11].
d
where ur (e, xd ) =
γ = 20,
−1 −1 − sin t
1 1 −1 0 e. 0 −1 − e1
(18)
Choose the Lyapunov functions 1 ˜ t) = V (e, t) + β˜ 2 , W (e, β, 2 where 10 0 0 P = 0 10 0 , 0 0 15
V (e, t) = eT P e, (19)
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