IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 10, OCTOBER 2012

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Synchronization Error Estimation and Controller Design for Delayed Lur’e Systems With Parameter Mismatches Wangli He, Member, IEEE, Feng Qian, Qing-Long Han, Member, IEEE, and Jinde Cao, Senior Member, IEEE

Abstract— This paper investigates the problem of master–slave synchronization of two delayed Lur’e systems in the presence of parameter mismatches. First, by analyzing the corresponding synchronization error system, synchronization with an error level, which is referred to as quasi-synchronization, is established. Some delay-dependent quasi-synchronization criteria are derived. An estimation of the synchronization error bound is given, and an explicit expression of error levels is obtained. Second, sufficient conditions on the existence of feedback controllers under a predetermined error level are provided. The controller gains are obtained by solving a set of linear matrix inequalities. Finally, a delayed Chua’s circuit is chosen to illustrate the effectiveness of the derived results. Index Terms— Delayed Lur’e systems, neural networks, parameter mismatches, synchronization.

I. I NTRODUCTION

N

EURAL networks have been extensively investigated in the past three decades and many applications have been found in different areas, such as optimization, pattern recognition, and associative memories. As dynamical behaviors of neural networks, such as stability, bifurcation, periodic attractors, and even chaotic attractors, have played an important role in such applications, synchronization of chaotic neural networks has attracted considerable attention

Manuscript received September 16, 2011; revised June 21, 2012; accepted June 21, 2012. Date of publication July 25, 2012; date of current version September 10, 2012. This work was supported in part by the National 973 Project under Grant 2012CB720500, the Key Program of National Natural Science Foundation of China under Grant U1162202, the Fundamental Research Funds for the Central Universities, the Shanghai Leading Academic Discipline Project under Grant B504, the Australian Research Council Discovery Projects under Grant DP1096780 and Grant DP0986376, and the Research Advancement Awards Scheme Program (from 2010 to 2012) at Central Queensland University, Australia. W. He is with the Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China, and also with the Centre for Intelligent and Networked Systems, Central Queensland University, Rockhampton 4702, Australia (e-mail: [email protected]). F. Qian is with the Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China (e-mail: [email protected]). Q.-L. Han is with the Centre for Intelligent and Networked Systems, Central Queensland University, Rockhampton 4702, Australia, and also with the School of Information and Communication Technology, Central Queensland University, Rockhampton 4702, Australia (e-mail: [email protected]). J. Cao is with the Department of Mathematics, Southeast University, Nanjing 210096, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2012.2205941

due to its theoretical importance and practical applications, such as secure communication, tracking control, and biological systems [1]– [4]. Notice that some nonlinear systems, such as Hopfield neural networks [5], [6], cellular neural networks [7], [8], Chua’s circuits [9], and genetic oscillators [10], can be represented in the form of Lur’e systems. The problem of master–slave synchronization of Lur’e systems has been studied well in the last decade [11], [12]. In the literature, master–slave synchronization of Lur’e systems can be classified into two categories: 1) master–slave synchronization of identical Lur’e systems and 2) master–slave synchronization of nonidentical Lur’e systems. In master–slave synchronization of identical Lur’e systems, a unified approach to chaos synchronization is to reformulate it as a generalized Lur’e system and then discuss the absolute stability of its error dynamics [13]. When a propagation delay is involved in the remote master– slave synchronization scheme, a constant time-delay output feedback control was adapted to synchronize master–slave Lur’e systems [14]. Liao and Chen [15] improved some of the results, presented in [14], by applying feedback control including both the current error state feedback and delayed static error output feedback. Moreover, a less conservative synchronization condition was derived, and the controller design problem was addressed in [16]. Furthermore, Han [17] further improved the results in [14], [16] by introducing a new Lyapunov–Krasovskii functional. Those studies provide important feedback schemes and useful techniques. Notice also that nonidentical-coupled systems widely exist in practice. Most biological systems have their own properties. The idealization that coupled systems are identical ignores the diversity that is common in biology. There are some individuals in any real population that are inherently faster or slower [18]. In the brain, neural assemblies consisting of distributed local networks of neurons in different brain regions interact with each other for large-scale integration to enable the emergence of coherent behavior and cognition [19]. In physiology, physiological rhythm deriving from the interactions of different cells in the body can be synchronized to external or internal stimuli [20]. The fact that diverse individuals coordinate with each other to achieve a unified behavior in those systems requires to take diversity into account. As a result, mathematical descriptions of nonidenticalcoupled systems with parameter mismatches are proposed. In the case of parameter mismatches, loss of complete

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 10, OCTOBER 2012

synchronization may occur [21] and approximate synchronization with a nonzero synchronization error is observed in [22], which shows the importance of parameter mismatches in synchronization implementations. Therefore, it is significant to investigate the effects of parameter mismatches on master– slave synchronization between nonidentical systems [23]–[25]. There are some studies contributing to master–slave synchronization of nonidentical Lur’e systems. In [26], both static error feedback and dynamic output error feedback were applied to synchronize Lur’e systems with parameter mismatches. Based on a quadratic Lyapunov function, robust synchronization criteria were derived in the form of linear matrix inequalities (LMIs), and the error bound was estimated within two balls and one ellipsoid. Suykens et al. [27] extended the idea in [26] to the case with an external input in which H∞ synchronization was considered. However, only delayfree Lur’e systems with current error feedback are taken into account in [26] and [27]. In [28], the authors proposed a time-delayed Lur’e system, and further studied its master– slave synchronization by using impulsive control. Several delay-independent algebraic criteria were given to ensure that the synchronization error converged to a bounded ball. A more general delayed system was proposed by Huang et al. [29]. By applying static error feedback control, master–slave synchronization with a bounded synchronization error was achieved, and some delay-independent criteria were derived. In our recent work [30], we improved the results in [29] and obtained some less conservative criteria by means of generalized Halanay inequality and matrix measure. It should be pointed out that although there are a number of published papers concerning delay-dependent conditions for master–slave synchronization of identical delayed systems, it is still challenging to extend the results to nonidentical delayed systems. On one hand, a direct extension is impossible because there is an additional term caused by parameter mismatches, which differs from synchronization of identical delayed systems. On the other hand, a Lyapunov functional containing delay information is required for delay-dependent less conservative criteria, which, however, brings difficulties in synchronization analysis, as a fundamental lemma (Lemma 1 in [25], [29], and [30]) cannot be applied directly. In [25], [29], and [30], only a quadratic Lyapunov function e T (t)Pe(t) or Pe(t) is used for synchronization with parameter mismatches. From a technical point of view, the lemma in [25], [29], and [30] can be effective when involving delaydependent Lyapunov functional by using some techniques. The drawback is that the estimation of the synchronization error bound will be much larger than the expected one. So, how to derive a delay-dependent criterion with a precise error level for synchronization of delayed systems remains a challenge. In terms of control design for master–slave synchronization of nonidentical systems, little work has been done. In [29], only sufficient quasi-synchronization criteria were derived. Suykens et al. [26] proposed a nonconvex optimization approach to handle the controller design issue. However, nondifferentiability might occur. Based on derived algebraic criteria, a controller design procedure was presented in [30],

but it was unapplicable for criteria in the form of LMIs. Although the controller design problem was well studied in the form of LMIs in [16], it is unable to deal with the case involving parameter mismatches, which is more difficult as it is required not only to design the controller for quasi-synchronization, but also to keep the error level small. It is even more challenging to design the controller under a predetermined error level using a convex optimization technique. In this paper, we will consider master–slave synchronization of delayed Lur’e systems in the presence of parameter mismatches by applying feedback control, including both the current error feedback and delayed error feedback. A new delay-dependent methodology, which does not rely on the fundamental lemma in [25], [29], and [30], will be developed to achieve two goals: 1) to ensure synchronization in the sense of a bound on the synchronization error and 2) to estimate the error bound, known as the error level. As a consequence, several delay-dependent criteria will be derived and explicit expressions of error levels will be given. In addition, the relationship among the error level, parameter mismatches, and time delays will be established. Then the controller design problem will be addressed. With a predetermined error level, the corresponding feedback gains can be obtained by solving a set of LMIs. Finally, a delayed Chua’s circuit will be used to validate the proposed method. Notation: Rn denotes the n-dimensional Euclidean space.  ·  stands for either the Euclidean vector norm or its induced matrix two-norm. C([−τ˜ , 0], Rn ) denotes the Banach space of continuous vector-valued functions mapping the interval [−τ˜ , 0] into Rn . λmax (A)(λmin (A)) represents the maximum (minimum) eigenvalue of the symmetric matrix A. For symmetric matrices P and Q, the notation P < Q means that matrix P − Q is negative definite. “∗” denotes the entries implied by symmetry of a matrix. I is an identity matrix of appropriate dimensions. II. P ROBLEM S TATEMENT Consider the following master–slave synchronization scheme using both current error feedback and delayed error feedback: M : x(t) ˙ = A1 x(t) + B1 x(t − τ1 ) + D1 f (C1T x) S : y˙ (t) = A2 y(t) + B2 y(t − τ1 ) +

D2 f (C2T y) + u(t)

(1) (2)

U : u(t) = K 1 (x(t) − y(t)) + K 2 (x(t − τ2 ) − y(t − τ2 )) (3) with master system M, slave system S, and controller U; the master and slave systems are delayed Lur’e systems with state vectors x(t), y(t) ∈ Rn , respectively, τ1 > 0 is the internal delay and τ2 > 0 is the transmittal delay; Ai ∈ Rn×n , Bi ∈ Rn×n , Ci = (ci1 , ci2 , . . . , cim ) ∈ Rn×m and Di ∈ Rn×m (i = 1, 2) are constant matrices, representing parameter mismatches caused by disturbances or measurement errors; cik is the k-th column of Ci (i = 1, 2); the nonlinear function f (C2T y) = ( f 1 (c21 y), f 2 (c22 y), . . . , fm (c2m y))T satisfies a sector condition with f k (z) (k = 1, 2, . . . , m) belonging to a sector [αk , βk ] (αk < βk ) [ f k (z) − αk z][ f k (z) − βk z] ≤ 0 ∀z ∈ R.

(4)

HE et al.: SYNCHRONIZATION ERROR ESTIMATION AND CONTROLLER DESIGN

Defining the error signal e(t) = y(t) − x(t), we have the following error systems: e(t) ˙ = (A2 − K 1 )e(t) + B2 e(t − τ1 ) − K 2 e(t − τ2 ) +D2 g(C2T e(t), x(t)) + F(x(t), x(t − τ1 ))

(6)

and F(x(t), x(t − τ1 )) = δ Ax(t) + δ Bx(t − τ1 ) + D2 f (C2T x) − D1 f (C1T x)

x(t − τ1 )) in (8) is rough. For a more precise value, we define sup F(x(t), x(t − τ1 )) = ω

(5)

where g(C2T e(t), x(t)) = f (C2T e(t) + C2T x(t)) − f (C2T x(t))

1553

(7)

where Tˆ > T is sufficiently large. The purpose of this paper is to develop a methodology for delay-dependent criteria ensuring quasi-synchronization between master–slave systems, and further give the explicit expression of the error level.

with δ A = A2 − A1 and δ B = B2 − B1 . The initial condition of (5) is defined by

III. S YNCHRONIZATION C RITERIA

e(θ ) = φ(θ ), − τ˜ ≤ θ ≤ 0 where τ˜ = max{τ1 , τ2 } and φ(θ ) ∈ C([−τ˜ , 0], Rn ). For the master–slave synchronization scheme (1)–(3), we need the following assumptions. Assumption 1: The nonlinearity g(C2T e(t), x(t)) belongs to the sector [α, β]

In this section, we will first derive some delay-dependent criteria for quasi-synchronization between the master system (1) and the slave system (2), and then estimate the error level with additional constraints. First, choose a Lyapunov–Krasovskii functional candidate as

[gk (c2Tk e(t), x(t)) − αk c2Tk e(t)][gk (c2Tk e(t), x(t)) −βk c2Tk e(t)] ≤ 0

V (t, et ) =

∀ e(t), x(t) ∈ Rn .

where α = diag{α1 , α2 , . . . , αm }, β = diag{β1 , β2 , . . . , βm }, k = 1, 2, . . . , m. Assumption 2: For the master system (1), there exists a positive constant γ such that for any initial condition ϕ(θ ) ∈ C([−τ1 , 0], Rn ), there exists T (ϕ(0)) x(t, ϕ(0)) ≤ γ ∀t ≥ T. As pointed out by [26], these assumptions are reasonable and practical, especially for chaotic Lur’e systems, as master systems with bounded trajectories are widely employed in practice. In this paper, complete synchronization fails between the master system (1) and the slave system (2) due to parameter mismatches. Instead, synchronization with a nonzero error level will be achieved. We refer to such synchronization as quasi-synchronization, which is defined in the following. Definition 1: The master system (1) and the slave system (2) are said to be quasi-synchronized with an error level > 0 if there exists a compact set M such that for any φ(θ ) ∈ C([−τ˜ , 0], Rn ), the error signal e(t) = y(t) − x(t) converges to M = {e ∈ Rn | e ≤ } as t goes to infinity. The nonlinear function F(x(t), x(t − τ1 )) in (7), which is caused by parameter mismatches contributes to the error level. By Assumption 2, it is clear that the nonlinear function F(x(t), x(t − τ1 )) is bounded F(x(t), x(t − τ1 ))

6 

where L = diag{l1 , l2 , . . . , lm } with lk = max{|αk |, |βk |}, k = 1, 2, . . . , m. However, the upper bound of F(x(t),

(10)

where V1 (t, et ) = e T (t)Pe(t)  2  e T (t)Wi V2 (t, et ) = 2 V3 (t, et ) = V4 (t, et ) = V5 (t, et ) = V6 (t, et ) =

i=1 2  t  t −τi i=1  2  t i=1 2  i=1 2 



t t −τi

 τi

t t −τi

i=1

t −τi

e(s)ds

e(s)ds

 Yi

(12) 

t t −τi

e(s)ds

e T (s)Q i e(s)ds

t −τi

τi

(11) t

T

(13)

(14)

(τi − t + s)e T (s)Ti e(s)ds

(15)

(τi − t + s)e˙ T (s)Ri e(s)ds ˙

(16)

with et = e(t + θ ), ∀θ ∈ [−τ˜ , 0], symmetric matrices P > 0, Yi > 0, Q i > 0, Ti > 0, Ri > 0 and constant matrices Wi (i = 1, 2). We have the following result. Theorem 1: Under Assumptions 1 and 2, the trajectory of the error system (5) converges exponentially to the set 



M = e ∈ R | e ≤

(8)

Vi (t, et )

i=1

n

≤ δ Ax(t) + δ Bx(t − τ1 ) +(D2 C2  + D1 C1 )Lx(t) ≤ (δ A + δ B + (D2 C2  + D1 C1 )L)γ

(9)

t ≥Tˆ

μ ω σ λmin (1 )

 (17)

if there exist symmetric matrices P > 0, H > 0,  > 0, N > 0, Yi > 0, Q i > 0, Ti > 0, Ri > 0,  j > 0 (i = 1, 2; j = 1, 2, 3, 4), constant matrices Wi (i = 1, 2), a diagonal matrix  > 0, and positive scalars σ, μ

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 10, OCTOBER 2012

positive scalars σ, μ such that

such that 1 > 0

(18)

1 < 2

(19)

σ 1 + 2 − H < 0 (e (e

σ τ1

− 1)τ12 R1

σ τ1

(20)

1 < 0

(21)

(eσ τ2 − 1)Q 2 − eσ τ2 2 < 0

(22)

+ (e

− 1)Q 1 − e

σ τ1

σ τ2

− 1)τ22 R2

ˆ1 > 0  ˆ2 ˆ1 <   ˆ ˆ σ 1 + 2 − H < 0

− 3 < 0

(23)

 − 4 < μI

(24)

(eσ τ − 1)Q − eσ τ 1 < 0 (eσ τ − 1)τ 2 R − 2 < 0  − 3 < μI where ˆ1 = 

where

ˆ2 = 

⎞ P W1 W2 1 = ⎝ ∗ Y1 0 ⎠ ∗ ∗ Y2 ⎛

and 1 is given in Appendix B. Remark 1: In the existing literature, there are few methods to estimate the synchronization error in the presence of parameter mismatches, and it is challenging to extend the delay-independent criteria to delay-dependent ones when parameter mismatches are involved. Theorem 1 provides us a general methodology to derive delay-dependent criteria for quasi-synchronization in the context of two delayed systems, regardless of the chosen Lyapunov functional. While in [23], [25], [26], and [29], only specific Lyapunov functions were employed and delay-independent sufficient synchronization conditions were obtained. One can see that the extension work is not trivial, and the results are general. The developed methodology can also be applied in the synchronization of nonidentical networks, as discussed in [31]–[35]. In the case of τ1 = τ2 = τ , the corresponding Lyapunov– Krasovskii functional becomes  t  t V (t, et ) = e T Pe + 2e T W e(s)ds + e T (s)Qe(s)ds + +τ +τ

t −τ

 t −τ t

T

t −τ t t −τ

e(s)ds

 >0

(eσ τ − 1)τ 2 T 0 ∗ 0



ˆ 1 is given in Appendix B. and  If the current error feedback is not available, which means that K 1 = 0 in (5), then the error system becomes

2 = diag{−H, −1, −2 , 0, −3 , −4 }

t



P W ∗ Y

ˆ 2 = diag{−H, −1, 0, −2 , −3 } 

2 = diag{(eσ τ1 − 1)τ12 T1 + (eσ τ2 − 1)τ22 T2 , 0, 0}





Y



t −τ

t

t −τ



e(t) ˙ = A2 e(t) + B2 e(t − τ1 ) − K 2 e(t − τ2 ) +D2 g(C2T e(t), x(t)) + F(x(t), x(t − τ1 )). One can immediately obtain sufficient conditions for quasisynchronization by setting K 1 = 0 in Theorem 1. The above results, including Theorem 1 and Corollary 1 are derived mainly based on the fact that internal delay and transmittal delay coexist. We now consider some specific cases in which one single delay is involved. If only transmittal delay is considered in master–slave Lur’e systems, which corresponds to the case of B1 = B2 = 0, then the systems become Hopfield neural networks [5], [6], cellular neural networks [8] without external inputs and similar models discussed in [15]–[17]. By assuming τ2 = τ , the master system and the slave system become M : x(t) ˙ = A1 x(t) + D1 f (C1T x) S : y˙ (t) = A2 y(t) + D2 f (C2T y) + u(t) U : u(t) = K 1 (x(t) − y(t)) + K 2 (x(t − τ ) −y(t − τ )).

(28)

Then the error system is

e(s)ds

e˙(t) = (A2 − K 1 )e(t) − K 2 e(t − τ ) +D2 g(C2T e(t), x(t)) + F(x(t))

(τ − t + s)e T (s)T e(s)ds (τ − t + s)e˙ T (s)R e(s)ds. ˙

(26) (27)

(25)

By Theorem 1, we have the following corollary. Corollary 1: Under Assumptions 1 and 2, the trajectory of the error system (5) converges exponentially to the set    μ Mˆ = e ∈ Rn | e ≤ ω ˆ 1) σ λmin ( if there exist symmetric matrices P > 0, H > 0,  > 0, N > 0, Y > 0, Q > 0, T > 0, R > 0,  j > 0 ( j = 1, 2, 3), a constant matrix W , a diagonal matrix  > 0, and

where F(x(t)) = δ Ax(t) + D2 f (C2T x) − D1 f (C1T x). The corresponding quasi-synchronization conditions are easily derived by setting B2 = 0 in Corollary 1. When parameter mismatches vanish, it reduces to the case as that in [15]–[17]. From Corollary 1, complete synchronization can be guaranteed if there exist symmetric matrices P > 0, N > 0, Y > 0, Q > 0, T > 0, R > 0, a constant matrix W and a diagonal matrix  > 0 such that   P W >0 ∗ Y

HE et al.: SYNCHRONIZATION ERROR ESTIMATION AND CONTROLLER DESIGN

and

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

(1, 1) ∗ ∗ ∗ ∗

where



Y (1, 3) (1, 4) (A2 − K 1 N ⎟ −T −Y 0 WT ⎟ T ⎟ 0,  > 0, N > 0, Q i > 0, Ri > 0,  j > 0 (i = 1, 2; j = 1, 2, 3, 4, 5), a diagonal matrix  > 0, and positive scalars σ, μ such that ˜1 <  ˜2  σ P − 1 < 0 (eσ τ1 − 1)Q 1 − eσ τ1 2 < 0 (eσ τ2 − 1)Q 2 − eσ τ2 3 < 0

e(t) ˙ = (A2 − K 1 )e(t) + B2 e(t − τ ) +D2 g(C2T e(t), x(t)) + F(x(t), x(t − τ ))

(eσ τ1 − 1)τ12 R1 + (eσ τ2 − 1)τ22 R2 − 4 < 0  − 5 < μI

where F(x(t), x(t − τ )) = δ Ax(t) + δ Bx(t − τ ) +D2 f (C2T x) − D1 f (C1T x).

D2 f (C2T

S : y˙ (t) = A2 y(t) + U : u(t) = K 1 (x(t) − y(t))

y) + u(t)

(29) (30) (31)

if there exist symmetric matrices P > 0,  > 0, 1 > 0, a diagonal matrix  > 0, and positive scalars σ, μ such that  < μI and



⎞ (1, 1) P D2 + C2 (α + β) P ⎝ ∗ −2 0 ⎠ ηI (40) to Corollary 3, where η is a positive constant. Then it is easy to √ obtain the corresponding error level (μ/σ η)ω. Given scalars μ, σ , and η, we establish the following result by applying Corollary 3 assuming α = 0. Theorem 2: Under Assumptions 1 and 2, for given scalars μ, σ, η, and ρi (i = 1, 2, 3, 4), the master system (1) and the slave system (2) achieve quasi-synchronization with the predetermined error level  μ ω (41) = ση

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if there exist symmetric matrices P˜ > 0,  > 0, N˜ > 0, 5 > 0, Q˜ i > 0, R˜ i > 0, ˜ j > 0 (i = 1, 2; j = 1, 2, 3, 4), ˜ > 0, and X, Y of appropriate dimensions a diagonal matrix  such that 

˜ > 0 and X, Y of appropriate dimensions such that matrix    ˜  ˜  ˜ 0, ˜ j > 0 ( j = 1, 2, 3), a diagonal



P˜ ⎜∗ ⎜ ˜ =⎜∗  ⎜ ⎝∗ ∗

P˜ ρ1 R˜ 0 0 ∗ 0 ∗ ∗ ∗ ∗

0 0 R˜ ρ2 Q˜ 0 0 ∗ 0 ∗ ∗

0 0 0 τ N˜ 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ N˜ ⎠ 0

and ˜ −˜ 1 , − R, ˜ −˜ 2 , − Q, ˜ − R, ˜ −˜ 3 }. ˜ = diag{− Q,  Moreover, the controller gains are given by K 1 = X P˜ −1 and K 2 = Y R˜ −1 . V. N UMERICAL S IMULATIONS In order to show the effectiveness of the derived results, we consider the following time-delay Chua’s circuit as the master system: ⎧ dx 1 ⎪ ⎨ dt = a(x 2 (t) − m 1 x 1 (t) + g(x 1 (t))) − cx 1(t − τ1 ) d x2 (50) dt = x 1 − x 2 + x 3 − cx 1 (t − τ1 ) ⎪ ⎩ d x3 = −bx (t) + c(2x (t − τ ) − x (t − τ )) 2 1 1 3 1 dt with nonlinear characteristic g(x 1 ) = 0.5(m 1 − m 0 )(|x 1 + 1| − |x 1 − 1|) and parameters m 0 = −1/7, m 1 = 2/7, a = 9, b = 14.286, c = 0.1, and τ1 = 1.

HE et al.: SYNCHRONIZATION ERROR ESTIMATION AND CONTROLLER DESIGN

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MASTER

1.6

2

1.4

||F(x(t), x(t−τ1))||

x3(t)

1.8 4

0 −2 −4 4 2

1.2 1 0.8 0.6

0.5

0.4

0

0

−2 −4

x (t) 1

−0.5

0.2 x2(t)

0 SLAVE

5 10

20

30

40

50

60

70

80

90

100

60

70

80

90

100

60

70

80

90

100

60

70

80

90

100

t Fig. 2.

4

Evolution of F(x(t), x(t − τ1 )). 0.05

e1(t)

2 y3(t)

1.4408

0

0 −0.05 5 10

−2

20

30

40

50 t

−4 4 2

0.5 0

1

−4

−0.5

−0.05 5 10

y (t) 2

and f (x) = (0.5(|x 1 + 1| − |x 1 − 1|), 0, 0)T belonging to a sector [0, 1]. We take the same slave system as that in [28] with ⎡ ⎤ −am 1 + 0.2 a + 0.1 0 1 −1 1⎦, A2 = ⎣ 0 −b + 0.2 0 ⎤ ⎡ −c − 0.1 0 0 B2 = ⎣ −c + 0.5 0 0 ⎦ , 2c 0 −c ⎡ ⎡ ⎤ ⎤ 100 a(m 1 − m 0 ) − 1 0 0 0 0 0 ⎦. C 2 = ⎣ 0 1 0 ⎦ , D2 = ⎣ 0 01 001 Fig. 1 shows the trajectories of the master system and the slave system with initial values x 1 (s) = −0.1, x 2 (s) = 0.2, x 3 (s) = 0.1, y1 (s) = −0.1, y2 (s) = 0.2, y3 (s) = 0.1, ∀s ∈ [−1, 0]. First, we consider quasi-synchronization. Let the controller gains be K 1 = 27.1I, K 2 = 0.1I

30

40

50

0.05

Master–slave systems without controller u(t).

The system can be represented in the Lur’e form with ⎡ ⎡ ⎤ ⎤ −am 1 a 0 −c 0 0 A1 = ⎣ 1 −1 1 ⎦ , B1 = ⎣ −c 0 0 ⎦ , 0 −b 0 2c 0 −c ⎡ ⎡ ⎤ ⎤ 100 a(m 1 − m 0 ) 0 0 0 0 0⎦ C 1 = ⎣ 0 1 0 ⎦ , D1 = ⎣ 0 00 001

20

t

e (t) 3

Fig. 1.

0

0

−2 y (t)

e (t)

0.05 2

0 −0.05 5

10

20

30

40

50 t

Fig. 3.

Time evolutions of synchronization error signals ei (t), i = 1, 2, 3.

and τ2 = 0.2, σ = 6.6. Fig. 2 depicts the evolution of F(x(t), x(t − τ1 )) defined by (7), from which we get ω = 1.4408 in (9). The sufficient conditions by Theorem 1 are satisfied and the error level is 0.1049. Thus, we can conclude that master–slave systems are in a state of quasi-synchronization with the error level 0.1049. Fig. 3 gives the simulation result for the error system. Fig. 4 compares the derived error level with simulated synchronization errors, which shows the effectiveness of quasi-synchronization criteria. We take the same setting as Example 3 in [30] by assuming B1 = B2 = 0. The error level derived by Corollary 2 is 0.0897, comparing with 0.12242 in [30] in the setting of norm-2 with K 1 = 20.2794I . Corollary 2 provides a tighter error level, which is closer to the maximum of simulation errors 0.0551. Second, we address the controller design in two cases. Case 1: Let τ1 = 1, τ2 = 0.2, μ = 0.05, σ = 2, and η = 0.1. It is easy to get the error level 0.7204. Choosing ρ1 = 0.01, ρ2 = 0.01, ρ3 = 0.5, ρ4 = 0.1, and using Theorem 2, the feedback gains are given by ⎡ ⎤ 28.5034 −3.7212 −5.7968 K 1 = ⎣ −9.2011 73.7076 16.6895 ⎦ −4.7261 1.6132 72.1839

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1.5×10

ERROR

5× 10−1

The error level 0.1049 −1

10

0.0579

The predetermined error level 0.3222

−1

||e(t)||

||e(t)||

10

0

5 10

−2

20

30

40

50

60

70

80

90

10

100

t

5 10

20

30

40

50

60

70

80

90

100

t

Fig. 4.

e(t) on logarithmic scale and the error level 0.1049.

Fig. 6. Master–slave quasi-synchronization of delayed Chua’s circuits for controller design Case 2 with the predetermined error level 0.3222.

ERROR

0

10

VI. C ONCLUSION The predetermined error level 0.7204

−1

||e(t)||

10

−2

10

5 10

20

30

40

50

60

70

80

90

100

t

Fig. 5. Master–slave quasi-synchronization of delayed Chua’s circuits for controller design Case 1 with the predetermined error level 0.7204.



⎤ 0.0001 0.0006 0.0007 K 2 = ⎣ −0.0002 0.0005 0.0016 ⎦. 0.0004 0.0009 0.0002 We give the simulation result of the error signal for the derived gains in Fig. 5, from which one can see that the master and slave systems are quasi-synchronized with the error level 0.7204. Case 2: Let τ1 = τ2 = 0.5, μ = 0.05, σ = 10, and η = 0.1. It is easy to obtain the error level 0.3222. Choosing ρ1 = 0.01 and ρ2 = 0.5, applying Theorem 3 yield ⎡

13.8435 K 1 = ⎣ 1.1132 −0.2423 ⎡ −0.1993 K 2 = ⎣ 0.3999 0.1999

The master–slave synchronization of coupled delayed Lur’e systems with parameter mismatches has been studied. By introducing a type of quasi-synchronization, a general methodology has been developed to derive several delay-dependent quasi-synchronization criteria and to give the explicit expressions of error levels. Based on these synchronization criteria, the controller design with a predetermined error level has been proposed. Finally, the effectiveness of obtained results has been illustrated by a delayed Chua’s circuit. The quasi-synchronization scheme discussed in this paper is practical, and the developed delay-dependent methodology provides us an effective way to solve the problem, in which the key point is to estimate the synchronization error and give an accurate error level. From the practical point of view, even if parameter mismatches are large, it is possible to control the synchronization error in a relative small level by using the proposed controller design method. On one hand, some optimization methods should be applied to derive the more accurate error level. On the other hand, this general methodology may be extended to deal with synchronization over heterogeneous networks. A PPENDIX A P ROOF OF T HEOREM 1 Proof: Taking the derivative of (10) with respect to t along the trajectory of (5) yields V˙ (t, et ) =



9.2338 −0.2101 11.6582 0.6709 ⎦ −14.4713 15.3014 ⎤ −0.0000 0.0000 0.0001 −0.0000 ⎦. 0.0000 −0.0995

The simulation result of the error signal for the derived gains is illustrated in Fig. 6. One can see that master and slave systems are quasi-synchronized with the predetermined error level 0.3222, which shows that the design method is effective.

6 

V˙i (t, et )

(51)

i=1

where V˙1 (t, et ) = 2e T (t)P[(A2 − K 1 )e(t) + B2 e(t − τ1 ) −K 2 e(t − τ2 ) + D2 g(C2T e(t), x(t)) +F(x(t), x(t − τ1 ))]  t  T T ˙ V2 (t, et ) = 2e˙ (t)W1 e(s)ds + 2e˙ (t)W2 t −τ1

t t −τ2

e(s)ds

+2e T (t)(W1 + W2 )e(t) − 2e T (t)W1 e(t − τ1 ) −2e T (t)W2 e(t − τ2 )

HE et al.: SYNCHRONIZATION ERROR ESTIMATION AND CONTROLLER DESIGN

V˙3 (t, et ) = 2



T

t

t −τ1  t

+2

Y1 (e(t) − e(t − τ1 ))

e(s)ds

1559

From (19), we have V˙ (t, et ) ≤ −ς T (t)H ς (t) − e T (t − τ1 )1 e(t − τ1 )

T

t −τ2

Y2 (e(t) − e(t − τ2 ))

e(s)ds

˙ −e T (t − τ2 )2 e(t − τ2 ) − e˙ T (t)3 e(t)

V˙4 (t, et ) = e T (t)(Q 1 + Q 2 )e(t) − e T (t − τ1 )Q 1 e(t − τ1 ) −e T (t − τ2 )Q 2 e(t − τ2 ) V˙5 (t, et ) = e T (t)(τ12 T1 + τ22 T2 )e(t)  t  T −τ1 e (s)T1 e(s)ds − τ2 V˙6 (t, et ) =

t

e T (s)T2 e(s)ds t −τ1 t −τ2 e˙ T (t)(τ12 R1 + τ22 R2 )e(t) ˙  t  t e˙ T (s)R1 e˙(s)ds − τ2 e˙ T (s)R2 e(s)ds. ˙ −τ1 t −τ1 t −τ2



≤−  t − τ2

t −τ1

t −τ2



≤− and

 − τ1

T

t

t −τ1

T1

t −τ1

T

t

e(s)ds

 T2

e(s)ds



t t −τ2

V˜ (t, et ) = eσ t V (t, et )

e(s)ds

Then integrating both sides from 0 to t, we have  t ˜ ˜ V (t, et ) − V (0, e0 ) = σ eσ s V (s, es ))ds 0  t + eσ s V˙ (s, es )ds.

≤ −[e(t) − e(t − τ1 )]T R1 [e(t) − e(t − τ1 )]  t − τ2 e˙ T (s)R2 e(s)ds ˙

It follows that:  t σ eσ s V (s, es )ds 0  t = σ eσ s ς T (s)1 ς (s)ds 0

t −τ2



≤ −[e(t) − e(t − τ2 )] R2 [e(t) − e(t − τ2 )]. T

2  t  0

i=1

According to Assumption 1, for  = diag{λ1 , . . . , λm } > 0, we have m  −2 λk [gk (c2Tk e(t), x(t)) − αk c2Tk e(t)]

+σ +σ

k=1

×[gk (c2Tk e(t), x(t)) − βk c2Tk e(t)] ≥ 0

2  i=1 2 

τi

s

 t 0

τi

eσ s e T (θ )Q i e(θ )dθ ds

s−τi s s−τi

 t 0

i=1

s s−τi

eσ s (τi − s + θ )eiT (θ )Ti e(θ )dθ ds eσ s (τi − s + θ )e˙iT (θ )Ri e(θ ˙ )dθ ds. (55)

which can be rewritten as Exchange integral orders to obtain

−2g T (C2T e(t), x(t))g(C2T e(t), x(t)) +2e T (t)C2 (α + β)g(C2T e(t), x(t)) −2e

(t)C2 αβC2T e(t)

≥ 0.

From (5), it is clear that 2[(A2 − K 1 )e(t) + B2 e(t − τ1 ) − K 2 e(t − τ2 )

σ

2  t  i=1

0

V˙ (t, et ) ≤ ξ (t)1 ξ(t) +F T (x(t), x(t − τ1 ))F(x(t), x(t − τ1 )).

s

eσ s e T (θ )Q i e(θ )dθ ds

s−τi

= σ

+D2 g(C2T e(t), x(t)) + F(x(t), x(t − τ1 )) − e(t)] ˙ T N e(t) ˙ = 0. t t Let ξ(t) = [e T (t), ( t −τ1 e(s)ds)T , ( t −τ2 e(s)ds)T , e T (t− τ1 ), e T (t −τ2 ), g T (C T e(t), x(t)), e˙T (t), F T (x(t), x(t −τ1 ))]T . Then we have T

(54)

0

e˙ T (s)R1 e(s)ds ˙

T

(53)

V˙˜ (t, et ) = σ eσ t V (t, et ) + eσ t V˙ (t, et ).



t

e T (s)T2 e(s)ds

t −τ2 t

e(s)ds



(52) ×F(x(t), x(t − τ1 )) T t t  where ς (t) = e T (t), ( t −τ1 e(s)ds)T , ( t −τ2 e(s)ds)T . Next, we give the convergence analysis of the error signal, in which an explicit expression of the error level is obtained. Define a new functional as

where σ > 0 is a constant to be determined. Taking the derivative of V˜ (t, et ) with respect t yields

Using Jensen’s inequality [36], we obtain  t − τ1 e T (s)T1 e(s)ds t −τ1

+F T (x(t), x(t − τ1 ))( − 4 )

2  



0

θ+τi



0 i=1 −τi  2 t −τ  i







θ

0



i=1 2  t  i=1

=

2   i=1

0 −τi

t −τi

 dθ

θ

t

eσ s e T (θ )Q i e(θ )ds θ+τi

eσ s e T (θ )Q i e(θ )ds

eσ s e T (θ )Q i e(θ )ds

(eσ (θ+τi ) − 1)e T (θ )Q i e(θ )dθ

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 2  + (eσ τi − 1)

σθ T

e e (θ )Q i e(θ )dθ

0

i=1

+

t −τi

2  

t

(eσ t − eσ θ )e T (θ )Q i e(θ )dθ

t −τi

i=1

(56)

and σ

2 

τi

 t 0

i=1

s

eσ s (τi − s + θ )e T (θ )Ti e(θ )dθ ds

2 

τi2

i=1



0

s−τi

≤σ ≤

 t ˜ eσ s ς T (s)(σ 1 + 2 − H )ς T (s)ds ≤ V (0, e0 ) + 0  t −τ1 σs T e e (s)[(eσ τ1 − 1)Q 1 − eσ τ1 1 ]e(s)ds + 0  t −τ2 + eσ s e T (s)[(eσ τ2 − 1)Q 2 − eσ τ2 2 ]e(s)ds 0  t eσ s e˙ T (s)[(eσ τ1 − 1)τ12 R1 + (eσ τ2 − 1)τ22 R2 +

2  i=1 2 

0



τi2

0

−τi

 τi2

2 

0

−τi

i=1

+

 t

s

−3 ]e(s)ds ˙ + eσ t

i=1

eσ s e T (θ )Ti e(θ )dθ ds

s−τi

+ϕ(0, e0 ) +

(eσ (θ+τi ) − 1)e T (θ )Ti e(θ )dθ (e

σ (θ+τi )

τi2 (eσ τi − 1)

ϕ(0, e0 ) =

− 1)e (θ )Ti e(θ )dθ eσ θ e T (θ )Ti e(θ )dθ

+

(57)

0

i=1

and σ

2 

τi

+

 t

i=1

0

s

+

2  i=1 2 

 τi2

0 −τi

τi2 (eσ τi

i=1

− 1)

t

eσ θ e˙ T (θ )Ri e(θ ˙ )dθ.

−eσ τ2

−τ1 0 −τ2

(58)

μω2 σ t (e − 1). σ

(59)

Referring to (54)–(59), we obtain

= V˜ (0, e0 ) +



t 0

t −τi

e T (θ )Q i e(θ )dθ

(eσ t − 1)

(60)

(eσ (θ+τi ) − 1)e T (θ )Q i e(θ )dθ 

τi2

0 −τi

 τi2 

0 −τi

0

−τ1  0 −τ2

(eσ (θ+τi ) − 1)e T (θ )Ti e(θ )dθ (eσ (θ+τi ) − 1)e˙ T (θ )Ri e(θ ˙ )dθ

eσ s e T (s)1 e(s)ds eσ s e T (s)2 e(s)ds.

(61)

From (20)–(23), (53), and (60), we have

0

eσ s e T (s)2 e(s)ds +

V˜ (t, et )

i=1 2 

−eσ τ2

From (24) and (52), we have  t eσ s V˙ (s, e(s))ds 0  t  t −τ1 ≤− eσ s ς T (s)H ς T (s)ds − eσ τ1 eσ s e T (s)1 e(s)ds 0 −τ1  t  t −τ1 −eσ τ2 eσ s e T (s)2 e(s)ds − eσ s e˙ T (s)3 e(s)ds ˙ 0 −τ2  t eσ s ds +μω2 0  t  t −τ1 σs T T σ τ1 ≤− e ς (s)H ς (s)ds − e eσ s e T (s)1 e(s)ds 0 0  t  t −τ2 σ τ2 σs T −e e e (s)2 e(s)ds − eσ s e˙ T (s)3 e(s)ds ˙ 0 0  0 eσ s e T (s)1 e(s)ds −eσ τ1 

−τi

2 

−eσ τ1

(eσ (θ+τi ) − 1)e˙ T (θ )Ri e(θ ˙ )dθ 

0

i=1

eσ s (τi − s + θ )e˙ T (θ )Ri e(θ ˙ )dθ ds

s−τi



σ

2   i=1

t

μω2

t

where

T



2  

σ eσ s V (s, es ))ds +



t 0

eσ s V˙ (s, es )ds

eσ t ς T (t)1 ς T (t) μω2 σ t (e − 1) ≤ V˜ (0, e0 ) + ϕ(0, e0 ) + σ

(62)

which implies that ς T (t)1 ς T (t) ≤ e−σ t V˜ (0, e0 ) + e−σ t ϕ(0, e0 ) +

μω2 (1 − e−σ t ). σ

(63)

Consequently λmin (1 )e T (t)e(t) μω2 . ≤ e−σ t V˜ (0, e0 )) + e−σ t ϕ(0, e0 ) + σ

(64)

As V˜ (0, e0 ) and ϕ(0, e0 ) are bounded, we can conclude that the error system (5) converges exponentially to    μ M = e ∈ Rn | e(t) ≤ ω σ λmin (1 ) which means that the master system (1) and the slave system (2) achieve quasi-synchronized with an error level √ (μ/σ λmin (1 ))ω according to Definition 1. This completes the proof.

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A PPENDIX B 1 in Theorem 1 ⎛

(1, 1) ⎜ ∗ ⎜ ⎜ ∗ ⎜ ⎜ ∗ 1 = ⎜ ⎜ ∗ ⎜ ⎜ ∗ ⎜ ⎝ ∗ ∗

Y1 Y2 P B2 − W1 + R1 −P K 2 − W2 + R2 P D2 + C2 (α + β) (A2 − K 1 )T N −T1 0 −Y1 0 0 W1T ∗ −T2 0 −Y2 0 W2T 0 0 B2T N ∗ ∗ −Q 1 − R1 ∗ ∗ ∗ −Q 2 − R2 0 −K 2T N ∗ ∗ ∗ ∗ −2 D2T N ∗ ∗ ∗ ∗ ∗ (7, 7) ∗ ∗ ∗ ∗ ∗ ∗

⎞ P 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ N ⎠ −

(1, 1) = P(A2 − K 1 ) + (A2 − K 1 )T P + Q 1 + Q 2 + W1 + W1T + W2 + W2T + τ12 T1 + τ22 T2 − R1 − R2 − C2 αβC2T − C2 βαC2T (7, 7) = −2N + τ12 R1 + τ22 R2 ˆ 1 in Corollary 1  ⎛

(1, 1) ⎜ ∗ ⎜ ⎜ ∗ ˆ 1 = ⎜ ⎜ ∗ ⎜ ⎝ ∗ ∗

⎞ Y P(B2 − K 2 ) − W + R P D2 + C2 (α + β) (A2 − K 1 )T N P 0 ⎟ −T −Y 0 WT ⎟ ∗ −Q − R 0 (B2 − K 2 )T N 0 ⎟ ⎟ 0 ⎟ ∗ ∗ −2 D2T N ⎟ ∗ ∗ ∗ −2N + τ 2 R N ⎠ ∗ ∗ ∗ ∗ −

(1, 1) = P(A2 − K 1 ) + (A2 − K 1 )T P + Q + W + W T + τ 2 T − R − C2 αβC2T − C2 βαC2T ˜ 1 in Corollary 3  ⎛

(1, 1) P B2 + R1 −P K 2 + R2 P D2 + C2 (α + β) (A2 − K 1 )T N ⎜ ∗ −Q 1 − R1 0 0 B2T N ⎜ ⎜ ∗ −Q 2 − R2 0 −K 2T N ˜1 =⎜ ∗  ⎜ ∗ ∗ ∗ −2 D2T N ⎜ ⎝ ∗ ∗ ∗ ∗ −2N + τ12 R1 + τ22 R2 ∗ ∗ ∗ ∗ ∗

⎞ P 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ N ⎠ −

(1, 1) = P(A2 − K 1 ) + (A2 − K 1 )T P + Q 1 + Q 2 − R1 − R2 − C2 αβC2T − C2 βαC2T , , and  in Theorem 2 ⎛

⎞ ˜ 2 β P˜ A T − X T ˜ + PC I (1, 1) B2 R˜ 1 + P˜ −Y + P˜ D2  2 ⎜ ∗ ⎟ 0 (2, 2) 0 0 R˜ 1 B2T ⎜ ⎟ T ⎜ ∗ ⎟ 0 ∗ (3, 3) 0 −Y ⎜ ⎟ =⎜ ⎟ T ˜ ˜ ∗ ∗ −2 D2 0 ⎜ ∗ ⎟ ⎝ ∗ ⎠ ∗ ∗ ∗ (5, 5) I ∗ ∗ ∗ ∗ ∗ − + 5 T T ˜ ˜ ˜ (1, 1) = A2 P + P A2 − X − X − 2(ρ1 + ρ2 ) P

(2, 2) = −(2ρ3 + 1) R˜ 1 (3, 3) = −(2ρ4 + 1) R˜ 2 (5, 5) = −2 N˜ ⎛ ˜ ˜ P P ⎜∗ 0 ⎜ ⎜∗ ∗ =⎜ ⎜∗ ∗ ⎜ ⎝∗ ∗ ∗ ∗

⎞ 0 0 0 0 0 0 P˜ ρ1 R˜ 1 ρ2 R˜ 2 0 0 0 0 R˜ 1 ρ3 Q˜ 1 0 0 0 0 0⎟ ⎟ ˜ ˜ 0 0⎟ 0 0 0 0 0 R2 ρ4 Q 2 0 ⎟ ∗ 0 0 0 0 0 0 0 0 0⎟ ⎟ ∗ ∗ 0 0 0 0 0 τ1 N˜ τ2 N˜ N˜ ⎠ ∗ ∗ ∗ 0 0 0 0 0 0 0  = diag{− Q˜ 1 , − Q˜ 2 , −˜ 1 , − R˜ 1 , − R˜ 2 , −˜ 2 , − Q˜ 1 , −˜ 3 , − Q˜ 2 , − R˜ 1 , − R˜ 2 , −˜ 4 }.

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 10, OCTOBER 2012

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Wangli He (M’11) received the B.S. degree in information and computing science and the Ph.D. degree in applied mathematics from Southeast University, Nanjing, China, in 2005 and 2010, respectively. She was a Post-Doctoral Research Fellow with the Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, School of Information Science and Engineering, East China University of Science and Technology, Shanghai, China, from January 2010 to November 2011. She was a Visiting Post-Doctoral Research Fellow with the Centre for Intelligent and Networked Systems and the School of Information and Communication Technology, Central Queensland University, Rockhampton, Australia, from July 2010 to July 2011. She is currently a Lecturer with the School of Information Science and Engineering, East China University of Science and Technology, Shanghai, China. Her current research interests include chaos synchronization and control, stability theory, synchronization of heterogeneous networks, and consensus in multiagent systems.

Feng Qian received the B.Sc. degree in chemical automation and meters from the Nanjing Institute of Chemical Technology, Nanjing, China, in 1982, and the M.S. and Ph.D. degrees in automation from the East China Institute of Chemical Technology, Shanghai, China, in 1988 and 1995, respectively. He was the Director of the Automation Institute, East China University of Science and Technology, from 1999 to 2001, and was the Head of the Scientific and Technical Department from 2001 to 2006. He is currently the Vice President of the East China University of Science and Technology, the Director of the Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, Shanghai, and the Director of the Process System Engineering Research Center, Ministry of Education, Shanghai. His current research interests include modeling, control, optimization and integration of petrochemical complex industrial processes and their industrial applications, neural network theory, and real-time intelligent control technology and its applications to the ethylene, PTA, PET, and refining industries. Dr. Qian is a member of the China Instrument and Control Society, the Chinese Association of Higher Education, and China’s PTA Industry Association.

HE et al.: SYNCHRONIZATION ERROR ESTIMATION AND CONTROLLER DESIGN

Qing-Long Han (M’09) received the B.Sc. degree in mathematics from Shandong Normal University, Jinan, China, in 1983, and the M.Eng. and the Ph.D. degrees in information science (electrical engineering) from the East China University of Science and Technology, Shanghai, China, in 1992 and 1997, respectively. From 1997 to 1998, he was a Post-Doctoral Researcher Fellow with the Laboratoire d’Auomatique et d’Informatique Industrielle (LAII), École Supérieure d’Ingénieurs de Poitiers (ESIP), Université de Poitiers, Poitiers, France. From 1999 to 2001, he was an Assistant Research Professor with the Department of Mechanical and Industrial Engineering, Southern Illinois University, Edwardsville. Since 2001, he has been with Central Queensland University, Rockhampton, Australia, where he is currently a Professor with the School of Information and Communication Technology, an Associate Dean (Research and Innovation) with the Faculty of Arts, Business, Informatics and Education, and the Director of the Center for Intelligent and Networked Systems. He was a Visiting Professor with LAII-ESIP, Université de Poitiers, a Guest Professor with the Huazhong University of Science and Technology, Hubei, China, and the East China University of Science and Technology. In 2010, he held the C. Jiang (Yangtze River) Scholar Chair Professorship, Ministry of Education, China. In October 2011, he held the 100 Talents Program Chair Professorship, Shanxi Province, China. His current research interests include time-delay systems, networked control systems, neural networks, and complex dynamical systems.

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Jinde Cao (M’07–SM’07) received the B.S. degree from Anhui Normal University, Wuhu, China, the M.S. degree from Yunnan University, Kunming, China, and the Ph.D. degree from Sichuan University, Chengdu, China, in 1986, 1989, and 1998, respectively, all in mathematics and applied mathematics. He was with Yunnan University from 1989 to 2000. Since 2000, he has been with the Department of Mathematics, Southeast University, Nanjing, China. From 2001 to 2002, he was a Post-Doctoral Research Fellow with the Department of Automation and Computer-Aided Engineering, Chinese University of Hong Kong, Shatin, Hong Kong. He was a Visiting Research Fellow and a Visiting Professor with the School of Information Systems, Computing and Mathematics, Brunel University, Middlesex, U.K., from 2006 to 2008. He is the author or co-author of more than 160 research papers and five edited books. His current research interests include nonlinear systems, neural networks, complex systems, complex networks, stability theory, and applied mathematics. Dr. Cao was an Associate Editor of the IEEE T RANSACTIONS ON N EURAL N ETWORKS from 2006 to 2009. He is an Associate Editor of the Journal of the Franklin Institute, Neurocomputing, the International Journal of Differential Equations, Discrete Dynamics in Nature and Society, and Differential Equations and Dynamical Systems.

Synchronization error estimation and controller design for delayed Lur'e systems with parameter mismatches.

This paper investigates the problem of master-slave synchronization of two delayed Lur'e systems in the presence of parameter mismatches. First, by an...
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