International Journal of Bifurcation and Chaos, Vol. 16, No. 10 (2006) 3087–3091 c World Scientific Publishing Company
DELAY-DEPENDENT SYNCHRONIZATION CRITERION FOR LUR’E SYSTEMS WITH DELAY FEEDBACK CONTROL YONG HE, GUILIN WEN and QING-GUO WANG∗ Department of Electrical and Computer Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 ∗
[email protected] Received March 9, 2005; Revised August 8, 2005 A master-slave synchronization scheme for Lur’e systems is studied for a known delay existing between master and slave systems. Based on the latest development of stability studies for time-delay systems, a new delay-dependent synchronization criterion is derived by the freeweighting matrix approach. The criterion shown by example is less conservative than the existing synchronization criteria. Keywords: Chaos; synchronization; time-delay feedback; delay-dependent criterion; linear matrix inequality (LMI); free-weighting matrix approach.
1. Introduction In recent years, the research on chaotic synchronization has received considerable attention [Pecora & Carroll, 1991; Curran & Chua, 1997; Curran et al., 1997; Suykens & Vandewalle, 1997; Chen & Dong, 1998; Suykens et al., 1999; Chen & Liu, 2000; Yalcin et al., 2001; Cao et al., 2005]. An overview of synchronization methods has been presented in [Chen & Dong, 1998]. A number of master-slave synchronization schemes for Lur’e systems have been proposed [Curran & Chua, 1997; Curran et al., 1997; Suykens & Vandewalle, 1997; Suykens et al., 1999]. Chen and Liu [2000] first handled propagation delay in master-slave synchronization schemes and introduced the possibility of applying synchronization to optical communication. For the two remote chaotic systems, the existence of a time-delay may destroy synchronization. Yalcin et al. [2001] studied master-slave schemes with identical Lur’e systems and supposed that the output of a master system
was received at the slave systems with delay, which is assumed to be a known value. Recently, Cao et al. [2005] added the linear feedback term into the delay feedback controller to assist the time-delay feedback control term to stabilize the error systems. However, it is clear that the time-delay feedback control term becomes useless if the nondelay feedback term is available and used because the latter can stabilize the error system directly. For the delay-dependent global asymptotic stability criterion presented in [Yalcin et al., 2001], the first-order model transformation for the error system was employed. However, there exist some additional eigenvalues in the transformed system when the first-order model transformation [Fridman & Shaked, 2003] was employed. Thus, it may be not equivalent to the original one, see e.g. [Gu & Niculescu, 2000]. Although Park [1999] and Moon et al. [2001]’s inequalities and descriptor model transformation [Fridman & Shaked, 2003] can overcome the conservatism of the first-order model
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transformation, there is room for further investigation. The free-weighting matrix approach proposed in [He et al., 2004a, 2004b] and [Wu et al., 2004] is the most efficient one to handle the delay-dependent problems. In this paper, a master-slave synchronization scheme for Lur’e systems is studied for a known delay existing between master and slave systems. The free-weighting matrix approach is employed to derive a delay-dependent synchronization criterion. Finally, a numerical example demonstrates the effectiveness and improvement over the existing results.
2. Time-Delay Synchronization Scheme Consider a general master-slave type of coupled Lur’e systems: x(t) ˙ = Ax(t) + Bσ(C(x(t))) M: p(t) = Hx(t) y(t) ˙ = Ay(t) + Bσ(C(y(t))) + u(t) (1) S: q(t) = Hy(t) C : u(t) = M(p(t − τ ) − q(t − τ )) with master system M, slave system S and controller C, where the time-delay τ > 0. The master and slave systems are Lur’e systems with state vectors x, y ∈ Rn , outputs of subsystems p, q ∈ Rl respectively, and matrices H ∈ Rl×n , A ∈ Rn×n , B ∈ Rn×nh , C ∈ Rnh ×n . σ(·) satisfies a sector condition with σj (·), j = 1, 2, . . . , nh , belonging to sectors [0, kj ], i.e. σj (ξ)(σj (ξ) − kj ξ) ≤ 0, ∀ ξ, for j = 1, 2, . . . , nh . (2) The scheme aims at synchronizing the master system to the slave system by applying full output error feedback to the slave systems with control signal u ∈ Rn with feedback gain matrix M ∈ Rn×l . Now, defining the synchronization error as e(t) = x(t) − y(t), one has the error-dynamics system of the form: e(t) ˙ = Ae(t) + Bη(Ce(t), y(t)) + Fe(t − τ ),
(3)
where F = −MH and η(Ce(t), y(t)) = σ(Ce +Cy) − σ(Cy). Let C = [c1 c2 . . . cnh ]T with cj ∈ Rn , j = 1, 2, . . . , nh . One assumes that the nonlinearity η(Ce(t), y(t)) belongs to sectors [0, kj ][Curran & Chua, 1997; Curran et al., 1997; Suykens & Vandewalle, 1997; Chen & Dong, 1998; Suykens
et al., 1999; Chen & Liu, 2000; Yalcin et al., 2001; Cao et al., 2005], i.e. 0≤
ηj (cTje, y) cTje
=
σ(cTje + cTjy) − σ(cTjy)
≤ kj , cTje ∀ e, y, j = 1, 2, . . . , nh .
(4)
This implies that ηj (cTje, y)(ηj (cTje, y) − kj cTje) ≤ 0, ∀ e, y, j = 1, 2, . . . , nh .
(5)
3. Delay-Dependent Synchronization Criterion In this section, the free-weighting matrices are employed to express the relationship between the terms in the Leibniz–Newton formula and the errordynamics systems, and a delay-dependent synchronization criterion is derived. Theorem 1. For a given scalar δ, the error system (3) has an unique and globally asymptotically stable equilibrium point e = 0 if there exist P = P T > 0, Q = QT > 0, Z = Z T > 0, Λ = diag(λ1 , λ2 , . . . , λnh ) ≥ 0, T = diag(t1 , t2 , . . . , tnh ) ≥ 0, S = diag(s1 , s2 , . . . , snh ) ≥ 0, and any appropriate dimensional matrices G, V and Nj , j = 1, . . . , 5, such that the following LMI (6 ) is feasible, 2 6 6 6 6 6 6 Φ=6 6 6 6 6 6 4
Φ11
Φ12
Φ13
Φ14
Φ15
ΦT 12
Φ22
C TΛ
−GB
Φ25
ΦT 13
ΛC
−2T
0
−N3
ΦT 14
T
0
−2S
−N4
T
−B G
ΦT 15
ΦT 25
−N T 3
−N T 4
Φ55
−τ N T 1
−τ N T 2
−τ N T 3
−τ N T 4
−τ N T 5
−τ N1
3
7 −τ N2 7 7 7 7 −τ N3 7 7 7 −τ N4 7 7 7 −τ N5 7 5 −τ Z
(6)
< 0,
where Φ11 Φ12 Φ13 Φ14 Φ15 Φ22 Φ25 Φ55 K
= −δGA − δATGT + Q + N1 + N T1 , = P + δG − ATGT + N T2 , = C TKT + N T3 , = −δGB + C TKS + N T4 , = δVH + N5T − N1 , = G + GT + τ Z, = VH − N2 , = −Q − N5 − N T5 , = diag{k1 , k2 , . . . , knh }.
Moreover, a delay feedback gain matrix is given by M = G−1 V .
Delay-Dependent Synchronization Criterion for Lur’e Systems with Delay Feedback Control
Construct the Krasovskii functional:
following
Proof.
V(e(t)) = eT(t)Pe(t) + 2
λj
j=1 t
+ t−τ
nh
0
≤ 2eT(t)P e(t) ˙ + 2σ T(Ce(t))ΛC e(t) ˙
Lyapunov–
+ [eT(t)Qe(t) − eT(t − τ )Qe(t − τ )] t T ˙ − e˙T(s)Z e(s)ds ˙ + τ e˙ (t)Z e(t)
σj (s)ds
t−τ
T
e (s)Qe(s)ds
+ 2eT(t)C TKT σ(Ce(t))
− 2σ T(Ce(t))T σ(Ce(t))
t
+ −τ
0
cT j e
t+θ
e˙T(s)Z e(s)dsdθ, ˙
PT
QT
+ 2eT(t)C TKSη T(Ce(t), y(t))
(7)
− 2η T(Ce(t), y(t))Sη(Ce (t), y(t))
ZT
> 0, Q = > 0, Z = >0 where P = and Λ = diag(λ1 , λ2 , . . . , λnh ) ≥ 0 are to be determined. For any appropriate dimensional matrices Ri , i = 1, 2, the following relationship holds through system (3), ˙ − Ae(t) 0 = [eT(t)R1 + e˙ T(t)R2 ] · [e(t) − Bη(Ce(t), y(t)) − Fe(t − τ )].
˙ + [eT(t)R1 + e˙T(t)R2 ] · [e(t) − Ae(t) − Bη(Ce(t), y(t)) − Fe(t − τ )] + [eT(t)N1 + e˙T(t)N2 + σ T(Ce(t))N3 + η T(Ce(t), y(t))N4 + eT(t − τ )N5 ] t e(s)ds ˙ · e(t) − e(t − τ ) −
(8)
Using the Leibniz–Newton formula, for any appropriate dimensional matrices Nj , j = 1, . . . , 5, the following is true, 0 = [eT(t)N1 + e˙T(t)N2 + σ T(Ce(t))N3 + η T(Ce(t), y(t))N4 + eT(t − τ )N5 ] t e(s)ds ˙ . · e(t) − e(t − τ ) −
[tj σj (cTje)(σj (cTje) − kj cTje)
j=1
(10)
Calculating the derivative of V(e(t)) along the solution of system (3) and adding the terms on the right ˙ of Eqs. (8)–(10) into V(e(t)) yield: nh j=1
λj σj (cTje)cTje(t) ˙
+ [eT(t)Qe(t) − eT(t − τ )Qe(t − τ )] t T ˙ − e˙T(s)Z e(s)ds ˙ + τ e˙ (t)Z e(t) t−τ
t
t−τ
ζ T(t, s)Ψζ(t, s)ds,
(11)
σ T(Ce(t))
η T(Ce(t), y(t))
eT(t − τ ) e˙T(s)]T ,
(9)
nh
˙ ˙ +2 V(e(t)) = 2eT(t)P e(t)
1 τ
t−τ
ζ(t) = [eT(t) e˙T(t)
On the other hand, for any T = diag(t1 , t2 , . . . , tnh ) ≥ 0 and S = diag(s1 , s2 , . . . , snh ) ≥ 0, it follows from (2) and (5) that
+ sj ηj (ηj − kj cTje)] = 2eT(t)C TKT σ(Ce(t)) − 2σ T (Ce(t))T σ(Ce(t)) + 2eT(t)C TKSη T(Ce(t), y(t)) − 2η T(Ce(t), y(t))Sη(Ce (t), y(t)).
=
where
t−τ
0 ≤ −2
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2 6 6 6 6 6 6 Ψ =6 6 6 6 6 4
Ψ11
Ψ12
Φ13
Ψ14
Ψ15
ΨT 12
Ψ22
C TΛ
−R2 B
Ψ25
ΦT 13
ΛC
−2T
0
−N3
T
0
−2S
−N4
−N T 3 −τ N T 3
−N T 4 −τ N T 4
Φ55
ΨT 14 ΨT 15 −τ N T 1
−B R2T ΨT 25 −τ N T 2
−τ N T 5
−τ N1
3
7 −τ N2 7 7 7 −τ N3 7 7 7, −τ N4 7 7 7 −τ N5 7 5 −τ Z
Ψ11 = −R1 A − ATR1T + Q + N1 + N T1 ,
Ψ12 = P + R1 − ATR2T + N T2 , Ψ14 = −R1 B + C TKS + N T4 , Ψ15 = −R1 F + N5T − N1 , Ψ22 = R2 + R2T + τ Z, Ψ25 = −R2 F − N2 .
˙ Thus, V(e(t)) < −εe(t)2 for a sufficiently small ε for Ψ < 0, which ensures the asymptotic stability of equilibrium point e = 0. Setting R2 = G and R1 = δG, and letting V = GM , Ψ becomes Φ. Φ < 0 implies that G + GT is negative definite, and G is nonsingular. Then, M = G−1 V . This completes the proof.
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Remark 1. The free-weighting matrix approach pro-
posed in [He et al., 2004a, 2004b] and [Wu et al., 2004] has been employed to derive the delaydependent synchronization criterion such that no model transformations are used in the procedure of proving Theorem 1. However, the first-order model transformation which may introduce some additional eigenvalues was employed in [Yalcin et al., 2001] and may lead to conservatism.
0 −a(m0 − m1 ) 1, B = 0 , 0 0 C=H = 1 0 0, (14)
−am1 A = 1 0
a −1 −b
and σ(ξ) = (1/2)(|ξ + c| − |ξ − c|) belongs to sector [0, k] with k = 1. Let δ = 2, LMI (6) is feasible for τ ∈ [0, 0.18] with M = [3.9125
4. An example Let us take the following representation of Chua’s Circuit [Yalcin et al., 2001] x˙ = a(y − h(x)), y˙ = x − y + z, (12) z˙ = −by, with nonlinear characteristic 1 h(x) = m1 x + (m0 − m1 )(|x + c| − |x − c|), 2
(13)
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
0.9545
−3.8273].
On the other hand, the error system (3) with M given in (15) is simulated and the synchronization is observed until the maximum delay τ = 0.25. When the initial conditions of the master and slave systems are x(0) = [−0.2 −0.33 0.2]T , y(0) = [0.5 −0.1 0.66]T , the synchronization is exhibited in Fig. 1 for the upper bound of delay τ = 0.18 and the maximum delay τ = 0.25 resulting from simulation. However, the upper bounds of delay τ derived by the matrix inequality in (9) in [Yalcin et al., 2001] and their simulation result are only 0.039 and 0.21, respectively. This implies that our criterion is less conservative than that in [Yalcin et al., 2001].
1
τ
ERROR for =0.18 0.5
||e(t)||
0
−0.5 −1
0
5
10
15
20
25
30
35
40
45
50
45
50
t 1
τ
ERROR for =0.25 0.5
||e(t)||
0
−0.5 −1
0
5
10
15
20
25
30
35
40
t Fig. 1.
(15)
The time history of error e(t) with τ = 0.18 and τ = 0.25.
Delay-Dependent Synchronization Criterion for Lur’e Systems with Delay Feedback Control
It should be pointed out that though the result in [Cao et al., 2005] is τ ∈ [0, 0.3), it is achieved with the control law: u(t) = −K(x(t) − y(t)) + M (p(t − τ ) − q(t − τ )), (16) where current state feedback is used. It is well known that different types of control laws will lead to different control performance, and the state and/or nondelay feedback will usually yield stronger results than the corresponding output and/or delay feedback. Therefore, the result in [Cao et al., 2005] is not comparable with ours. To our best knowledge, our result of τ ∈ [0, 0.18) is the strongest one in the literature so far. Furthermore, we are of opinion that the delay feedback alone should be used in the synchronization problem for which the output of the master system is supposed to be received at the slave system with delay [Yalcin et al., 2001], and that in other applications where the current feedback is available, the current feedback alone should be used since then the delay feedback is redundant information and becomes useless.
5. Conclusion In this paper, a master-slave synchronization scheme for Lur’e systems is studied for a known delay existing between master and slave systems. The free-weighting matrix approach is employed to derive a delay-dependent synchronization criterion. Finally, a numerical example demonstrates the effectiveness and improvement over the existing results.
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Chen, H. F. & Liu, J. M. [2000] “Open-loop chaotic synchronization of injection-locked semiconductor lasers with Gigahertz range modulation,” IEEE J. Quant. Electron. 36, 27–34. Curran, P. F. & Chua, L. O. [1997] “Absolute stability theory and the synchronization problem,” Int. J. Bifurcation and Chaos 7, 1375–1382. Curran, P. F., Suykens, J. A. K. & Chua, L. O. [1997] “Absolute stability theory and master-slave synchronization,” Int. J. Bifurcation and Chaos 7, 2891–2896. Fridman, E. & Shaked, U. [2003] “Delay-dependent stability and H∞ control: Constant and time-varying delays,” Int. J. Contr. 76, 48–60. Gu, K. & Niculescu, S. I. [2000] “Additional dynamics in transformed time delay systems,” IEEE Trans. Automat. Contr. 45, 572–575. He, Y., Wu, M., She, J. H. & Liu, G. P. [2004a] “Delaydependent robust stability criteria for uncertain neutral systems with mixed delays,” Syst. Contr. Lett. 51, 57–65. He, Y., Wu, M., She, J. H. & Liu, G. P. [2004b] “Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties,” IEEE Trans. Automat. Contr. 49, 828–832. Moon, Y. S., Park, P., Kwon, W. H. & Lee, Y. S. [2001] “Delay-dependent robust stabilization of uncertain state-delayed systems,” Int. J. Contr. 74, 1447–1455. Park, P. [1999] “A delay-dependent stability criterion for systems with uncertain time-invariant delays,” IEEE Trans. Automat. Contr. 44, 876–877. Pecora, L. M. & Carroll, T. L. [1991] “Synchronization in chaotic systems,” Phys. Rev. Lett. 64, 821–824. Suykens, J. A. K. & Vandewalle, J. [1997] “Master-slave synchronization of Lur’e systems,” Int. J. Bifurcation and Chaos 7, 665–669. Suykens, J. A. K., Curran, P. F. & Chua, L. O. [1999] “Robust synthesis for master-slave synchronization of Lur’e Systems,” IEEE Trans. Circuits Syst.-I 46, 841–850. Wu, M., He, Y., She, J. H. & Liu, G. P. [2004] “Delaydependent criteria for robust stability of time-varying delay systems,” Automatica 40, 1435–1439. Yalcin, M. E., Suykens, J. A. K. & Vandewalle, J. [2001] “Master-slave synchronization of Lur’e systems with time-delay,” Int. J. Bifurcation and Chaos 11, 1707–1722.
