IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 8, AUGUST 2013
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Sampled-Data Exponential Synchronization of Complex Dynamical Networks With Time-Varying Coupling Delay Zheng-Guang Wu, Peng Shi, Hongye Su, and Jian Chu
Abstract— This paper studies the problem of sampled-data exponential synchronization of complex dynamical networks (CDNs) with time-varying coupling delay and uncertain sampling. By combining the time-dependent Lyapunov functional approach and convex combination technique, a criterion is derived to ensure the exponential stability of the error dynamics, which fully utilizes the available information about the actual sampling pattern. Based on the derived condition, the design method of the desired sampled-data controllers is proposed to make the CDNs exponentially synchronized and obtain a lowerbound estimation of the largest sampling interval. Simulation examples demonstrate that the presented method can significantly reduce the conservatism of the existing results, and lead to wider applications. Index Terms— Complex dynamical networks (CDNs), exponential synchronization, sampled-data control, time-varying coupling delay.
I. I NTRODUCTION
S
AMPLED-DATA systems have been a hot research topic and investigated extensively in the last decades [1]–[4]. In such systems, a digital computer is adopted to sample and quantize a continuous-time measurement signal to produce a discrete-time signal, and then produce a discrete-time control input signal, which is further converted back into a continuous-time control input signal using a zero-order hold (ZOH) [5]. Until now, three main approaches have been used to the analysis and synthesis of sampled-data systems. The first one is based on discrete-time models [5]. The second approach is based on the representation of the sampled-data system in the form of impulsive model [6], and the third one is the input delay approach [7], where the system is modeled as a continuous-time system with a time-varying sawtooth delay in the control input induced by sample-and-hold. The input
Manuscript received July 31, 2012; revised November 22, 2012 and January 28, 2013; accepted March 10, 2013. Date of publication April 8, 2013; date of current version June 28, 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: nashwzhg@ gmail.com; [email protected]; [email protected]). P. Shi is with the College of Engineering and Science, Victoria University, Melbourne, VIC 8001, Australia, and also with the School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, SA 5005, Australia (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.2013.2253122
delay approach is popular and has been widely used to sampled-data systems and networked control systems [8]–[11]. In [12], the time-dependent Lyapunov functional method has been introduced to improve the input delay approach. The most significant advantage of the time-dependent Lyapunov functional method is that the sawtooth evolution of the time-varying delay induced by sample-and-hold is used. Thus, in recent years the time-dependent Lyapunov functional method has been applied to all sorts of sampled-data systems and networked control systems, and some useful results have been obtained, see for example, [13]–[20] and the references therein. On the other hand, much attention has been drawn to the study of complex dynamical networks (CDNs) over the last decade, because CDNs are successfully applicable to describe a variety of real world systems including Internet networks, biological networks, epidemic spreading networks, collaborative networks, and social networks [21], [22]. Particularly, the synchronization of CDNs has been one of the focal points in many research and application fields. Accordingly, a great number of important and interesting research results have been published on this topic [21]–[32]. To mention a few representative works, the global exponential synchronization of delayed CDNs with nonidentical nodes and stochastic perturbations has been studied in [23], where a less conservative synchronization criterion has been obtained. In [24], the networked synchronization control problem for the CDNs with time-varying delay has been considered, and a delay-dependent stochastic synchronization criterion has been proposed by the usage of the Kronecker product and the stochastic Lyapunov stability theory. In the framework of the input delay approach, the sampled-data synchronization control problem has been investigated for CDNs with time-varying coupling delay in [33] and [34], where the desired sampled-data feedback controllers have been designed in terms of the solution to certain linear matrix inequalities (LMIs). However, it should be pointed out that in [33] and [34], the sawtooth structure of the time-varying delay induced by sample-and-hold and all available information about the actual sampling pattern are neglected, because the induced delay is simply treated as a bounded fast varying delay (it is time-varying delay without any constraint on the delay derivative). It is clear that this treatment inevitably leads to the conservatism of the obtained results [12]. Thus, it is necessary to further investigate the problem of sampled-data exponential synchronization of CDNs with time-varying coupling delay
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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 8, AUGUST 2013
and uncertain sampling to get some less conservative results, which is the motivation for this paper, as it is theoretical and practical significance. In this paper, the problem of sampled-data exponential synchronization of CDNs with time-varying coupling delay and variable sampling is investigated. First, based on the timedependent Lyapunov functional approach [12] and convex combination technique, a novel stability criterion is derived for the error systems, which fully utilizes the available information about the actual sampling pattern. Then, the problem of sampled-data exponential synchronization is solved, and the explicit expression of the desired sampled-data controllers is also given. Simulation results are provided to demonstrate the effectiveness and less conservativeness of the developed approaches. 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 semi-definite). I and 0 are the identity matrix and a zero matrix, respectively. The superscript “T” is the transpose, and diag{· · · } is a block-diagonal matrix. || · || is the Euclidean norm of a vector and its induced norm of a matrix. λmax (Q) (λmin (Q)) is the maximum (minimum) of the eigenvalue of a real symmetric matrix Q. For an arbitrary matrix B and two A B symmetric matrices A and C, is a symmetric matrix, ∗ C where “∗” is the term that 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 CDN consisting of N identical coupled nodes with each node being an n-dimensional dynamical system: x˙i (t) = f (x i (t)) + c
N
G i j Ax j (t − τ (t)) + u i (t)
(1)
j =1
where i = 1, 2, . . . , N, x i (t) and u i (t) are, respectively, the state variable and the control input of the node i , c is a constant denoting the coupling strength, A = (ai j )n×n ∈ Rn×n is a constant inner-coupling matrix between two connected nodes, and G = (G i j ) N×N is an outer-coupling configuration matrix representing the topological structure of the network, where G i j is defined as follows: if there exists a connection from node j to node i ( j = i ), then G i j > 0; otherwise, G i j = 0, and the diagonal elements of matrix G are defined by G ii = −
N
Gi j ,
i = 1, 2, . . . , N.
(2)
j =1, j =i
τ˙ (t) ν
[ f (x) − f (y) − U (x − y)]T × [ f (x) − f (y) − V (x − y)] 0
(3)
where μ > 0 and ν are known constants. f : Rn → Rn is a continuous vector-valued function and satisfies the following
∀x, y ∈ Rn
(4)
where U and V are constant matrices of appropriate dimensions. It is noted that the nonlinear description in (4) is quite general and covers the usual Lipschitz condition as a special case. Let ri (t) = x i (t) − s(t) be the error vectors, where s(t) ∈ Rn is the state trajectory of the unforced isolate node s˙ (t) = f (s(t)). Then, the error dynamics of CND (1) can be obtained as follows: r˙i (t) = g(ri (t)) + c
N
G i j Ar j (t − τ (t)) + u i (t)
(5)
j =1
where i = 1, 2, . . . , N, and g(ri (t)) = f (x i (t)) − f (s(t)). It is assumed that the state variables of error system (5) are measurable at time instants 0 = t0 < t1 < · · · < tk < · · · , and only ri (tk ) are available for interval tk t < tk+1 . Then, for error dynamic (5), we are interested in designing a set of sampled-data state feedback controllers in the form of u i (t) = K i ri (tk ), tk t < tk+1 ,
i = 1, 2, . . . , N
(6)
where K i is the state feedback controller gain matrix to be determined. It is assumed here that the sampling of the measurement is synchronized with the holding of the control signal generated by using a ZOH function, and the intervals between any two sampling instants satisfy tk+1 − tk = h k p
∀k 0
(7)
where p > 0 is the largest sampling interval, i.e., the sampling interval is bounded. Note that it does not require the sampling to be periodic, and the designed controllers in the form of (6) should be effective for any sampling interval not larger than p. By substituting (6) into (5), we obtain r˙i (t) = g(ri (t)) + c
N
G i j Ar j (t − τ (t) ) + K i ri (tk )
j =1
tk t < tk+1 (8) where i = 1, 2, . . . , N. It is clear that (8) can be rewritten as r(t) ˙ = g(r ¯ (t)) + c(G ⊗ A)r (t − τ (t)) + K r (tk ) tk t < tk+1
(9)
where K = diag{K 1 , K 2 , . . . , K N }, and ⎤ ⎡ ⎤ ⎡ g(r1 (t)) r1 (t) ⎢ g(r2 (t)) ⎥ ⎢ r2 (t) ⎥ ⎥ ⎢ ⎥ ⎢ ¯ (t)) = ⎢ r (t) = ⎢ . ⎥, g(r ⎥. .. ⎣ ⎦ ⎣ .. ⎦ . r N (t)
The scalar τ (t) denotes the time-varying delay satisfying 0 τ (t) μ,
sector-bounded condition [35]:
g(r N (t))
The following lemma and definition will be used to derive the main results in this paper. M1 S Lemma 1 [36]: For any matrix 0, scalars ∗ M2 τ > 0, τ (t) satisfying 0 τ (t) τ , and vector function
WU et al.: SAMPLED-DATA EXPONENTIAL SYNCHRONIZATION OF CDNs
x(t ˙ + ·) : [ −τ, 0 ] → Rn such that the concerned integrations are well defined, then t x(α) ˙ T M1 x(α) ˙ dα
−τ t −τ (t )
t −τ (t )
x(α) ˙ T M2 x(α) ˙ dα (t)T Ω (t) (10)
−τ t −τ
where
T (t) = x(t)T x(t − τ (t))T x(t − τ )T ⎡ ⎤ −M1 M1 − S S Ω = ⎣ ∗ −M1 − M2 + S + S T −S + M2 ⎦. ∗ ∗ −M2 Remark 1: Note that Lemma 1 is a special case of [36, Th. 1], which is presented in a form more convenient for the present application. Definition 1: The CDN (1) is said to be exponentially synchronized if the error dynamic (9) is exponentially stable, i.e., there exist two constants α > 0 and β > 0 such that ||r (t)|| βe−αt
sup {||r (θ )||, ||˙r(θ )||}
−μθ 0
(11)
where α and β are the decay rate and decay coefficient, respectively. We are now in a position to formulate the sampled-data exponential synchronization problem to be addressed in this paper as follows. Design sampled-data controllers in the form of (6) such that the error system (9) is exponentially stable, that is, CDN (1) is exponentially synchronized.
III. M AIN R ESULTS In this section, the exponential stability of error system (9) is first investigated based on the time-dependent Lyapunov functional approach, and sufficient condition is derived to guarantee the system stability and synthesize the sampled-data controllers in the form of (6). For brevity, we denote (I N ⊗ U )T (I N ⊗ V ) (I N ⊗ V )T (I N ⊗ U ) + U¯ = 2 2 T + (I ⊗ V )T ⊗ U ) (I N N V¯ = − 2 μ, if ν < 1 ρ = 0, if ν 1. Theorem 1: Given a scalar α > 0, if there exist matrices U1 U2 > 0, P > 0, Q 1 > 0, Q 2 > 0, Z 1 > 0, Z 2 > 0, ∗ U3
X, X 1 , H = H1 H2 H3 , S, F1 , F2 , and a scalar ε > 0 such
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that ⎤ X + XT −2αp +e U1 Π1 U2 ⎥ ⎢P+ p 2 ⎥ ⎢ ⎣ ∗ Π2 −U2 ⎦ > 0 (12) ∗ ∗ e2αp U3 ⎡ ⎤ Ξ11 Ξ12 Ξ13 Ξ14 S Ξ16 ⎢ ∗ Ξ22 Ξ23 Ξ24 0 F2 ⎥ ⎢ ⎥ ⎢ ∗ ∗ Ξ33 0 0 0 ⎥ ⎢ ⎥ 0.
(21)
Due to the fact that P > 0 and (21), we can obtain that there exists a sufficiently small scalar δ > 0 such that P > δ I and Γˆ + p(H + U) > δ I . Thus V (t) δe2αt ||r (t)||2 .
(22)
Therefore, V (t) defined in (17) is a valid Lyapunov functional for (9). Taking the time derivative of V (t) along the trajectory of (9) for t ∈ [tk , tk+1 ) yields (23) V˙1 (t) = 2e2αt r (t)T P r˙ (t) + 2αe2αt r (t)T Pr (t) 2α(t −τ (t )) T ˙ r (t − τ (t)) Q 1 r (t − τ (t)) V2 (t) = −(1 − τ˙ (t))e +e2αt r (t)T Q 1 r (t) −e2αt (1 − ν)e−2αρ r (t − τ (t))T Q 1r (t − τ (t)) +e2αt r (t)T Q 1 r (t)
(24) 2αt T 2αt −2αμ T ˙ V3 (t) = e r (t) Q 2r (t) − e e r (t − μ) Q 2r (t − μ) (25) t e2αs r˙ (s)T Z 1r˙ (s) ds V˙4 (t) = μ2 e2αt r˙ (t)T Z 1r˙ (t) − μ t −μ
e2αt μ2r˙ (t)T Z 1r˙ (t) t 2αt e−2αμr˙ (s)T Z 1r˙ (s) ds −e μ t −μ
V˙5 (t) = μτ (t)e2αt r˙ (t)T Z 2r˙ (t) t −μ e2αs r˙ (s)T Z 2r˙ (s) ds t −τ (t )
(26)
WU et al.: SAMPLED-DATA EXPONENTIAL SYNCHRONIZATION OF CDNs
t
where
e2αs r˙ (s)T Z 2r˙ (s) ds
+μτ˙ (t) t −τ (t ) 2 2αt T
μ e
r˙ (t) Z 2r˙ (t) t 2αt −μ(1 − ν)e e−2αρ r˙ (s)T Z 2r(s) ˙ ds
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(27)
T χ(t) = r (t)T r (t − τ (t))T r (t − μ)T ⎡ ⎤ −φ φ−S S Θ = ⎣ ∗ −φ − e−2αμ Z 1 + S + S T −S + e−2αμ Z 1 ⎦ ∗ ∗ −e−2αμ Z 1 φ = e−2αμ Z 1 + (1 − ν)e−2αρ Z 2 .
t −τ (t )
V˙6 (t) = −
t e
2αs
tk
r˙ (s) r (tk )
T U1 U2 r˙ (s) ds ∗ U3 r (tk )
On the other hand, based on Schur complement, for any appropriately dimensioned matrix H , the following inequality holds: T 2αp −1 HT H e U1 H 0. (31) ∗ e−2αp U1
T r˙ (t) U1 U2 r˙ (t) +(tk+1 − t)e ∗ U3 r (tk ) r (tk ) t T r˙ (s) U1 U2 r˙ (s) ds −e2αt e−2αp r (tk ) ∗ U3 r (tk ) 2αt
tk
Thus, we can easily find T T 2αp −1 t ω(t) ω(t) H e U1 H HT ds 0 r˙ (s) ∗ e−2αp U1 r˙ (s)
T r˙ (t) U1 U2 r˙ (t) +(tk+1 − t)e ∗ U3 r (tk ) r (tk ) T r˙ (t) U1 U2 r˙ (t) (tk+1 − t)e2αt ∗ U3 r (tk ) r (tk ) t −e2αt r˙ (s)T e−2αp U1r˙ (s) ds 2αt
tk
T where ω(t) = r (t)T r˙ (t)T r (tk )T . From (32), we can immediately get that t −
tk 2αt −2αp
X + XT r˙ (t) 2 +2(tk+1 − t)e2αt r (tk )T (−X T + X 1T )˙r (t).
(t − tk )e2αp ω(t)T H TU1−1 H ω(t) +2ω(t)T H T (r (t) − r (tk )).
(28)
[g(ri (t)) − Uri (t)]T [g(ri (t)) − V ri (t)] 0 which is equivalent to T ri (t) ri (t) Uˆ Vˆ 0 g(ri (t)) g(ri (t)) ∗ I Uˆ =
(29)
e−2αμr˙ (s)T Z 1r˙ (s) ds
t −τ (t )
r˙ (s)T e−2αμ Z 1r˙ (s) ds
r˙ (s)T φr˙ (s) ds − μ
χ(t) Θχ(t)
(35)
V TU 2 ,
T T Vˆ = − U +V . 2
(36)
Furthermore, based on descriptor systems method [38], we can get from (9) that for any appropriately dimensioned matrices F1 and F2 , the following equation holds: ¯ (t)) 0 = 2e2αt r (t)T F1 + r˙ (t)T F2 [−˙r(t) + g(r (37)
Then, adding the right-hand side of (37) to V˙ (t), we have from (23) to (30), (33), and (36) that for t ∈ [tk , tk+1 )
e−2αρ r˙ (s)T Z 2r˙ (s) ds
t −τ (t )
t −τ (t ) T
+
+ c(G ⊗ A)r (t − τ (t)) + K r (tk )].
t = −μ
UTV 2
It can be found from (35) that T r (t) r (t) U¯ V¯ ψ(t) = 0. g(r ¯ (t)) g(r ¯ (t)) ∗ I
t −μ
−μ(1 − ν)
(34)
where
It is noted that if (16) is satisfied, then by Lemma 1, we have
t
(33)
On the other hand, we have from (4) that
+2(tk+1 − t)e2αt r (t)T
t
r˙ (s)T e−2αp U1r˙ (s) ds
tk
−2e e (r (t) − r (tk ))T U2r (tk ) −e2αt e−2αp (t − tk )r (tk )T U3r (tk ) T r (t) r (t) V˙7 (t) = −e2αt H r (tk ) r (tk ) T r (t) r (t) +2α(tk+1 − t)e2αt H r (tk ) r (tk ) T r˙ (t) 2αt r (t) H +2(tk+1 − t)e 0 r (tk ) T r (t) r (t) = −e2αt H r (tk ) r (tk ) T r (t) r (t) +2α(tk+1 − t)e2αt H r (tk ) r (tk )
−μ
(32)
t −μ
(30)
V˙ (t)
7
V˙i (t) − εe2αt ψ(t)
i=1
e
2αt
ξ(t)
T
tk+1 − t t − tk Y1 + Y2 ξ(t) (38) hk hk
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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 8, AUGUST 2013
where
Based on (22), (43), and (44), we have
T ξ(t) = ω(t)T r (t − τ (t))T r (t − μ)T g(r ¯ (t))T
δe
2αt
2
||r (t)|| (a1 + a2 )
sup {||r (θ )||, ||˙r(θ )||}
−μθ 0
and Y1 and Y2 are given as follows. On the other hand, it can be shown from (13) and (15) that Y1 =
hk p − hk Ξ3 + Ξ1 < 0. p p
(39)
Furthermore, based on Schur complement, we obtain from (14) that Y2 | h k = p < 0 (40) thus Y2 =
hk p − hk Y2 | h k = p + Ξ1 < 0. p p
(41)
It is clear from (38), (39), and (41) that V˙ (t) 0,
t ∈ [tk , tk+1 ).
(42)
Thus V (t) V (tk ) V (tk−1 ) · · · V (0).
(43)
On the other hand, it is noted that V6 (0) = 0 and V7 (0) = 0, and thus V (0) =
5
Vi (0)
i=1
λmax (P) r (0)2 + μλmax (Q 1 ) +μλmax (Q 2 )
−μθ 0
+μ λmax (Z 1 ) +μ3 λmax (Z 2 )
2
sup {||˙r (θ )|| }
−μθ 0
sup {||˙r(θ )||2 }
−μθ 0
sup {||r (θ )||2 } + a2
−μθ 0
(a1 + a2 )
sup {||˙r(θ )||2 }
−μθ 0
2
sup {||r (θ )||, ||˙r(θ )||}
−μθ 0
where a1 = λmax (P) + μλmax (Q 1 ) + μλmax (Q 2 ) a2 = μ3 λmax (Z 1 ) + μ3 λmax (Z 2 ).
(45) which implies ||r (t)||
a1 + a2 −αt sup {||r (θ )||, ||˙r(θ )||}. e δ −μθ 0
(46)
According to Definition 1, we can get from (46) that (9) is exponentially stable with decay rate α. This completes the proof. Remark 2: It is noted that based on the time-dependent Lyapunov functional method [12], two (tk , tk+1 )-dependent terms V6 (t) and V7 (t) are introduced in the Lyapunov functional (17), which make good use of the available information about the actual sampling pattern, and thus can efficiently reduce the conservatism of the proposed results. However, the two (tk , tk+1 )-dependent terms are overlooked in [33] and [34]. Thus, Theorem 1 proposed here is more effective and practical than the ones of [33] and [34]. Remark 3: It can be found that when U2 = 0 and U3 = σ I > 0 (σ → 0), V6 (t) in the Lyapunov functional (17) reduces to t V¯6 (t) = (tk+1 − t) e2αs r˙ (s)T U1r˙ (s) ds. (47) tk
sup {||r (θ )||2}
3
a1
sup {||r (θ )||2 }
−μθ 0
2
(44)
It should be pointed out that V¯6 (t) was first proposed in [12], and has also been applied in [14]. On the other hand, it is noted that in the proof of Theorem 1, V1 (t), V6 (t), and V7 (t) are applied to ensure V (t) δe2αt ||r (t)||2 (δ > 0). While in [12] and [14], only V1 (t) and V7 (t) are applied, and V¯6 (t) is overlooked. Thus, the Lyapunov functional and method employed here have advantages over those of [12] and [14] in terms of conservatism reduction. Based on Theorem 1, we can obtain the following corollary. Corollary 1: If (12)–(16) are feasible for α = 0, then the error system (9) is exponentially stable with a small enough decay rate. Next, we will design the sampled-data controllers in the form of (6) to make CDN (1) exponentially synchronized.
⎡ ⎤ Ξ11 + Θ11 | p=h k Ξ12 + Θ12 | p=h k Ξ13 + Θ13 | p=h k Ξ14 Ξ15 Ξ16 ⎢ ∗ Ξ22 + h k U1 Ξ23 + Θ23 | p=h k Ξ24 0 Ξ26 ⎥ ⎢ ⎥ ⎢ ∗ ∗ Ξ33 + Θ33 | p=h k 0 0 0 ⎥ ⎢ ⎥ Y1 = ⎢ ∗ ∗ ∗ Ξ44 Ξ45 0 ⎥ ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ Ξ55 0 ⎦ ∗ ∗ ∗ ∗ ∗ −ε I ⎡ T ⎤T ⎡ ⎤ ⎡ T⎤ Ξ13 Ξ14 Ξ15 Ξ16 H1 Ξ11 Ξ12 H1 T T⎥ ⎢ ⎥ ⎢ ∗ Ξ22 ⎥ ⎢ Ξ Ξ 0 Ξ 23 24 26 ⎥ ⎢ H2 ⎥ ⎢ ⎢ H2 ⎥ T ⎢ ⎥ ⎢ ∗ ∗ Ξ33 − h k e−2αp U3 0 0 ⎥ ⎢ 0 ⎥ ⎢ H3 ⎥ H3T⎥ ⎥ Y2 = ⎢ + ⎢ ⎥ h k e2αp U1−1 ⎢ ⎢ ⎥ ⎢ ∗ ∗ ⎥ ∗ Ξ44 Ξ45 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ ⎣ ⎦ ⎣ ∗ ∗ ⎦ ⎣ ∗ ∗ Ξ55 0 0 0 ⎦ ∗ ∗ ∗ ∗ ∗ −ε I 0 0
WU et al.: SAMPLED-DATA EXPONENTIAL SYNCHRONIZATION OF CDNs
The following theorem presents a sufficient condition of the existence of the desired sampled-data controllers based on Theorem 1. Theorem 2: Given scalars α > 0 and κ, if there exist matriU1 U2 > 0, ces P > 0, Q 1 > 0, Q 2 > 0, Z 1 > 0, Z 2 > 0, ∗ U3
X, X 1 , H = H1 H2 H3 , S, F = diag{F1 , F2 . . . , F N }, L = diag{L 1 , L 2 . . . , L N }, and a scalar ε > 0 such that (12) and (16) and the following LMIs hold: ⎤ ⎡ Ξ11 Ξ˜ 12 Ξ˜ 13 Ξ˜ 14 S Ξ˜ 16 ⎢ ∗ Ξ˜ 22 Ξ˜ 23 Ξ˜ 24 0 κF ⎥ ⎥ ⎢ ⎢ ∗ ∗ Ξ33 0 0 0 ⎥ ⎥ ⎢ (48) ⎢ ∗ ∗ ∗ Ξ44 Ξ45 0 ⎥ < 0 ⎥ ⎢ ⎣ ∗ ∗ ∗ ∗ Ξ55 0 ⎦ ∗ ∗ ∗ ∗ ∗ −ε I ⎤ ⎡ Ξ11 Ξ˜ 12 Ξ˜ 13 Ξ˜ 14 S Ξ˜ 16 peαp H1T ⎢ ∗ Ξ˜ 22 Ξ˜ 23 Ξ˜ 24 0 κF peαp H T ⎥ ⎢ 2⎥ ⎢ ∗ ∗ Ξ¯ 33 0 0 0 peαp H3T ⎥ ⎥ ⎢ ⎢ ∗ ∗ ∗ Ξ44 Ξ (49) 0 ⎥ 45 0 ⎥
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Sampled-Data Exponential Synchronization of Complex Dynamical Networks With Time-Varying Coupling Delay Zheng-Guang Wu, Peng Shi, Hongye Su, and Jian Chu
Abstract— This paper studies the problem of sampled-data exponential synchronization of complex dynamical networks (CDNs) with time-varying coupling delay and uncertain sampling. By combining the time-dependent Lyapunov functional approach and convex combination technique, a criterion is derived to ensure the exponential stability of the error dynamics, which fully utilizes the available information about the actual sampling pattern. Based on the derived condition, the design method of the desired sampled-data controllers is proposed to make the CDNs exponentially synchronized and obtain a lowerbound estimation of the largest sampling interval. Simulation examples demonstrate that the presented method can significantly reduce the conservatism of the existing results, and lead to wider applications. Index Terms— Complex dynamical networks (CDNs), exponential synchronization, sampled-data control, time-varying coupling delay.
I. I NTRODUCTION
S
AMPLED-DATA systems have been a hot research topic and investigated extensively in the last decades [1]–[4]. In such systems, a digital computer is adopted to sample and quantize a continuous-time measurement signal to produce a discrete-time signal, and then produce a discrete-time control input signal, which is further converted back into a continuous-time control input signal using a zero-order hold (ZOH) [5]. Until now, three main approaches have been used to the analysis and synthesis of sampled-data systems. The first one is based on discrete-time models [5]. The second approach is based on the representation of the sampled-data system in the form of impulsive model [6], and the third one is the input delay approach [7], where the system is modeled as a continuous-time system with a time-varying sawtooth delay in the control input induced by sample-and-hold. The input
Manuscript received July 31, 2012; revised November 22, 2012 and January 28, 2013; accepted March 10, 2013. Date of publication April 8, 2013; date of current version June 28, 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: nashwzhg@ gmail.com; [email protected]; [email protected]). P. Shi is with the College of Engineering and Science, Victoria University, Melbourne, VIC 8001, Australia, and also with the School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, SA 5005, Australia (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.2013.2253122
delay approach is popular and has been widely used to sampled-data systems and networked control systems [8]–[11]. In [12], the time-dependent Lyapunov functional method has been introduced to improve the input delay approach. The most significant advantage of the time-dependent Lyapunov functional method is that the sawtooth evolution of the time-varying delay induced by sample-and-hold is used. Thus, in recent years the time-dependent Lyapunov functional method has been applied to all sorts of sampled-data systems and networked control systems, and some useful results have been obtained, see for example, [13]–[20] and the references therein. On the other hand, much attention has been drawn to the study of complex dynamical networks (CDNs) over the last decade, because CDNs are successfully applicable to describe a variety of real world systems including Internet networks, biological networks, epidemic spreading networks, collaborative networks, and social networks [21], [22]. Particularly, the synchronization of CDNs has been one of the focal points in many research and application fields. Accordingly, a great number of important and interesting research results have been published on this topic [21]–[32]. To mention a few representative works, the global exponential synchronization of delayed CDNs with nonidentical nodes and stochastic perturbations has been studied in [23], where a less conservative synchronization criterion has been obtained. In [24], the networked synchronization control problem for the CDNs with time-varying delay has been considered, and a delay-dependent stochastic synchronization criterion has been proposed by the usage of the Kronecker product and the stochastic Lyapunov stability theory. In the framework of the input delay approach, the sampled-data synchronization control problem has been investigated for CDNs with time-varying coupling delay in [33] and [34], where the desired sampled-data feedback controllers have been designed in terms of the solution to certain linear matrix inequalities (LMIs). However, it should be pointed out that in [33] and [34], the sawtooth structure of the time-varying delay induced by sample-and-hold and all available information about the actual sampling pattern are neglected, because the induced delay is simply treated as a bounded fast varying delay (it is time-varying delay without any constraint on the delay derivative). It is clear that this treatment inevitably leads to the conservatism of the obtained results [12]. Thus, it is necessary to further investigate the problem of sampled-data exponential synchronization of CDNs with time-varying coupling delay
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and uncertain sampling to get some less conservative results, which is the motivation for this paper, as it is theoretical and practical significance. In this paper, the problem of sampled-data exponential synchronization of CDNs with time-varying coupling delay and variable sampling is investigated. First, based on the timedependent Lyapunov functional approach [12] and convex combination technique, a novel stability criterion is derived for the error systems, which fully utilizes the available information about the actual sampling pattern. Then, the problem of sampled-data exponential synchronization is solved, and the explicit expression of the desired sampled-data controllers is also given. Simulation results are provided to demonstrate the effectiveness and less conservativeness of the developed approaches. 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 semi-definite). I and 0 are the identity matrix and a zero matrix, respectively. The superscript “T” is the transpose, and diag{· · · } is a block-diagonal matrix. || · || is the Euclidean norm of a vector and its induced norm of a matrix. λmax (Q) (λmin (Q)) is the maximum (minimum) of the eigenvalue of a real symmetric matrix Q. For an arbitrary matrix B and two A B symmetric matrices A and C, is a symmetric matrix, ∗ C where “∗” is the term that 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 CDN consisting of N identical coupled nodes with each node being an n-dimensional dynamical system: x˙i (t) = f (x i (t)) + c
N
G i j Ax j (t − τ (t)) + u i (t)
(1)
j =1
where i = 1, 2, . . . , N, x i (t) and u i (t) are, respectively, the state variable and the control input of the node i , c is a constant denoting the coupling strength, A = (ai j )n×n ∈ Rn×n is a constant inner-coupling matrix between two connected nodes, and G = (G i j ) N×N is an outer-coupling configuration matrix representing the topological structure of the network, where G i j is defined as follows: if there exists a connection from node j to node i ( j = i ), then G i j > 0; otherwise, G i j = 0, and the diagonal elements of matrix G are defined by G ii = −
N
Gi j ,
i = 1, 2, . . . , N.
(2)
j =1, j =i
τ˙ (t) ν
[ f (x) − f (y) − U (x − y)]T × [ f (x) − f (y) − V (x − y)] 0
(3)
where μ > 0 and ν are known constants. f : Rn → Rn is a continuous vector-valued function and satisfies the following
∀x, y ∈ Rn
(4)
where U and V are constant matrices of appropriate dimensions. It is noted that the nonlinear description in (4) is quite general and covers the usual Lipschitz condition as a special case. Let ri (t) = x i (t) − s(t) be the error vectors, where s(t) ∈ Rn is the state trajectory of the unforced isolate node s˙ (t) = f (s(t)). Then, the error dynamics of CND (1) can be obtained as follows: r˙i (t) = g(ri (t)) + c
N
G i j Ar j (t − τ (t)) + u i (t)
(5)
j =1
where i = 1, 2, . . . , N, and g(ri (t)) = f (x i (t)) − f (s(t)). It is assumed that the state variables of error system (5) are measurable at time instants 0 = t0 < t1 < · · · < tk < · · · , and only ri (tk ) are available for interval tk t < tk+1 . Then, for error dynamic (5), we are interested in designing a set of sampled-data state feedback controllers in the form of u i (t) = K i ri (tk ), tk t < tk+1 ,
i = 1, 2, . . . , N
(6)
where K i is the state feedback controller gain matrix to be determined. It is assumed here that the sampling of the measurement is synchronized with the holding of the control signal generated by using a ZOH function, and the intervals between any two sampling instants satisfy tk+1 − tk = h k p
∀k 0
(7)
where p > 0 is the largest sampling interval, i.e., the sampling interval is bounded. Note that it does not require the sampling to be periodic, and the designed controllers in the form of (6) should be effective for any sampling interval not larger than p. By substituting (6) into (5), we obtain r˙i (t) = g(ri (t)) + c
N
G i j Ar j (t − τ (t) ) + K i ri (tk )
j =1
tk t < tk+1 (8) where i = 1, 2, . . . , N. It is clear that (8) can be rewritten as r(t) ˙ = g(r ¯ (t)) + c(G ⊗ A)r (t − τ (t)) + K r (tk ) tk t < tk+1
(9)
where K = diag{K 1 , K 2 , . . . , K N }, and ⎤ ⎡ ⎤ ⎡ g(r1 (t)) r1 (t) ⎢ g(r2 (t)) ⎥ ⎢ r2 (t) ⎥ ⎥ ⎢ ⎥ ⎢ ¯ (t)) = ⎢ r (t) = ⎢ . ⎥, g(r ⎥. .. ⎣ ⎦ ⎣ .. ⎦ . r N (t)
The scalar τ (t) denotes the time-varying delay satisfying 0 τ (t) μ,
sector-bounded condition [35]:
g(r N (t))
The following lemma and definition will be used to derive the main results in this paper. M1 S Lemma 1 [36]: For any matrix 0, scalars ∗ M2 τ > 0, τ (t) satisfying 0 τ (t) τ , and vector function
WU et al.: SAMPLED-DATA EXPONENTIAL SYNCHRONIZATION OF CDNs
x(t ˙ + ·) : [ −τ, 0 ] → Rn such that the concerned integrations are well defined, then t x(α) ˙ T M1 x(α) ˙ dα
−τ t −τ (t )
t −τ (t )
x(α) ˙ T M2 x(α) ˙ dα (t)T Ω (t) (10)
−τ t −τ
where
T (t) = x(t)T x(t − τ (t))T x(t − τ )T ⎡ ⎤ −M1 M1 − S S Ω = ⎣ ∗ −M1 − M2 + S + S T −S + M2 ⎦. ∗ ∗ −M2 Remark 1: Note that Lemma 1 is a special case of [36, Th. 1], which is presented in a form more convenient for the present application. Definition 1: The CDN (1) is said to be exponentially synchronized if the error dynamic (9) is exponentially stable, i.e., there exist two constants α > 0 and β > 0 such that ||r (t)|| βe−αt
sup {||r (θ )||, ||˙r(θ )||}
−μθ 0
(11)
where α and β are the decay rate and decay coefficient, respectively. We are now in a position to formulate the sampled-data exponential synchronization problem to be addressed in this paper as follows. Design sampled-data controllers in the form of (6) such that the error system (9) is exponentially stable, that is, CDN (1) is exponentially synchronized.
III. M AIN R ESULTS In this section, the exponential stability of error system (9) is first investigated based on the time-dependent Lyapunov functional approach, and sufficient condition is derived to guarantee the system stability and synthesize the sampled-data controllers in the form of (6). For brevity, we denote (I N ⊗ U )T (I N ⊗ V ) (I N ⊗ V )T (I N ⊗ U ) + U¯ = 2 2 T + (I ⊗ V )T ⊗ U ) (I N N V¯ = − 2 μ, if ν < 1 ρ = 0, if ν 1. Theorem 1: Given a scalar α > 0, if there exist matrices U1 U2 > 0, P > 0, Q 1 > 0, Q 2 > 0, Z 1 > 0, Z 2 > 0, ∗ U3
X, X 1 , H = H1 H2 H3 , S, F1 , F2 , and a scalar ε > 0 such
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that ⎤ X + XT −2αp +e U1 Π1 U2 ⎥ ⎢P+ p 2 ⎥ ⎢ ⎣ ∗ Π2 −U2 ⎦ > 0 (12) ∗ ∗ e2αp U3 ⎡ ⎤ Ξ11 Ξ12 Ξ13 Ξ14 S Ξ16 ⎢ ∗ Ξ22 Ξ23 Ξ24 0 F2 ⎥ ⎢ ⎥ ⎢ ∗ ∗ Ξ33 0 0 0 ⎥ ⎢ ⎥ 0.
(21)
Due to the fact that P > 0 and (21), we can obtain that there exists a sufficiently small scalar δ > 0 such that P > δ I and Γˆ + p(H + U) > δ I . Thus V (t) δe2αt ||r (t)||2 .
(22)
Therefore, V (t) defined in (17) is a valid Lyapunov functional for (9). Taking the time derivative of V (t) along the trajectory of (9) for t ∈ [tk , tk+1 ) yields (23) V˙1 (t) = 2e2αt r (t)T P r˙ (t) + 2αe2αt r (t)T Pr (t) 2α(t −τ (t )) T ˙ r (t − τ (t)) Q 1 r (t − τ (t)) V2 (t) = −(1 − τ˙ (t))e +e2αt r (t)T Q 1 r (t) −e2αt (1 − ν)e−2αρ r (t − τ (t))T Q 1r (t − τ (t)) +e2αt r (t)T Q 1 r (t)
(24) 2αt T 2αt −2αμ T ˙ V3 (t) = e r (t) Q 2r (t) − e e r (t − μ) Q 2r (t − μ) (25) t e2αs r˙ (s)T Z 1r˙ (s) ds V˙4 (t) = μ2 e2αt r˙ (t)T Z 1r˙ (t) − μ t −μ
e2αt μ2r˙ (t)T Z 1r˙ (t) t 2αt e−2αμr˙ (s)T Z 1r˙ (s) ds −e μ t −μ
V˙5 (t) = μτ (t)e2αt r˙ (t)T Z 2r˙ (t) t −μ e2αs r˙ (s)T Z 2r˙ (s) ds t −τ (t )
(26)
WU et al.: SAMPLED-DATA EXPONENTIAL SYNCHRONIZATION OF CDNs
t
where
e2αs r˙ (s)T Z 2r˙ (s) ds
+μτ˙ (t) t −τ (t ) 2 2αt T
μ e
r˙ (t) Z 2r˙ (t) t 2αt −μ(1 − ν)e e−2αρ r˙ (s)T Z 2r(s) ˙ ds
1181
(27)
T χ(t) = r (t)T r (t − τ (t))T r (t − μ)T ⎡ ⎤ −φ φ−S S Θ = ⎣ ∗ −φ − e−2αμ Z 1 + S + S T −S + e−2αμ Z 1 ⎦ ∗ ∗ −e−2αμ Z 1 φ = e−2αμ Z 1 + (1 − ν)e−2αρ Z 2 .
t −τ (t )
V˙6 (t) = −
t e
2αs
tk
r˙ (s) r (tk )
T U1 U2 r˙ (s) ds ∗ U3 r (tk )
On the other hand, based on Schur complement, for any appropriately dimensioned matrix H , the following inequality holds: T 2αp −1 HT H e U1 H 0. (31) ∗ e−2αp U1
T r˙ (t) U1 U2 r˙ (t) +(tk+1 − t)e ∗ U3 r (tk ) r (tk ) t T r˙ (s) U1 U2 r˙ (s) ds −e2αt e−2αp r (tk ) ∗ U3 r (tk ) 2αt
tk
Thus, we can easily find T T 2αp −1 t ω(t) ω(t) H e U1 H HT ds 0 r˙ (s) ∗ e−2αp U1 r˙ (s)
T r˙ (t) U1 U2 r˙ (t) +(tk+1 − t)e ∗ U3 r (tk ) r (tk ) T r˙ (t) U1 U2 r˙ (t) (tk+1 − t)e2αt ∗ U3 r (tk ) r (tk ) t −e2αt r˙ (s)T e−2αp U1r˙ (s) ds 2αt
tk
T where ω(t) = r (t)T r˙ (t)T r (tk )T . From (32), we can immediately get that t −
tk 2αt −2αp
X + XT r˙ (t) 2 +2(tk+1 − t)e2αt r (tk )T (−X T + X 1T )˙r (t).
(t − tk )e2αp ω(t)T H TU1−1 H ω(t) +2ω(t)T H T (r (t) − r (tk )).
(28)
[g(ri (t)) − Uri (t)]T [g(ri (t)) − V ri (t)] 0 which is equivalent to T ri (t) ri (t) Uˆ Vˆ 0 g(ri (t)) g(ri (t)) ∗ I Uˆ =
(29)
e−2αμr˙ (s)T Z 1r˙ (s) ds
t −τ (t )
r˙ (s)T e−2αμ Z 1r˙ (s) ds
r˙ (s)T φr˙ (s) ds − μ
χ(t) Θχ(t)
(35)
V TU 2 ,
T T Vˆ = − U +V . 2
(36)
Furthermore, based on descriptor systems method [38], we can get from (9) that for any appropriately dimensioned matrices F1 and F2 , the following equation holds: ¯ (t)) 0 = 2e2αt r (t)T F1 + r˙ (t)T F2 [−˙r(t) + g(r (37)
Then, adding the right-hand side of (37) to V˙ (t), we have from (23) to (30), (33), and (36) that for t ∈ [tk , tk+1 )
e−2αρ r˙ (s)T Z 2r˙ (s) ds
t −τ (t )
t −τ (t ) T
+
+ c(G ⊗ A)r (t − τ (t)) + K r (tk )].
t = −μ
UTV 2
It can be found from (35) that T r (t) r (t) U¯ V¯ ψ(t) = 0. g(r ¯ (t)) g(r ¯ (t)) ∗ I
t −μ
−μ(1 − ν)
(34)
where
It is noted that if (16) is satisfied, then by Lemma 1, we have
t
(33)
On the other hand, we have from (4) that
+2(tk+1 − t)e2αt r (t)T
t
r˙ (s)T e−2αp U1r˙ (s) ds
tk
−2e e (r (t) − r (tk ))T U2r (tk ) −e2αt e−2αp (t − tk )r (tk )T U3r (tk ) T r (t) r (t) V˙7 (t) = −e2αt H r (tk ) r (tk ) T r (t) r (t) +2α(tk+1 − t)e2αt H r (tk ) r (tk ) T r˙ (t) 2αt r (t) H +2(tk+1 − t)e 0 r (tk ) T r (t) r (t) = −e2αt H r (tk ) r (tk ) T r (t) r (t) +2α(tk+1 − t)e2αt H r (tk ) r (tk )
−μ
(32)
t −μ
(30)
V˙ (t)
7
V˙i (t) − εe2αt ψ(t)
i=1
e
2αt
ξ(t)
T
tk+1 − t t − tk Y1 + Y2 ξ(t) (38) hk hk
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where
Based on (22), (43), and (44), we have
T ξ(t) = ω(t)T r (t − τ (t))T r (t − μ)T g(r ¯ (t))T
δe
2αt
2
||r (t)|| (a1 + a2 )
sup {||r (θ )||, ||˙r(θ )||}
−μθ 0
and Y1 and Y2 are given as follows. On the other hand, it can be shown from (13) and (15) that Y1 =
hk p − hk Ξ3 + Ξ1 < 0. p p
(39)
Furthermore, based on Schur complement, we obtain from (14) that Y2 | h k = p < 0 (40) thus Y2 =
hk p − hk Y2 | h k = p + Ξ1 < 0. p p
(41)
It is clear from (38), (39), and (41) that V˙ (t) 0,
t ∈ [tk , tk+1 ).
(42)
Thus V (t) V (tk ) V (tk−1 ) · · · V (0).
(43)
On the other hand, it is noted that V6 (0) = 0 and V7 (0) = 0, and thus V (0) =
5
Vi (0)
i=1
λmax (P) r (0)2 + μλmax (Q 1 ) +μλmax (Q 2 )
−μθ 0
+μ λmax (Z 1 ) +μ3 λmax (Z 2 )
2
sup {||˙r (θ )|| }
−μθ 0
sup {||˙r(θ )||2 }
−μθ 0
sup {||r (θ )||2 } + a2
−μθ 0
(a1 + a2 )
sup {||˙r(θ )||2 }
−μθ 0
2
sup {||r (θ )||, ||˙r(θ )||}
−μθ 0
where a1 = λmax (P) + μλmax (Q 1 ) + μλmax (Q 2 ) a2 = μ3 λmax (Z 1 ) + μ3 λmax (Z 2 ).
(45) which implies ||r (t)||
a1 + a2 −αt sup {||r (θ )||, ||˙r(θ )||}. e δ −μθ 0
(46)
According to Definition 1, we can get from (46) that (9) is exponentially stable with decay rate α. This completes the proof. Remark 2: It is noted that based on the time-dependent Lyapunov functional method [12], two (tk , tk+1 )-dependent terms V6 (t) and V7 (t) are introduced in the Lyapunov functional (17), which make good use of the available information about the actual sampling pattern, and thus can efficiently reduce the conservatism of the proposed results. However, the two (tk , tk+1 )-dependent terms are overlooked in [33] and [34]. Thus, Theorem 1 proposed here is more effective and practical than the ones of [33] and [34]. Remark 3: It can be found that when U2 = 0 and U3 = σ I > 0 (σ → 0), V6 (t) in the Lyapunov functional (17) reduces to t V¯6 (t) = (tk+1 − t) e2αs r˙ (s)T U1r˙ (s) ds. (47) tk
sup {||r (θ )||2}
3
a1
sup {||r (θ )||2 }
−μθ 0
2
(44)
It should be pointed out that V¯6 (t) was first proposed in [12], and has also been applied in [14]. On the other hand, it is noted that in the proof of Theorem 1, V1 (t), V6 (t), and V7 (t) are applied to ensure V (t) δe2αt ||r (t)||2 (δ > 0). While in [12] and [14], only V1 (t) and V7 (t) are applied, and V¯6 (t) is overlooked. Thus, the Lyapunov functional and method employed here have advantages over those of [12] and [14] in terms of conservatism reduction. Based on Theorem 1, we can obtain the following corollary. Corollary 1: If (12)–(16) are feasible for α = 0, then the error system (9) is exponentially stable with a small enough decay rate. Next, we will design the sampled-data controllers in the form of (6) to make CDN (1) exponentially synchronized.
⎡ ⎤ Ξ11 + Θ11 | p=h k Ξ12 + Θ12 | p=h k Ξ13 + Θ13 | p=h k Ξ14 Ξ15 Ξ16 ⎢ ∗ Ξ22 + h k U1 Ξ23 + Θ23 | p=h k Ξ24 0 Ξ26 ⎥ ⎢ ⎥ ⎢ ∗ ∗ Ξ33 + Θ33 | p=h k 0 0 0 ⎥ ⎢ ⎥ Y1 = ⎢ ∗ ∗ ∗ Ξ44 Ξ45 0 ⎥ ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ Ξ55 0 ⎦ ∗ ∗ ∗ ∗ ∗ −ε I ⎡ T ⎤T ⎡ ⎤ ⎡ T⎤ Ξ13 Ξ14 Ξ15 Ξ16 H1 Ξ11 Ξ12 H1 T T⎥ ⎢ ⎥ ⎢ ∗ Ξ22 ⎥ ⎢ Ξ Ξ 0 Ξ 23 24 26 ⎥ ⎢ H2 ⎥ ⎢ ⎢ H2 ⎥ T ⎢ ⎥ ⎢ ∗ ∗ Ξ33 − h k e−2αp U3 0 0 ⎥ ⎢ 0 ⎥ ⎢ H3 ⎥ H3T⎥ ⎥ Y2 = ⎢ + ⎢ ⎥ h k e2αp U1−1 ⎢ ⎢ ⎥ ⎢ ∗ ∗ ⎥ ∗ Ξ44 Ξ45 0 ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ ⎣ ⎦ ⎣ ∗ ∗ ⎦ ⎣ ∗ ∗ Ξ55 0 0 0 ⎦ ∗ ∗ ∗ ∗ ∗ −ε I 0 0
WU et al.: SAMPLED-DATA EXPONENTIAL SYNCHRONIZATION OF CDNs
The following theorem presents a sufficient condition of the existence of the desired sampled-data controllers based on Theorem 1. Theorem 2: Given scalars α > 0 and κ, if there exist matriU1 U2 > 0, ces P > 0, Q 1 > 0, Q 2 > 0, Z 1 > 0, Z 2 > 0, ∗ U3
X, X 1 , H = H1 H2 H3 , S, F = diag{F1 , F2 . . . , F N }, L = diag{L 1 , L 2 . . . , L N }, and a scalar ε > 0 such that (12) and (16) and the following LMIs hold: ⎤ ⎡ Ξ11 Ξ˜ 12 Ξ˜ 13 Ξ˜ 14 S Ξ˜ 16 ⎢ ∗ Ξ˜ 22 Ξ˜ 23 Ξ˜ 24 0 κF ⎥ ⎥ ⎢ ⎢ ∗ ∗ Ξ33 0 0 0 ⎥ ⎥ ⎢ (48) ⎢ ∗ ∗ ∗ Ξ44 Ξ45 0 ⎥ < 0 ⎥ ⎢ ⎣ ∗ ∗ ∗ ∗ Ξ55 0 ⎦ ∗ ∗ ∗ ∗ ∗ −ε I ⎤ ⎡ Ξ11 Ξ˜ 12 Ξ˜ 13 Ξ˜ 14 S Ξ˜ 16 peαp H1T ⎢ ∗ Ξ˜ 22 Ξ˜ 23 Ξ˜ 24 0 κF peαp H T ⎥ ⎢ 2⎥ ⎢ ∗ ∗ Ξ¯ 33 0 0 0 peαp H3T ⎥ ⎥ ⎢ ⎢ ∗ ∗ ∗ Ξ44 Ξ (49) 0 ⎥ 45 0 ⎥
