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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 3, MARCH 2013
Sampled-Data Synchronization of Chaotic Lur’e Systems With Time Delays Zheng-Guang Wu, Peng Shi, Senior Member, IEEE, Hongye Su, and Jian Chu
Abstract— This paper studies the problem of sampled-data control for master–slave synchronization schemes that consist of identical chaotic Lur’e systems with time delays. It is assumed that the sampling periods are arbitrarily varying but bounded. In order to take full advantage of the available information about the actual sampling pattern, a novel Lyapunov functional is proposed, which is positive definite at sampling times but not necessarily positive definite inside the sampling intervals. Based on the Lyapunov functional, an exponential synchronization criterion is derived by analyzing the corresponding synchronization error systems. The desired sampled-data controller is designed by a linear matrix inequality approach. The effectiveness and reduced conservatism of the developed results are demonstrated by the numerical simulations of Chua’s circuit and neural network. Index Terms— Chaotic systems, exponential synchronization, Lur’e systems, neural networks, sampled-data control.
I. I NTRODUCTION
S
INCE the pioneering work of [1], the study on the synchronization of chaotic systems has gradually become an active area of research (see, [1]–[7] and the references therein). This stems from the fact that chaos synchronization has been widely applied in various fields, including chaos generator design, secure communication, chemical reactions, biological systems, and information science. It has been shown that several nonlinear systems, including neural networks and Chua’s circuits, can be represented in the form of Lur’e systems [6], [8]–[11]. Therefore, the master–slave synchronization of chaotic Lur’e systems has been an important topic, and a number of master–slave synchronization criteria have been proposed. For example, in [12], a master–slave type of chaos synchronization problem has been investigated for a general form of Lur’e systems by a time-delay feedback control technique, and some simple algebraic conditions have been derived for easy verification, facilitating the design and application of Manuscript received September 2, 2012; revised November 9, 2012; accepted December 12, 2012. Date of publication January 9, 2013; date of current version January 30, 2013. This work was supported in part by the National Natural Science Foundation of China under Grant 61174029 and Grant 61174058, the National Key Basic Research Program, China, under Grant 2012CB215202, and the 111 Project B12018. Z.-G. Wu, H. Su, and J. Chu are with the National Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou 310027, China (e-mail: [email protected]; [email protected]; [email protected]). P. Shi is with the School of Engineering and Science, Victoria University, Melbourne 8001, Australia, and also with the School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide 5005, Australia (e-mail: [email protected]). Digital Object Identifier 10.1109/TNNLS.2012.2236356
chaos systems. The problem of designing time-varying delay feedback controllers for master–slave synchronization of Lur’e systems has been discussed in [13], where, based on Lyapunov functional approach, some delay-dependent synchronization criteria have been obtained and formulated in the form of linear matrix inequalities (LMIs). It should be noted that two cases of time-varying delays have been fully considered in [13], and thus the results proposed therein are very general and powerful. In [14], the results of [13] were extended to the fault-tolerant master–slave synchronization of Lur’e systems. It is to be noted that time delay is an inherent feature of many physical processes, such as chemical processes, nuclear reactors, and biological systems, and may lead to instability or significantly deteriorate the performance of the corresponding closed-loop systems [15]–[20]. Thus, recently, some attention has also been directed to the synchronization of chaotic Lur’e systems with time delays. For example, in [10], based on the invariant principle of functional differential equations, an adaptive approach has been proposed for the master– slave synchronization of chaotic Lur’e systems with timevarying delays that does not require any prior knowledge of the systems’ parameters. The master–slave synchronization of coupled time-delay Lur’e systems with parameter mismatches has been studied in [21], where, by introducing a type of quasisynchronization, a general methodology has been developed to derive several delay-dependent quasi-synchronization criteria and to give the explicit expression of error levels. Also, the last decade has seen a wealth of research on sampled-data control systems, because modern control systems usually employ digital technology for controller implementation [22]–[29]. Recently, based on the input delay approach [24], the sampled-data master–slave synchronization schemes that consist of identical chaotic Lur’e systems have been studied in [30], where sufficient conditions for global asymptotic synchronization of chaotic Lur’e systems have been obtained using the free-weighting matrix approach and expressed in terms of LMIs. It is to be noted that an obvious and important merit of sampled-data synchronization for chaotic systems is that only the samples of the state variables of the master systems and the slave systems at discrete time instants are needed, and thus the amount of transmitted synchronization information greatly reduces and the efficiency of bandwidth usage increases, which makes the synchronization of chaotic systems more efficient and useful in real-life applications. However, most of the available information about the actual sampling pattern has not been used in [30], because the input delay induced by sample-and-hold is simply treated
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WU et al.: SYNCHRONIZATION OF CHAOTIC LUR’E SYSTEMS WITH TIME DELAYS
as a bounded fast-varying delay (i.e., a time-varying delay without any constraint on the delay derivative), and thus the obtained synchronization conditions have considerable conservatism. The work in [31] and [32] has improved upon the results presented in [30] by applying some piecewise Lyapunov functionals. Compared to the results in [30], the main advantage of the results in [31] and [32] is that the discussed synchronization error system is no longer regarded as a system with fast-varying delay, and the characteristic of the discussed system is captured. However, there still exists room for improving the synchronization criteria of [31] and [32], because some important and useful terms have been ignored in the Lyapunov functionals proposed by [31] and [32], and the available information about the actual sampling pattern has not been fully adopted, which might lead to conservatism to some extent. Moreover, only delay-free chaotic Lur’e systems are taken into account in [30]–[32]. To the best of our knowledge, there is little information in the published literature about sampled-data master–slave synchronization for chaotic Lur’e systems with time delays. Therefore, it is necessary to further study the sampled-data master–slave synchronization for chaotic Lur’e systems with or without time delays, which is the motivation for this paper. In this paper, we consider master–slave synchronization of Lur’e systems with time delays by applying sampleddata control. In order to make full use of the available information about the actual sampling pattern, a novel Lyapunov functional is proposed. The positive definitiveness of the given Lyapunov functional is required only at sampling times but not necessarily inside the sampling intervals, which implies that the traditional continuous-time Lyapunov theorem cannot directly lead to the desired exponential synchronization criterion. Thus, we first introduce a new inequality on the synchronization error system. Based on the new inequality and Lyapunov functional, a criterion is proposed to ensure the exponential stability of the synchronization error systems, and thus the master systems exponentially synchronize with the slave systems. By means of the numerical simulations of Chua’s circuit and neural network, it is shown that the proposed results are effective and can significantly improve the existing ones. Notation: The notations used throughout this paper are fairly standard. Rn and Rm×n denote the n-dimensional Euclidean space and the set of all m × n real matrices, respectively. The notation X > Y (X Y ), where X and Y are symmetric matrices, means that X − Y is positive definite (positive semidefinite). For integers a and b with a < b, N[a, b] = {a, a + 1, . . . , b − 1, b}. λmax (Q) (λmin (Q)) denotes the maximum (minimum) of the eigenvalue of a real symmetric matrix Q. I and 0 represent the identity matrix and a zero matrix, respectively. The superscript “T” represents the transpose, and diag{· · · } stands for a block-diagonal matrix. || · || denotes the Euclidean norm of a vector and its induced norm of a matrix. For an arbitrary matrix B and two symmetric matrices A and C A B ∗ C denotes a symmetric matrix, where “∗” denotes the term that
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is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations. II. P RELIMINARIES Consider the following master–slave synchronization scheme with sampled-data control: x(t) ˙ =Ax(t) + Bx(t − d) + W ϕ(Dx(t)) M : p(t) =C x(t) z˙ (t) =Az(t) + Bz(t − d) + W ϕ(Dz(t)) + u(t) S : q(t) =Cz(t) C : u(t) = K ( p(tk ) − q(tk )),
tk t < tk+1
(1)
which consists of the master system M , the slave system S , and the controller C . M and S with u(t) = 0 are identical chaotic time-delay Lur’e systems with state vectors x(t), z(t) ∈ Rn , outputs of subsystems p(t), q(t) ∈ Rl , respectively. A ∈ Rn×n , B ∈ Rn×n , C ∈ Rl×n , D ∈ Rnh ×n , and W ∈ Rn×nh are known constant matrices, and d > 0 is the constant time delay. u(t) ∈ Rn is the slave system control input, and K ∈ Rn×l is the sampled-data controller gain matrix to be designed. It is assumed that ϕ(·) : Rnh → Rnh is a diagonal nonlinearity with ϕi (·) belonging to sector [0, si ] for i = 1, 2, · · · , n h . For sampled-data synchronization, only discrete measurements of p(t) and q(t) can be used for synchronization purposes, that is, we only have the measurements p(tk ) and q(tk ) at the sampling instant tk . In this paper, the control signal is assumed to be generated by using a zero-order-hold (ZOH) function with a sequence of hold times 0 t0 < t1 < · · · < tk < · · · < limk→+∞ tk = +∞. Also, the sampling is not required to be periodic, and the only assumption is that the distance between any two consecutive sampling instants is less than a given bound [33]. Specifically, it is assumed that tk+1 − tk = h k h
(2)
for all k 0, where h > 0 represents the upper bound of the sampling periods. Given the synchronization scheme (1), the synchronization error is defined as r (t) = x(t) − z(t), and we can get the following synchronization error system: r(t) ˙ = Ar (t) + Br (t − d) +W η(Dr (t), z(t)) − K Cr (tk ),
tk t < tk+1 (3)
where η(Dr (t), z(t)) = ϕ(D(r (t) + z(t))) − ϕ(Dz(t)). As ϕi (·) belongs to sector [0, si ], it can be found that for any i = 1, 2, . . . , n h , and ∀r, z 0
ηi (diTr, z) diTr
=
ϕi [diT (r + z)] − ϕi (diTr ) diTr
si ,
diTr = 0 (4)
where diT denotes the i th row vector of D. It is easily found from (4) that for any i = 1, 2, . . . , n h , and ∀r, z ηi (diT r, z)[ηi (diTr, z) − si diTr ] 0.
(5)
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Based on (5), we have for any matrices Vl = diag {vl1 , vl2 , . . . , vlnh } 0, l = 1, 2, the following inequalities hold: −
nh
vl j η j (d Tj r (t), z(t))(η j (d Tj r (t), z(t))−s j d Tj r (t)) 0
j =1
(6) which are equivalent to T
T
r (t) D SVl η(Dr (t), z(t)) −η(Dr (t), z(t))T Vl η(Dr (t), z(t)) 0
(7)
where S = diag{s1 , s2 , . . . , snh }. In this paper, the following definition and lemmas will be used. Definition 1: The master system M and the slave system S in (1) are said to be exponentially synchronous if the synchronization error system (3) is exponentially stable; that is, there exist two constants α > 0 and β > 0 such that ||r (t)|| βe−αt ||r0 ||c
∀t 0
(8)
where ||rt ||c = sup−d θ 0 {||r (t + θ )||, ||˙r(t + θ )||}. Lemma 1 (Jensen Inequality) [13], [34]: For any matrix W > 0, scalars γ1 and γ2 satisfying γ2 > γ1 , and a vector function ω : [ γ1 , γ2 ] → Rn such that the integrations concerned are well defined, then γ2 ω(α)T W ω(α) dα
(γ2 − γ1 ) γ1
⎡
⎣
γ2
⎤T
⎡
ω(α) dα ⎦ W ⎣
γ1
γ2
⎤ ω(α) dα ⎦.
(9)
γ1
Lemma 2: Consider the synchronization error system (3). Then the following inequality holds: tk 2
2
||r (t)|| θ1 ||r (tk )|| + θ2
2
||r (α)|| , dα, tk t < tk+1
tk −d
(10)
synchronous exist matrices P > 0, Q > 0, Z > 0, if there R1 R2 > 0, X 1 , X 2 , X 3 , X 4 , X 5 , G, L, H = U > 0, ∗ R 3 H1 H2 H3 H4 , and diagonal matrices V1 > 0, and V2 > 0 such that (11) and (12) (see the bottom of next page) hold where Ξ11 = 2α P + Q −
X 1 + X 1T e−2αd − Z + H1 + H1T 2 d
+εG A + ε AT G T Ξ12 = P + H2 − εG + AT G T Ξ13 = X 1 − X 2 + H3 − H1T − εLC + γ AT G T Ξ14 = −X 3 + H4 e−2αd Z + εG B Ξ16 = d Ξ22 = d Z − G − G T Ξ23 = −H2T − LC − γ G T X 1 + X 1T − H3 − H3T − γ LC − γ C T L T Ξ33 = X 2 + X 2T − 2 Ξ34 = −X 4 − e−2αh R2T − H4 X 5 + X 5T e−2αh − R1 Ξ44 = − 2 h −2αd e Ξ66 = −e−2αd Q − Z d ¯ = α h(X ¯ 1 + X 1T ) + h¯ X 3 + h¯ X 3T + h¯ R1 Θ11 (h) X + X 1T ¯ = h¯ 1 Θ12 (h) 2 T ¯ = 2α h(−X ¯ Θ13 (h) 1 + X 2 ) + h¯ X 4 + h¯ R2 X + X 5T ¯ = 2α h¯ X 3 + h¯ 5 Θ14 (h) 2 ¯ = h(−X ¯ Θ23 (h) + X ) 2
1 X 1 + X 1T T ¯ ¯ + h¯ R3 Θ33 (h) = 2α h −X 2 − X 2 + 2 ¯ = α h(X ¯ 5 + X 5T ). Θ44 (h) Furthermore, the sampled-data controller gain matrix in (1) is given by K = G −1 L.
where 2 +||W ||2 ||S D||2 +||B||2 )h 2
θ1 = 5(1 + ||K C||2 h 2 )e5(||A|| 2
θ2 = 5||B|| he
5(||A||2 +||W ||2 ||S D||2 +||B||2 )h 2
.
Proof: See Appendix. The objective of this paper is to design sampled-data controller C in (1) to exponentially synchronize the master system M and the slave system S in (1). III. M AIN R ESULTS In this section, we discuss the sampled-data synchronization problem for chaotic Lur’e systems with time delays. Given the master–slave synchronization scheme in (1), we have the following result. Theorem 1: Given scalars α > 0, ε, and γ , the master system M and the slave system S in (1) are exponentially
(13)
Proof: Consider the following Lyapunov functional for the synchronization error system (3): V (t) =
6
Vi (t),
t ∈ [tk , tk+1 )
i=1
where V1 (t) = e2αt r (t)T Pr (t) t V2 (t) = e2αs r (s)T Qr (s) ds t −d 0
t e2αs r˙ (s)T Z r˙ (s) ds dθ
V3 (t) = −d t +θ
(14)
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413
t V4 (t) = (tk+1 − t)
t e
2αs
T
r˙ (s) U r˙ (s) ds
(tk+1 −t)e
2αt
T
r˙ (t) U r˙ (t) − e
2αt
tk
tk
t V5 (t) = (tk+1 − t)
e2αs tk
r (s) r (tk )
T
R1 R2 ∗ R3
r (s) ds r (tk )
⎡
⎤T ⎡ ⎤ r (t) r (t) ⎢ r (tk ) ⎥ ⎢ r (t ) ⎥ ⎥ H⎢ t k ⎥ V6 (t) = (tk+1 − t)e2αt ⎢ ⎣t ⎣ ⎦ ⎦ r (s) ds r (s) ds tk
X 1 + X 1T −X 1 + X 2 X3 ⎢ 2 ⎢ T ⎢ X1 + X1 H=⎢ X4 ∗ −X 2 − X 2T + ⎢ 2 ⎣ X 5 + X 5T ∗ ∗ 2
⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦
V˙1 (t) = 2e r (t) P r˙ (t) + 2αe r (t) Pr (t) V˙2 (t) = e2αt r (t)TQr (t)−e2αt e−2αd r (t −d)T Qr (t −d) t 2αt T ˙ e2αs r˙ (s)T Z r˙ (s) ds V3 (t) = de r˙ (t) Z r˙ (t) − 2αt
t de
r˙ (t) Z r˙ (t) − e T
2αt
T
(15)
V˙4 (t) = (tk+1 − t)e
e−2αd r˙ (s)T Z r˙ (s) ds
t r˙ (t) U r˙ (t) − e2αs r˙ (s)T U r˙ (s) ds (17) T
tk
t (tk+1 − t)e
2αt
r˙ (t) U r˙ (t) −
e2αtk r˙ (s)T U r˙ (s) ds
T
tk
r (s) ds r (tk )
T r (t) R1 R2 r (t) +(tk+1 − t)e ∗ R3 r (tk ) r (tk ) t T r (s) R1 R2 r (s) ds −e2αt e−2αh ∗ R3 r (tk ) r (tk ) 2αt
+(tk+1 − t)e2αt t = −e
2αt
−2e
r (t) r (tk )
T
R1 R2 ∗ R3
r (t) r (tk )
e−2αh r (s)T R1r (s) ds
tk 2αt
T −2αh
r (tk ) e
t r (s) ds
R2T
tk T −2αh
(16)
t −d 2αt
R1 R2 ∗ R3
tk
t −d 2αt
r (s) r (tk )
e2αs
T r (t) R1 R2 r (t) ∗ R3 r (tk ) r (tk ) T t R1 R2 r (s) 2αtk r (s) − e ds ∗ R3 r (tk ) r (tk )
It is noted that V4 (t), V5 (t), and V6 (t) vanish before tk and after tk . Therefore, V (t) is continuous in time since limt →tk V (t) = V (tk ). Calculating the time derivative of V (t) along the trajectories of (3) gives the following result: T
V˙5 (t) = −
T
+(tk+1 − t)e2αt
tk
⎡
(18)
t tk
and
2αt
e−2αh r˙ (s)T U r˙ (s) ds
−e2αt (t − tk )r (tk ) e R3r (tk ) T r (t) R1 R2 r (t) +(tk+1 − t)e2αt ∗ R3 r (tk ) r (tk ) ⎡ ⎤T ⎡ ⎤ r (t) r (t) ⎢ r (tk ) ⎥ ⎢ r (t ) ⎥ ⎥ H⎢ t k ⎥ V˙6 (t) = −e2αt ⎢ ⎣t ⎣ ⎦ ⎦ r (s) ds r (s) ds tk
(19)
tk
⎤T ⎡ ⎤ r (t) r (t) ⎢ r (t ) ⎥ ⎢ r (tk ) ⎥ ⎥ H⎢ t k ⎥ +2α(tk+1 − t)e2αt ⎢ ⎣ ⎣t ⎦ ⎦ r (s) ds r (s) ds ⎡
tk
tk
tk
⎡
¯ Ξ12 + Θ12 (h) ¯ Ξ13 + Θ13 (h) ¯ Ξ14 + Θ14 (h) ¯ εGW + D T SV1 Ξ11 + Θ11 (h) ⎢ ¯ ¯ ¯ ∗ Ξ22 + hU Ξ23 + Θ23 (h) h X3 GW ⎢ ⎢ ¯ Ξ34 + 2α h¯ X 4 ∗ ∗ Ξ + Θ ( h) γ GW 33 33 ¯ = ⎢ Ξ1 (h) ⎢ ¯ ∗ ∗ ∗ Ξ + Θ ( h) 0 44 44 ⎢ ⎣ ∗ ∗ ∗ ∗ −2V1 ∗ ∗ ∗ ∗ ∗ √ ⎡ ⎤ T T Ξ13 Ξ14 εGW + D SV2 Ξ16 √h¯ H1 Ξ11 Ξ12 ⎢ ⎥ Ξ23 0 GW G B √h¯ H2T ⎥ ⎢ ∗ Ξ22 ⎢ ⎥ ⎢ ∗ ∗ Ξ33 − e−2αh h¯ R3 Ξ34 γ GW γ G B √h¯ H3T ⎥ ⎢ ⎥ ¯ = ⎢ Ξ2 (h) < 0, h¯ H4T ⎥ ∗ Ξ44 0 0 ⎢ ∗ ∗ ⎥ ⎢ ∗ ∗ ⎥ ∗ ∗ −2V2 0 0 ⎢ ⎥ ⎣ ∗ ∗ ⎦ ∗ ∗ ∗ Ξ66 0 −2αh U ∗ ∗ ∗ ∗ ∗ ∗ −e
⎤ Ξ16 GB ⎥ ⎥ γ G B⎥ ⎥ 0 (λ → 0), X 3 = X 4 = X 5 = 0 ∗ R3 then V4 (t) + V5 (t) + V6 (t) reduces to T r (t) r (t) H V4 (t) + (tk+1 − t)e2αt r (tk ) r (tk ) where
⎤ X 1 + X 1T −X 1 + X 2 ⎥ ⎢ 2 H =⎣ ⎦ X 1 + X 1T T ∗ −X 2 − X 2 + 2 which was first proposed for linear sampled-data systems in [35], and thereafter has also been applied to sampled-data synchronization of neural networks in [36]. It is also worth mentioning that, in [35] and [36], the proposed Lyapunov functionals should be positive definite on the whole sampling intervals. Different from [35] and [36], the Lyapunov functional (14) is positive definite only at sampling times but not necessarily positive definite inside the sampling intervals thanks to the use of Lemma 2. Thus, the Lyapunov functional used here is more general and desirable than those applied in [35] and [36]. To end this section, we consider the following sampled-data master–slave synchronization scheme without time delay: x(t) ˙ = Ax(t) + W ϕ(Dx(t)) M : p(t) = C x(t) z˙ (t) = Az(t) + W ϕ(Dz(t)) + u(t) S : q(t) = Cz(t) ⎡
C : u(t) = K ( p(tk ) − q(tk )),
tk t < tk+1
(43)
which has been studied in [30]–[32]. Correspondingly, the synchronization error system (3) reduces to r˙ (t) = Ar (t) + W η[Dr (t), z(t)] − K Cr (tk ), tk t < tk+1 (44) and the Lyapunov functional (14) reduces to V (t) = V1 (t) + V4 (t) + V5 (t) + V6 (t), t ∈ [tk , tk+1 ) (45) where V1 (t), V4 (t), V5 (t), and V6 (t) are given in (14). By using a method similar to the one employed in the proof of Theorem 1, we have the following corollary. Corollary 1: Given scalars α > 0, ε, and γ , the master system M and the slave system S in (43) are exponentially R1 R2 synchronous if there exist matrices P > 0, U > 0, > ∗ R 3 0, X 1 , X 2 , X 3 , X 4 , X 5 , G, L, H = H1 H2 H3 H4 , and diagonal matrices V1 > 0, and V2 > 0 such that (46) and (47) (see the bottom of the next page) hold, where X 1 + X 1T + H1 + H1T + εG A + ε AT G T 2 and the other parameters are given in Theorem 1. Furthermore, the sampled-data controller gain matrix in (43) is given by (13). Ξˆ 11 = 2α P −
Remark 3: It is noted that Corollary 1 provides a new synchronization criterion for the master system M and the slave system S in (43). It should be pointed out that the terms V4 (t), V5 (t), and V6 (t) in the Lyapunov functional (45) are neglected in [30], the terms t V5 (t) and V6 (t) are overlooked in [31], and the terms tk e2αs r (s)T R1r (s) ds and t tk r (s) ds are ignored in [32]. Thus, the available information about the actual sampling pattern has not been fully used in [30]–[32]. On the other hand, the Lyapunov functionals proposed in [30] and [31] are required to be positive definite on the whole sampling intervals. As mentioned in Remark 2, such a requirement is not necessary any more for the Lyapunov functional (45). Thus the scheme in this paper has overcome the shortcomings in [30]–[32] and is less conservative. IV. N UMERICAL E XAMPLES In this section, we will adopt two illustrative examples to show the validity and reduced conservatism of the proposed synchronization scheme. Example 1: Consider the following time-delay Chua’s circuit as the master system: ⎧ ⎪ ⎨ x˙1 (t) = a(x 2(t) − m 1 x 1 (t) + g(x 1 (t))) − cx 1 (t − d) x˙2 (t) = x 1 (t) − x 2 (t) + x 3 (t) − cx 1 (t − d) (48) ⎪ ⎩ x˙3 (t) = − b2 x 2 (t) + c(2x 1(t − d) − x 3 (t − d)) with the nonlinear characteristics 1 g(x 1(t)) = (m 1 − m 0 )(|x 1 (t) + 1| − |x 1 (t) − 1|) 2 and parameters m 0 = −1/7, m 1 = 2/7, a = 9, b = 14.28, c = 0.1, and the constant time delay d = 1. It can be found that Chua’s circuit can be represented in the time-delay Lur’e form with ⎡ ⎤ ⎡ ⎤ −am 1 a 0 −c 0 0 A = ⎣ 1 −1 1⎦ , B = ⎣−c 0 0 ⎦ 0 −b 0 2c 0 −c ⎤ ⎡ ⎡ ⎤ 100 a(m 1 − m 0 ) 0 0 0 0 0⎦ , D = ⎣0 1 0⎦ W = ⎣ 0 00 001 with ϕ1 (x 1 (t)) = 21 (|x 1 (t) + 1| − |x 1 (t) − 1|) belonging to sector [0, 1], and ϕ2 (x 2 (t)) = ϕ3 (x 3 (t)) = 0. In this example, we choose C = 1 0 0 , ε = 2, and γ = 0. The initial conditions of the slave sys master and T and z(t) = tems are chosen as x(t) = 0.2 0.3 0.2 T −0.3 −0.1 0.4 , t ∈ [−1, 0]. Figs. 1 and 2 show the master system states x(t) and the slave system states z(t), respectively. Applying Theorem 1, we can obtain the different maximum values of the upper bound h for different α, which is the convergence rate of the synchronization error r (t), as shown in Table I. From Table I, we can find the influence of the choice of α on the value of the upper bound h. To be specific, a larger value of α, which implies the faster synchronization of the master and slave systems, corresponds to a smaller value of the upper bound h. On the other hand, the influence of the choice of the upper bound h on the value of α can be found in
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TABLE I M AXIMUM VALUES OF THE U PPER B OUND h FOR D IFFERENT α 4
α h
0.1 0.3247
0.2 0.2941
3
x (t)
2
0.4 0.2396
0.5 0.2154
TABLE II M AXIMUM VALUES OF α FOR D IFFERENT U PPER B OUNDS h
0
h α
−2
0.10 1.1650
0.15 0.8240
0.20 0.5683
0.25 0.3596
0.30 0.1802
1.5
−4 0.5 0 x2(t)
Fig. 1.
0.3 0.2658
−0.5
−4
0
−2
2
r1 (t) r2 (t) r3 (t)
4
1
x1(t)
0.5
Master system M in (1). 0
−0.5 4
−1
3
z (t)
2 0
−1.5
0
2
−2
Fig. 3.
4
6 Time t
8
10
12
State response of error system (3).
−4 0.5 0 z (t) 2
Fig. 2.
−0.5
−4
0
−2
2
4
z (t) 1
Slave system S with u(t) = 0 in (1).
Table II, from which we can find that, in order to achieve the fast synchronization of the master and slave systems, the value of the upper bound h should be chosen as small as possible. Choosing α = 0.2 and h = 0.2941, and using the M ATLAB LMI Toolbox to solve the LMIs (11) and (12), we can get the
following gain matrix in (1): K = 3.4727 1.1028
T −2.1042 .
That is, there exists a sampled-data controller such that the master and slave systems are of exponential synchronization for any sampling period h k 0.2941. For the above gain matrix, the response curves of error system (3) are given in Fig. 3, which shows that the synchronization error is tending to zero. Thus we can synchronize successfully the master and slave systems by the proposed sampled-data controller. To show the reduced conservatism of the proposed condition, we choose c = 0; that is, the time-delay Chua’s circuit
⎡ ⎤ ¯ Ξ12 + Θ12 (h) ¯ Ξ13 + Θ13 (h) ¯ Ξ14 + Θ14 (h) ¯ εGW + D T SV1 Ξˆ 11 + Θ11 (h) ⎢ ⎥ ¯ Ξ23 + Θ23 (h) ¯ ∗ −G − G T + hU h¯ X 3 GW ⎢ ⎥ ⎢ ⎥
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 3, MARCH 2013
Sampled-Data Synchronization of Chaotic Lur’e Systems With Time Delays Zheng-Guang Wu, Peng Shi, Senior Member, IEEE, Hongye Su, and Jian Chu
Abstract— This paper studies the problem of sampled-data control for master–slave synchronization schemes that consist of identical chaotic Lur’e systems with time delays. It is assumed that the sampling periods are arbitrarily varying but bounded. In order to take full advantage of the available information about the actual sampling pattern, a novel Lyapunov functional is proposed, which is positive definite at sampling times but not necessarily positive definite inside the sampling intervals. Based on the Lyapunov functional, an exponential synchronization criterion is derived by analyzing the corresponding synchronization error systems. The desired sampled-data controller is designed by a linear matrix inequality approach. The effectiveness and reduced conservatism of the developed results are demonstrated by the numerical simulations of Chua’s circuit and neural network. Index Terms— Chaotic systems, exponential synchronization, Lur’e systems, neural networks, sampled-data control.
I. I NTRODUCTION
S
INCE the pioneering work of [1], the study on the synchronization of chaotic systems has gradually become an active area of research (see, [1]–[7] and the references therein). This stems from the fact that chaos synchronization has been widely applied in various fields, including chaos generator design, secure communication, chemical reactions, biological systems, and information science. It has been shown that several nonlinear systems, including neural networks and Chua’s circuits, can be represented in the form of Lur’e systems [6], [8]–[11]. Therefore, the master–slave synchronization of chaotic Lur’e systems has been an important topic, and a number of master–slave synchronization criteria have been proposed. For example, in [12], a master–slave type of chaos synchronization problem has been investigated for a general form of Lur’e systems by a time-delay feedback control technique, and some simple algebraic conditions have been derived for easy verification, facilitating the design and application of Manuscript received September 2, 2012; revised November 9, 2012; accepted December 12, 2012. Date of publication January 9, 2013; date of current version January 30, 2013. This work was supported in part by the National Natural Science Foundation of China under Grant 61174029 and Grant 61174058, the National Key Basic Research Program, China, under Grant 2012CB215202, and the 111 Project B12018. Z.-G. Wu, H. Su, and J. Chu are with the National Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou 310027, China (e-mail: [email protected]; [email protected]; [email protected]). P. Shi is with the School of Engineering and Science, Victoria University, Melbourne 8001, Australia, and also with the School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide 5005, Australia (e-mail: [email protected]). Digital Object Identifier 10.1109/TNNLS.2012.2236356
chaos systems. The problem of designing time-varying delay feedback controllers for master–slave synchronization of Lur’e systems has been discussed in [13], where, based on Lyapunov functional approach, some delay-dependent synchronization criteria have been obtained and formulated in the form of linear matrix inequalities (LMIs). It should be noted that two cases of time-varying delays have been fully considered in [13], and thus the results proposed therein are very general and powerful. In [14], the results of [13] were extended to the fault-tolerant master–slave synchronization of Lur’e systems. It is to be noted that time delay is an inherent feature of many physical processes, such as chemical processes, nuclear reactors, and biological systems, and may lead to instability or significantly deteriorate the performance of the corresponding closed-loop systems [15]–[20]. Thus, recently, some attention has also been directed to the synchronization of chaotic Lur’e systems with time delays. For example, in [10], based on the invariant principle of functional differential equations, an adaptive approach has been proposed for the master– slave synchronization of chaotic Lur’e systems with timevarying delays that does not require any prior knowledge of the systems’ parameters. The master–slave synchronization of coupled time-delay Lur’e systems with parameter mismatches has been studied in [21], where, by introducing a type of quasisynchronization, a general methodology has been developed to derive several delay-dependent quasi-synchronization criteria and to give the explicit expression of error levels. Also, the last decade has seen a wealth of research on sampled-data control systems, because modern control systems usually employ digital technology for controller implementation [22]–[29]. Recently, based on the input delay approach [24], the sampled-data master–slave synchronization schemes that consist of identical chaotic Lur’e systems have been studied in [30], where sufficient conditions for global asymptotic synchronization of chaotic Lur’e systems have been obtained using the free-weighting matrix approach and expressed in terms of LMIs. It is to be noted that an obvious and important merit of sampled-data synchronization for chaotic systems is that only the samples of the state variables of the master systems and the slave systems at discrete time instants are needed, and thus the amount of transmitted synchronization information greatly reduces and the efficiency of bandwidth usage increases, which makes the synchronization of chaotic systems more efficient and useful in real-life applications. However, most of the available information about the actual sampling pattern has not been used in [30], because the input delay induced by sample-and-hold is simply treated
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WU et al.: SYNCHRONIZATION OF CHAOTIC LUR’E SYSTEMS WITH TIME DELAYS
as a bounded fast-varying delay (i.e., a time-varying delay without any constraint on the delay derivative), and thus the obtained synchronization conditions have considerable conservatism. The work in [31] and [32] has improved upon the results presented in [30] by applying some piecewise Lyapunov functionals. Compared to the results in [30], the main advantage of the results in [31] and [32] is that the discussed synchronization error system is no longer regarded as a system with fast-varying delay, and the characteristic of the discussed system is captured. However, there still exists room for improving the synchronization criteria of [31] and [32], because some important and useful terms have been ignored in the Lyapunov functionals proposed by [31] and [32], and the available information about the actual sampling pattern has not been fully adopted, which might lead to conservatism to some extent. Moreover, only delay-free chaotic Lur’e systems are taken into account in [30]–[32]. To the best of our knowledge, there is little information in the published literature about sampled-data master–slave synchronization for chaotic Lur’e systems with time delays. Therefore, it is necessary to further study the sampled-data master–slave synchronization for chaotic Lur’e systems with or without time delays, which is the motivation for this paper. In this paper, we consider master–slave synchronization of Lur’e systems with time delays by applying sampleddata control. In order to make full use of the available information about the actual sampling pattern, a novel Lyapunov functional is proposed. The positive definitiveness of the given Lyapunov functional is required only at sampling times but not necessarily inside the sampling intervals, which implies that the traditional continuous-time Lyapunov theorem cannot directly lead to the desired exponential synchronization criterion. Thus, we first introduce a new inequality on the synchronization error system. Based on the new inequality and Lyapunov functional, a criterion is proposed to ensure the exponential stability of the synchronization error systems, and thus the master systems exponentially synchronize with the slave systems. By means of the numerical simulations of Chua’s circuit and neural network, it is shown that the proposed results are effective and can significantly improve the existing ones. Notation: The notations used throughout this paper are fairly standard. Rn and Rm×n denote the n-dimensional Euclidean space and the set of all m × n real matrices, respectively. The notation X > Y (X Y ), where X and Y are symmetric matrices, means that X − Y is positive definite (positive semidefinite). For integers a and b with a < b, N[a, b] = {a, a + 1, . . . , b − 1, b}. λmax (Q) (λmin (Q)) denotes the maximum (minimum) of the eigenvalue of a real symmetric matrix Q. I and 0 represent the identity matrix and a zero matrix, respectively. The superscript “T” represents the transpose, and diag{· · · } stands for a block-diagonal matrix. || · || denotes the Euclidean norm of a vector and its induced norm of a matrix. For an arbitrary matrix B and two symmetric matrices A and C A B ∗ C denotes a symmetric matrix, where “∗” denotes the term that
411
is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations. II. P RELIMINARIES Consider the following master–slave synchronization scheme with sampled-data control: x(t) ˙ =Ax(t) + Bx(t − d) + W ϕ(Dx(t)) M : p(t) =C x(t) z˙ (t) =Az(t) + Bz(t − d) + W ϕ(Dz(t)) + u(t) S : q(t) =Cz(t) C : u(t) = K ( p(tk ) − q(tk )),
tk t < tk+1
(1)
which consists of the master system M , the slave system S , and the controller C . M and S with u(t) = 0 are identical chaotic time-delay Lur’e systems with state vectors x(t), z(t) ∈ Rn , outputs of subsystems p(t), q(t) ∈ Rl , respectively. A ∈ Rn×n , B ∈ Rn×n , C ∈ Rl×n , D ∈ Rnh ×n , and W ∈ Rn×nh are known constant matrices, and d > 0 is the constant time delay. u(t) ∈ Rn is the slave system control input, and K ∈ Rn×l is the sampled-data controller gain matrix to be designed. It is assumed that ϕ(·) : Rnh → Rnh is a diagonal nonlinearity with ϕi (·) belonging to sector [0, si ] for i = 1, 2, · · · , n h . For sampled-data synchronization, only discrete measurements of p(t) and q(t) can be used for synchronization purposes, that is, we only have the measurements p(tk ) and q(tk ) at the sampling instant tk . In this paper, the control signal is assumed to be generated by using a zero-order-hold (ZOH) function with a sequence of hold times 0 t0 < t1 < · · · < tk < · · · < limk→+∞ tk = +∞. Also, the sampling is not required to be periodic, and the only assumption is that the distance between any two consecutive sampling instants is less than a given bound [33]. Specifically, it is assumed that tk+1 − tk = h k h
(2)
for all k 0, where h > 0 represents the upper bound of the sampling periods. Given the synchronization scheme (1), the synchronization error is defined as r (t) = x(t) − z(t), and we can get the following synchronization error system: r(t) ˙ = Ar (t) + Br (t − d) +W η(Dr (t), z(t)) − K Cr (tk ),
tk t < tk+1 (3)
where η(Dr (t), z(t)) = ϕ(D(r (t) + z(t))) − ϕ(Dz(t)). As ϕi (·) belongs to sector [0, si ], it can be found that for any i = 1, 2, . . . , n h , and ∀r, z 0
ηi (diTr, z) diTr
=
ϕi [diT (r + z)] − ϕi (diTr ) diTr
si ,
diTr = 0 (4)
where diT denotes the i th row vector of D. It is easily found from (4) that for any i = 1, 2, . . . , n h , and ∀r, z ηi (diT r, z)[ηi (diTr, z) − si diTr ] 0.
(5)
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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 3, MARCH 2013
Based on (5), we have for any matrices Vl = diag {vl1 , vl2 , . . . , vlnh } 0, l = 1, 2, the following inequalities hold: −
nh
vl j η j (d Tj r (t), z(t))(η j (d Tj r (t), z(t))−s j d Tj r (t)) 0
j =1
(6) which are equivalent to T
T
r (t) D SVl η(Dr (t), z(t)) −η(Dr (t), z(t))T Vl η(Dr (t), z(t)) 0
(7)
where S = diag{s1 , s2 , . . . , snh }. In this paper, the following definition and lemmas will be used. Definition 1: The master system M and the slave system S in (1) are said to be exponentially synchronous if the synchronization error system (3) is exponentially stable; that is, there exist two constants α > 0 and β > 0 such that ||r (t)|| βe−αt ||r0 ||c
∀t 0
(8)
where ||rt ||c = sup−d θ 0 {||r (t + θ )||, ||˙r(t + θ )||}. Lemma 1 (Jensen Inequality) [13], [34]: For any matrix W > 0, scalars γ1 and γ2 satisfying γ2 > γ1 , and a vector function ω : [ γ1 , γ2 ] → Rn such that the integrations concerned are well defined, then γ2 ω(α)T W ω(α) dα
(γ2 − γ1 ) γ1
⎡
⎣
γ2
⎤T
⎡
ω(α) dα ⎦ W ⎣
γ1
γ2
⎤ ω(α) dα ⎦.
(9)
γ1
Lemma 2: Consider the synchronization error system (3). Then the following inequality holds: tk 2
2
||r (t)|| θ1 ||r (tk )|| + θ2
2
||r (α)|| , dα, tk t < tk+1
tk −d
(10)
synchronous exist matrices P > 0, Q > 0, Z > 0, if there R1 R2 > 0, X 1 , X 2 , X 3 , X 4 , X 5 , G, L, H = U > 0, ∗ R 3 H1 H2 H3 H4 , and diagonal matrices V1 > 0, and V2 > 0 such that (11) and (12) (see the bottom of next page) hold where Ξ11 = 2α P + Q −
X 1 + X 1T e−2αd − Z + H1 + H1T 2 d
+εG A + ε AT G T Ξ12 = P + H2 − εG + AT G T Ξ13 = X 1 − X 2 + H3 − H1T − εLC + γ AT G T Ξ14 = −X 3 + H4 e−2αd Z + εG B Ξ16 = d Ξ22 = d Z − G − G T Ξ23 = −H2T − LC − γ G T X 1 + X 1T − H3 − H3T − γ LC − γ C T L T Ξ33 = X 2 + X 2T − 2 Ξ34 = −X 4 − e−2αh R2T − H4 X 5 + X 5T e−2αh − R1 Ξ44 = − 2 h −2αd e Ξ66 = −e−2αd Q − Z d ¯ = α h(X ¯ 1 + X 1T ) + h¯ X 3 + h¯ X 3T + h¯ R1 Θ11 (h) X + X 1T ¯ = h¯ 1 Θ12 (h) 2 T ¯ = 2α h(−X ¯ Θ13 (h) 1 + X 2 ) + h¯ X 4 + h¯ R2 X + X 5T ¯ = 2α h¯ X 3 + h¯ 5 Θ14 (h) 2 ¯ = h(−X ¯ Θ23 (h) + X ) 2
1 X 1 + X 1T T ¯ ¯ + h¯ R3 Θ33 (h) = 2α h −X 2 − X 2 + 2 ¯ = α h(X ¯ 5 + X 5T ). Θ44 (h) Furthermore, the sampled-data controller gain matrix in (1) is given by K = G −1 L.
where 2 +||W ||2 ||S D||2 +||B||2 )h 2
θ1 = 5(1 + ||K C||2 h 2 )e5(||A|| 2
θ2 = 5||B|| he
5(||A||2 +||W ||2 ||S D||2 +||B||2 )h 2
.
Proof: See Appendix. The objective of this paper is to design sampled-data controller C in (1) to exponentially synchronize the master system M and the slave system S in (1). III. M AIN R ESULTS In this section, we discuss the sampled-data synchronization problem for chaotic Lur’e systems with time delays. Given the master–slave synchronization scheme in (1), we have the following result. Theorem 1: Given scalars α > 0, ε, and γ , the master system M and the slave system S in (1) are exponentially
(13)
Proof: Consider the following Lyapunov functional for the synchronization error system (3): V (t) =
6
Vi (t),
t ∈ [tk , tk+1 )
i=1
where V1 (t) = e2αt r (t)T Pr (t) t V2 (t) = e2αs r (s)T Qr (s) ds t −d 0
t e2αs r˙ (s)T Z r˙ (s) ds dθ
V3 (t) = −d t +θ
(14)
WU et al.: SYNCHRONIZATION OF CHAOTIC LUR’E SYSTEMS WITH TIME DELAYS
413
t V4 (t) = (tk+1 − t)
t e
2αs
T
r˙ (s) U r˙ (s) ds
(tk+1 −t)e
2αt
T
r˙ (t) U r˙ (t) − e
2αt
tk
tk
t V5 (t) = (tk+1 − t)
e2αs tk
r (s) r (tk )
T
R1 R2 ∗ R3
r (s) ds r (tk )
⎡
⎤T ⎡ ⎤ r (t) r (t) ⎢ r (tk ) ⎥ ⎢ r (t ) ⎥ ⎥ H⎢ t k ⎥ V6 (t) = (tk+1 − t)e2αt ⎢ ⎣t ⎣ ⎦ ⎦ r (s) ds r (s) ds tk
X 1 + X 1T −X 1 + X 2 X3 ⎢ 2 ⎢ T ⎢ X1 + X1 H=⎢ X4 ∗ −X 2 − X 2T + ⎢ 2 ⎣ X 5 + X 5T ∗ ∗ 2
⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦
V˙1 (t) = 2e r (t) P r˙ (t) + 2αe r (t) Pr (t) V˙2 (t) = e2αt r (t)TQr (t)−e2αt e−2αd r (t −d)T Qr (t −d) t 2αt T ˙ e2αs r˙ (s)T Z r˙ (s) ds V3 (t) = de r˙ (t) Z r˙ (t) − 2αt
t de
r˙ (t) Z r˙ (t) − e T
2αt
T
(15)
V˙4 (t) = (tk+1 − t)e
e−2αd r˙ (s)T Z r˙ (s) ds
t r˙ (t) U r˙ (t) − e2αs r˙ (s)T U r˙ (s) ds (17) T
tk
t (tk+1 − t)e
2αt
r˙ (t) U r˙ (t) −
e2αtk r˙ (s)T U r˙ (s) ds
T
tk
r (s) ds r (tk )
T r (t) R1 R2 r (t) +(tk+1 − t)e ∗ R3 r (tk ) r (tk ) t T r (s) R1 R2 r (s) ds −e2αt e−2αh ∗ R3 r (tk ) r (tk ) 2αt
+(tk+1 − t)e2αt t = −e
2αt
−2e
r (t) r (tk )
T
R1 R2 ∗ R3
r (t) r (tk )
e−2αh r (s)T R1r (s) ds
tk 2αt
T −2αh
r (tk ) e
t r (s) ds
R2T
tk T −2αh
(16)
t −d 2αt
R1 R2 ∗ R3
tk
t −d 2αt
r (s) r (tk )
e2αs
T r (t) R1 R2 r (t) ∗ R3 r (tk ) r (tk ) T t R1 R2 r (s) 2αtk r (s) − e ds ∗ R3 r (tk ) r (tk )
It is noted that V4 (t), V5 (t), and V6 (t) vanish before tk and after tk . Therefore, V (t) is continuous in time since limt →tk V (t) = V (tk ). Calculating the time derivative of V (t) along the trajectories of (3) gives the following result: T
V˙5 (t) = −
T
+(tk+1 − t)e2αt
tk
⎡
(18)
t tk
and
2αt
e−2αh r˙ (s)T U r˙ (s) ds
−e2αt (t − tk )r (tk ) e R3r (tk ) T r (t) R1 R2 r (t) +(tk+1 − t)e2αt ∗ R3 r (tk ) r (tk ) ⎡ ⎤T ⎡ ⎤ r (t) r (t) ⎢ r (tk ) ⎥ ⎢ r (t ) ⎥ ⎥ H⎢ t k ⎥ V˙6 (t) = −e2αt ⎢ ⎣t ⎣ ⎦ ⎦ r (s) ds r (s) ds tk
(19)
tk
⎤T ⎡ ⎤ r (t) r (t) ⎢ r (t ) ⎥ ⎢ r (tk ) ⎥ ⎥ H⎢ t k ⎥ +2α(tk+1 − t)e2αt ⎢ ⎣ ⎣t ⎦ ⎦ r (s) ds r (s) ds ⎡
tk
tk
tk
⎡
¯ Ξ12 + Θ12 (h) ¯ Ξ13 + Θ13 (h) ¯ Ξ14 + Θ14 (h) ¯ εGW + D T SV1 Ξ11 + Θ11 (h) ⎢ ¯ ¯ ¯ ∗ Ξ22 + hU Ξ23 + Θ23 (h) h X3 GW ⎢ ⎢ ¯ Ξ34 + 2α h¯ X 4 ∗ ∗ Ξ + Θ ( h) γ GW 33 33 ¯ = ⎢ Ξ1 (h) ⎢ ¯ ∗ ∗ ∗ Ξ + Θ ( h) 0 44 44 ⎢ ⎣ ∗ ∗ ∗ ∗ −2V1 ∗ ∗ ∗ ∗ ∗ √ ⎡ ⎤ T T Ξ13 Ξ14 εGW + D SV2 Ξ16 √h¯ H1 Ξ11 Ξ12 ⎢ ⎥ Ξ23 0 GW G B √h¯ H2T ⎥ ⎢ ∗ Ξ22 ⎢ ⎥ ⎢ ∗ ∗ Ξ33 − e−2αh h¯ R3 Ξ34 γ GW γ G B √h¯ H3T ⎥ ⎢ ⎥ ¯ = ⎢ Ξ2 (h) < 0, h¯ H4T ⎥ ∗ Ξ44 0 0 ⎢ ∗ ∗ ⎥ ⎢ ∗ ∗ ⎥ ∗ ∗ −2V2 0 0 ⎢ ⎥ ⎣ ∗ ∗ ⎦ ∗ ∗ ∗ Ξ66 0 −2αh U ∗ ∗ ∗ ∗ ∗ ∗ −e
⎤ Ξ16 GB ⎥ ⎥ γ G B⎥ ⎥ 0 (λ → 0), X 3 = X 4 = X 5 = 0 ∗ R3 then V4 (t) + V5 (t) + V6 (t) reduces to T r (t) r (t) H V4 (t) + (tk+1 − t)e2αt r (tk ) r (tk ) where
⎤ X 1 + X 1T −X 1 + X 2 ⎥ ⎢ 2 H =⎣ ⎦ X 1 + X 1T T ∗ −X 2 − X 2 + 2 which was first proposed for linear sampled-data systems in [35], and thereafter has also been applied to sampled-data synchronization of neural networks in [36]. It is also worth mentioning that, in [35] and [36], the proposed Lyapunov functionals should be positive definite on the whole sampling intervals. Different from [35] and [36], the Lyapunov functional (14) is positive definite only at sampling times but not necessarily positive definite inside the sampling intervals thanks to the use of Lemma 2. Thus, the Lyapunov functional used here is more general and desirable than those applied in [35] and [36]. To end this section, we consider the following sampled-data master–slave synchronization scheme without time delay: x(t) ˙ = Ax(t) + W ϕ(Dx(t)) M : p(t) = C x(t) z˙ (t) = Az(t) + W ϕ(Dz(t)) + u(t) S : q(t) = Cz(t) ⎡
C : u(t) = K ( p(tk ) − q(tk )),
tk t < tk+1
(43)
which has been studied in [30]–[32]. Correspondingly, the synchronization error system (3) reduces to r˙ (t) = Ar (t) + W η[Dr (t), z(t)] − K Cr (tk ), tk t < tk+1 (44) and the Lyapunov functional (14) reduces to V (t) = V1 (t) + V4 (t) + V5 (t) + V6 (t), t ∈ [tk , tk+1 ) (45) where V1 (t), V4 (t), V5 (t), and V6 (t) are given in (14). By using a method similar to the one employed in the proof of Theorem 1, we have the following corollary. Corollary 1: Given scalars α > 0, ε, and γ , the master system M and the slave system S in (43) are exponentially R1 R2 synchronous if there exist matrices P > 0, U > 0, > ∗ R 3 0, X 1 , X 2 , X 3 , X 4 , X 5 , G, L, H = H1 H2 H3 H4 , and diagonal matrices V1 > 0, and V2 > 0 such that (46) and (47) (see the bottom of the next page) hold, where X 1 + X 1T + H1 + H1T + εG A + ε AT G T 2 and the other parameters are given in Theorem 1. Furthermore, the sampled-data controller gain matrix in (43) is given by (13). Ξˆ 11 = 2α P −
Remark 3: It is noted that Corollary 1 provides a new synchronization criterion for the master system M and the slave system S in (43). It should be pointed out that the terms V4 (t), V5 (t), and V6 (t) in the Lyapunov functional (45) are neglected in [30], the terms t V5 (t) and V6 (t) are overlooked in [31], and the terms tk e2αs r (s)T R1r (s) ds and t tk r (s) ds are ignored in [32]. Thus, the available information about the actual sampling pattern has not been fully used in [30]–[32]. On the other hand, the Lyapunov functionals proposed in [30] and [31] are required to be positive definite on the whole sampling intervals. As mentioned in Remark 2, such a requirement is not necessary any more for the Lyapunov functional (45). Thus the scheme in this paper has overcome the shortcomings in [30]–[32] and is less conservative. IV. N UMERICAL E XAMPLES In this section, we will adopt two illustrative examples to show the validity and reduced conservatism of the proposed synchronization scheme. Example 1: Consider the following time-delay Chua’s circuit as the master system: ⎧ ⎪ ⎨ x˙1 (t) = a(x 2(t) − m 1 x 1 (t) + g(x 1 (t))) − cx 1 (t − d) x˙2 (t) = x 1 (t) − x 2 (t) + x 3 (t) − cx 1 (t − d) (48) ⎪ ⎩ x˙3 (t) = − b2 x 2 (t) + c(2x 1(t − d) − x 3 (t − d)) with the nonlinear characteristics 1 g(x 1(t)) = (m 1 − m 0 )(|x 1 (t) + 1| − |x 1 (t) − 1|) 2 and parameters m 0 = −1/7, m 1 = 2/7, a = 9, b = 14.28, c = 0.1, and the constant time delay d = 1. It can be found that Chua’s circuit can be represented in the time-delay Lur’e form with ⎡ ⎤ ⎡ ⎤ −am 1 a 0 −c 0 0 A = ⎣ 1 −1 1⎦ , B = ⎣−c 0 0 ⎦ 0 −b 0 2c 0 −c ⎤ ⎡ ⎡ ⎤ 100 a(m 1 − m 0 ) 0 0 0 0 0⎦ , D = ⎣0 1 0⎦ W = ⎣ 0 00 001 with ϕ1 (x 1 (t)) = 21 (|x 1 (t) + 1| − |x 1 (t) − 1|) belonging to sector [0, 1], and ϕ2 (x 2 (t)) = ϕ3 (x 3 (t)) = 0. In this example, we choose C = 1 0 0 , ε = 2, and γ = 0. The initial conditions of the slave sys master and T and z(t) = tems are chosen as x(t) = 0.2 0.3 0.2 T −0.3 −0.1 0.4 , t ∈ [−1, 0]. Figs. 1 and 2 show the master system states x(t) and the slave system states z(t), respectively. Applying Theorem 1, we can obtain the different maximum values of the upper bound h for different α, which is the convergence rate of the synchronization error r (t), as shown in Table I. From Table I, we can find the influence of the choice of α on the value of the upper bound h. To be specific, a larger value of α, which implies the faster synchronization of the master and slave systems, corresponds to a smaller value of the upper bound h. On the other hand, the influence of the choice of the upper bound h on the value of α can be found in
WU et al.: SYNCHRONIZATION OF CHAOTIC LUR’E SYSTEMS WITH TIME DELAYS
417
TABLE I M AXIMUM VALUES OF THE U PPER B OUND h FOR D IFFERENT α 4
α h
0.1 0.3247
0.2 0.2941
3
x (t)
2
0.4 0.2396
0.5 0.2154
TABLE II M AXIMUM VALUES OF α FOR D IFFERENT U PPER B OUNDS h
0
h α
−2
0.10 1.1650
0.15 0.8240
0.20 0.5683
0.25 0.3596
0.30 0.1802
1.5
−4 0.5 0 x2(t)
Fig. 1.
0.3 0.2658
−0.5
−4
0
−2
2
r1 (t) r2 (t) r3 (t)
4
1
x1(t)
0.5
Master system M in (1). 0
−0.5 4
−1
3
z (t)
2 0
−1.5
0
2
−2
Fig. 3.
4
6 Time t
8
10
12
State response of error system (3).
−4 0.5 0 z (t) 2
Fig. 2.
−0.5
−4
0
−2
2
4
z (t) 1
Slave system S with u(t) = 0 in (1).
Table II, from which we can find that, in order to achieve the fast synchronization of the master and slave systems, the value of the upper bound h should be chosen as small as possible. Choosing α = 0.2 and h = 0.2941, and using the M ATLAB LMI Toolbox to solve the LMIs (11) and (12), we can get the
following gain matrix in (1): K = 3.4727 1.1028
T −2.1042 .
That is, there exists a sampled-data controller such that the master and slave systems are of exponential synchronization for any sampling period h k 0.2941. For the above gain matrix, the response curves of error system (3) are given in Fig. 3, which shows that the synchronization error is tending to zero. Thus we can synchronize successfully the master and slave systems by the proposed sampled-data controller. To show the reduced conservatism of the proposed condition, we choose c = 0; that is, the time-delay Chua’s circuit
⎡ ⎤ ¯ Ξ12 + Θ12 (h) ¯ Ξ13 + Θ13 (h) ¯ Ξ14 + Θ14 (h) ¯ εGW + D T SV1 Ξˆ 11 + Θ11 (h) ⎢ ⎥ ¯ Ξ23 + Θ23 (h) ¯ ∗ −G − G T + hU h¯ X 3 GW ⎢ ⎥ ⎢ ⎥
