International Journal of Bifurcation and Chaos, Vol. 17, No. 5 (2007) 1753–1758 c World Scientific Publishing Company
CHAOS SYNCHRONIZATION VIA MULTIVARIABLE PID CONTROL GUILIN WEN State Key Laboratory of Advanced Design and Manufactory for Vehicle Body, College of Mechanical and Automotive Engineering, Hunan University, Changsha, Hunan 410082, P. R. China QING-GUO WANG∗ and CHONG LIN Department of Electrical and Computer Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
[email protected] GUANGYAO LI and XU HAN State Key Laboratory of Advanced Design and Manufactory for Vehicle Body, Hunan University, Changsha, Hunan 420082, P. R. China Received October 20, 2005; Revised September 5, 2006 Synchronization via multivariable PID control is studied. Based on the descriptor approach, the problem of PID controller design is transformed to that of static output feedback (SOF) controller design. The improvement of the solvability of the Linear Matrix Inequality (LMI) is achieved, in comparison with the existing literature on designing PID controller based on the LMI technique. With the aid of the free-weighting matrix approach and the S-procedure, the synchronization criterion for a general Lur’e system is established based on the LMI technique. The feasibility of the methodology is illustrated by the well-known Chua’s circuit. Keywords: Chaos synchronization; multivariable PID control; the descriptor approach; static output feedback; linear matrix inequality(LMI); free-weighting matrix approach.
1. Introduction The seminal paper on chaos synchronization [Pecora & Carroll, 1990] has stimulated a wide range of research activity including both theoretical studies [Chen & Dong, 1998; Wen et al., 2006; Femat & Solis-Perales, 1999; Suykens et al., 1999; Curran et al., 1997] and practical applications [Cuomo & Oppenheim, 1993; Itoh & Murakami, 1995; Kawata
∗
et al., 1999]. Chaos synchronization has been addressed using observers with linear output feedback [Morgul & Solak, 1996], PI observers [Hua & Guan, 2005; Hector & Alvarez-Ramirez, 2000; Busawon & Kabore, 2001] and nonlinear observers [Grassi & Mascolo, 1997; Wen & Xu, 2004, 2005]. Yassen [2005] investigated chaos synchronization under adaptive control. Jiang and Zheng [2005]
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proposed a linear-state-feedback synchronization criterion based on linear matrix inequality (LMI). To our best knowledge, no work is reported in the literature on chaos synchronization via full proportional-integral-derivative (PID) control. This may result from the fact that in PID control studies many prevalent PID controller design methods were established on basis of frequency response methods. The intrinsic characteristic of synchronization implies that the state-space approach is preferable for serving our purpose. However, PID control has a long history in control engineering [˚ Astr¨om & H¨agglund, 1995]. Its simplicity in architecture makes it much easier to be understood and used by the control engineer [Luyben, 1990] than advanced controllers. Hence, in most real industrial applications, the PID controller is still widely used even though a large number of advanced control techniques have been proposed. Furthermore, it is well known in the area of PID control that the integral control is mainly employed to improve the steady state tracking accuracy while the derivative control enhances stability and speed the system response [˚ Astr¨om & H¨agglund, 1995]. In other words, the derivative control is desirable to increase synchronization speed and the integral control is available for disturbance attenuation. These characteristics imply that if there had existed appropriate design methods of PID controller, PID control for synchronization should have prevailed. In this paper, our aim is to propose a multivariable PID controller design to achieve the masterslave synchronization of Lur’e systems [Khalil, 1993]. Due to the fact that measuring all the state variables of a system is inconvenient or even impossible in many practical situation [Luyben, 1990], output feedback control is considered. The control system is transformed to a descriptor system [Lin, 2004] such that the LMI technique, with the help of the free-weighting matrix approach [He et al., 2004a, 2004b, 2005; Wu et al., 2004a, 2004b] and the S-procedure [Boyd et al., 1994], is applicable to derive the robust synchronization criterion. The paper is organized as follows. Section 2 describes the Lur’e system and the master-slave synchronization scheme. The PID controller design problem is formulated in Sec. 3. In Sec. 4, the feasibility of the synchronization criterion is demonstrated by the paradigm in nonlinear physics — the Chua’s circuit. Section 5 concludes this paper.
2. Problem Formulation Consider the following master-slave synchronization scheme x(t) ˙ = Ax(t) + Bσ(C T x(t)) M: y(t) = Hx(t) (1) ˆ(t)) + u(t) x ˆ˙ (t) = Aˆ x(t) + Bσ(C T x S: yˆ(t) = H x ˆ(t) with PID controller of the form u(t) = Kp (y(t) − yˆ(t)) + Ki
t
(y(θ) 0
˙ − yˆ˙ (t)) − yˆ(θ))dθ + Kd (y(t)
(2)
where the master system M and slave system S are Lur’e systems with control input u ∈ Rn , state vectors x, x ˆ ∈ Rn , outputs y, yˆ ∈ Rl respectively, and matrices H ∈ Rl×n , A ∈ Rn×n , B ∈ Rn×nh , C ∈ Rn×nh , satisfying C T = [c1 c2 · · · cnh ]T with cj ∈ Rn , j = 1, 2, . . . , nh . The superscript T stands for the matrix transpose. σ(·) satisfies a sector condition with σj (·), j = 1, 2, . . . , nh , belonging to sectors [0, gj ], i.e. σj (ν)(σj (ν) − gj ν) ≤ 0,
∀ν, for j = 1, 2, . . . , nh . (3)
For the controller (2), Kp , Ki and Kd are the proportional, integral and derivative gain matrices, respectively. Let the synchronization error of system (1) be e(t) = x(t) − x ˆ(t). The error dynamic system is given by ˙ = Ae(t) + Bη(C T e(t), xˆ(t)) − u(t) e(t) t He(θ)dθ + Kd H e(t) ˙ u(t) = Kp He(t) + Ki 0
(4) ˆ) − σ(C T x ˆ). where η(C T e(t), xˆ(t)) = σ(C T e + C T x Assume that for ∀ e, x ˆ, j = 1, 2, . . . , nh , the nonlinearity η(C T e(t), xˆ(t)) belongs to sector [0, gj ], i.e. 0≤
ηj (cTj e, xˆ) cTj e
=
σ(cTj e + cTj x ˆ) − σ(cTj x ˆ) cTj e
≤ gj . (5)
Equation (5) implies that ˆ)(ηj (cTj e, x ˆ) − gj cTj e) ≤ 0, ηj (cTj e, x ∀ e, x ˆ, j = 1, 2, . . . , nh .
(6)
In this paper, we will study robust synthesis of output feedback PID control to achieve synchronization of system (1), i.e. e(t) → 0 as t → ∞.
Chaos Synchronization via Multivariable PID Control
3. Algebraic Condition for Synchronization In order to apply the LMI technique [Boyd et al., 1994] for PID controller design, we transform system (1) to a system in descriptor form [Lin, 2004]. Introduce a new state variable e(t) z1 (t) t (7) z(t) = z2 (t) = He(t)dθ . 0 z3 (t) e(t) ˙ Then system (1) with (2) is transformed into the following SOF control system in the descriptor form, ˜ ˜ ˆ E z(t) ˙ = Az(t) + Bη(t) + Bu(t), ˜ ye (t) = Hz(t),
(8)
u(t) = [Kp , Ki , Kd ]ye (t), where
0 0 , 0
0 A˜ = H A
In E =0 0
0 Il 0
0 ˆ = 0 , B −In
0 ˜ = 0 , B −B
0 In 0 0 , 0 −In
H ˜ H = 0 0
0 Il 0
0 0 . H
Let us now investigate robust synchronization terion which is applicable for the derivation of control parameters Kp , Ki and Kd . Construct following Lyapunov function:
P P z1 11 12 V (t) = [z1T , z2T ] T P12 P22 z2 n h cTj z1 λj σj (s)ds +2 j=1
crithe the
(9)
0
where P12 ∈ Rn×l , P11 ∈ Rn×n , P22 ∈ Rl×l , hP P12 i 11 > 0 and Λ = diag(λ1 , λ2 , . . . , λnh ) ≥ 0 P T P22 12
are to be determined. For any W = diag(w1 , w2 , . . . , wnh ) ≥ 0 and S = diag(s1 , s2 , . . . , snh ) ≥ 0, it follows from (3) and (6) that ˜ 0 ≤ S(t) nh [wj σj (σj − gj cTj z1 ) + sj ηj (ηj − gj cTj z1 )] = −2 j=1
= 2(z1T CGW σ − σ T W σ + z1T CGSη − η T Sη), (10)
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where G = diag(g1 , g2 , . . . , gnh ). Inequality (10) is a standard application of the S-procedure [Boyd et al., 1994]. Note that 0 = e(t) ˙ − z3 (t) = (A − Kp H)z1 − Ki z2 − (I + Kd H)z3 + Bη.
(11)
It is clear from (11) that for two given scalars δ1 and δ2 and any appropriately dimensional matrix N , the following relationship holds, ˜ 0 = L(t) = 2[z1T δ1 N + z2T Hδ2 N + z3T N ] · [(A − Kp H)z1 − Ki z2 − (I + Kd H)z3 + Bη],
(12)
where N, δj N, j = 1, 2, are considered as freeweighting matrices [He et al., 2004a, 2004b, 2005]. Let ξ T (t) = [z1T , z2T , z3T , σ T , η T ], M1 = N Kp , M2 = N Ki , M3 = N Kd . Taking the time derivative of V (t) and adding the terms on the right-hand side of Eqs. (10) and (12) into V˙ (t), one obtains ˜ + L(t) ˜ V˙ (t) ≤ S(t)
+ 2[z1T , z2T ] =
P11 T P12
P12 P22
z3 Hz1
(13)
+ 2σ T ΛC T z3 ξ T (t)Ψξ(t),
where Ψ = (ψij ), i, j = 1, . . . , 5, is a symmetric matrix with ψ11 = P12 H + δ1 NA − δ1 M1 H + (P12 H + δ1 NA − δ1 M1 H)T , T − δ1 M2 + (δ2 HNA − δ2 HM 1 H)T , ψ12 = H T P22 ψ13 = P11 − δ1 N − δ1 M3 H + (NA − M1 H)T , ψ14 = CGW, ψ15 = δ1 NB + CGS, ψ22 = −δ2 HM 2 − (δ2 HM 2 )T , T − δ2 HN − δ2 HM 3 H − M2T , ψ23 = P12 ψ24 = 0, ψ25 = δ2 HNB, ψ33 = −N − M3 H − (N + M3 H)T , ψ34 = CΛ, ψ35 = NB, ψ44 = −2W, ψ45 = 0, ψ55 = −2S.
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Thus, V˙ (t) < −εe(t)2 for a sufficiently small ε if Ψ < 0,
(14)
which ensures the asymptotic stability of equilibrium point e = 0. Note that for two given scalars δ1 and δ2 , LMI (14) is solvable for the variables P11 , P12 , P22 , Λ, W, S, N and Mj , j = 1, 2, 3, by using the LMI technique. Thus, one can obtain the gains Kp = N −1 M1 , Ki = N −1 M2 and Kd = N −1 M3 . Remark 1. It should be noted that if one takes t
z2 (t) = 0 e(θ)dθ as in [Lin, 2004], the solution set of LMI (14) becomes empty if matrix H is not of full column rank. This property is strongly conservative for output feedback control. However, if the new state variable takes the form in (7), i.e. t z2 (t) = 0 He(t)dθ, the aforesaid conservativeness is fully overcome. Moreover, in comparison to the design of PID control in [Lin, 2004], the dimension of the descriptor system (8) is reduced by (m − l).
δ1 = δ2 = 2.4. It is easy to compute LMI (14) and the original PID gains in (2) are given by −3750.6 −4846.4 Kp = 66.8 87.5 , 161.4 198.0 −3730.9 65.8 Ki = 161.4
−1566.9 27.4 Kd = 67.2
−4865.3 88.5 , 212.3
(17)
−2020.3 35.8 . 88.5
4. An Example To illustrate the merits and effectiveness of our results, consider the paradigm in nonlinear physics — Chua’s circuit [Yalcin et al., 2001]: x˙ = a(y − h(x)), y˙ = x − y + z, (15) z˙ = −by, with nonlinear characteristic 1 (16) h(x) = m1 x + (m0 − m1 )(|x + c| − |x − c|), 2 and parameters a = 9, b = 14.28, c = 1, m0 = −(1/7), m1 = 2/7. The system can be represented in Lur’e form by 27 18 9 0 7 − 7 A= , B = , −1 1 0 1 0 −14.28 0 0
(a)
1 C = 0 , 0 and σ(ν) = (1/2)(|ν + c| − |ν − c|) belongs to sector [0, g1 ] with g1 = 1, i.e. σ(ν)(σ(ν) − g1 ν) ≤ 0 and nh = 1. Suppose that the output matrix 1 0 0 H = 0 1 0 , which is not of full column rank. Let
(b) Fig. 1. Synchronization results under PID control: the time history of (a) the synchronization error e(t); (b) the control signal u(t).
Chaos Synchronization via Multivariable PID Control
and the S-procedure are used to derive the robust synchronization criterion, by which the merits of derivative and integral parts of the controller are applicable for speeding synchronization and disturbance attenuation, respectively. On comparing with the existing literature on designing PID controller based on LMI, improvement of the solvability is achieved.
2
10
1
10
0
variance of |e(t)|
10
−1
10
−2
10
Acknowledgments
−3
10
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10
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−2
10
−1
10
γ
0
10
1
10
Fig. 2. The log–log plot of the variances of |e(t)| via the noise level γ.
With the gains, the master-slave synchronization of chaotic system (1) under PID control is simulated. In Fig. 1(a), the time history of the error system (4) shows that synchronization is achieved quickly. Figure 1(b) exhibits the control signal. We finally discuss how noises affect the error dynamics. Assume that for Chua’s circuit (15) and (16), there exist random noises in the output signals of the master system M and the slave system S as follows: ˘ 2 (t), ˘ 1 (t), yˆ(t) = H x ˆ(t) + Bγ y(t) = Hx(t) + Bγ (18) ˘ = [1, 1]T , the positive constants γ1 and where B γ2 represent the noise levels (magnitudes), and (t) stands for the uniformly distributed random signal, bounded by | (t)| ≤ 1. It follows from Fig. 1 that with the gains Kp , Ki and Kd given in (17), synchronization in the noiseless case can be achieved before t = 5. Let γ = γ1 − γ2 = 0.01, 0.1, 1 and 10, respectively. During t ∈ [5, 100], the results of the variances of |e(t)| (briefly, var(|e(t)|)) via the noise level γ are shown by the log–log plot in Fig. 2. It implies that the error during synchronization will not be asymptotically stable, but remain bounded in the presence of noise. When 0 < γ < 1, we have var(|e(t)|) < γ. However, var(|e(t)|) exponentially increases and var(|e(t)|) > γ if γ > 1.
5. Conclusion We have investigated the master-slave synchronization of Lur’e system under PID control. The LMI technique, the free-weighting matrix approach
This work was supported by the national 973 program of China under the grant number 2004CB719402 and by the academic research fund of National University of Singapore (No: R-263-000306-112).
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