Neural Networks 54 (2014) 85–94
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Neural Networks journal homepage: www.elsevier.com/locate/neunet
Impulsive synchronization schemes of stochastic complex networks with switching topology: Average time approach Chaojie Li a , Wenwu Yu b , Tingwen Huang c,∗ a
Platform Technologies Research Institute, RMIT University, Melbourne, VIC 3001, Australia
b
Department of Mathematics, Southeast University, Nanjing 210096, PR China
c
Texas A&M University at Qatar, Doha, P.O. Box 23874, Qatar
article
info
Article history: Received 1 November 2013 Received in revised form 29 January 2014 Accepted 27 February 2014 Available online 12 March 2014 Keywords: Impulsive synchronization Stochastic complex networks Switching topology Average dwell time Average impulsive interval
abstract In this paper, a novel impulsive control law is proposed for synchronization of stochastic discrete complex networks with time delays and switching topologies, where average dwell time and average impulsive interval are taken into account. The side effect of time delays is estimated by Lyapunov–Razumikhin technique, which quantitatively gives the upper bound to increase the rate of Lyapunov function. By considering the compensation of decreasing interval, a better impulsive control law is recast in terms of average dwell time and average impulsive interval. Detailed results from a numerical illustrative example are presented and discussed. Finally, some relevant conclusions are drawn. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Consensus has received increasing attention since the collective dynamical behaviors of complex networks have been a subject of intensive research with potential applications in physical, social, biological, and technological fields. In general, complex networks are modeled by a graph with non-trivial topological features where every node is an individual element of the whole system with certain pattern of connections, which are neither entirely regular nor entirely random (Barabási & Albert, 1999; Strogatz, 2001; Watts & Strogatz, 1998). These features do not occur in the mathematical models of the networks that have been studied in the past, such as lattices or random graphs, but they do truly exist in nature. However, the phenomenon of synchronization of large populations is a challenging problem and requires different hypothesis to be solved. Synchronization processes in populations of locally interacting elements are in the focus of intensive research. The analysis of synchronizability has benefited not only from the advance in the understanding of complex networks, but also has it contributed to the understanding of general emergent properties of networked systems, such as the Internet, Human and Robot interaction networks, human collaboration networks, etc. As
∗
Corresponding author. Tel.: +974 44230174. E-mail address: [email protected] (T. Huang).
http://dx.doi.org/10.1016/j.neunet.2014.02.013 0893-6080/© 2014 Elsevier Ltd. All rights reserved.
a consequence, many results for synchronization of different complex networks have been extensively studied in the physics and mathematics literature; e.g. Wang and Chen (2002a), Wu (2003), Olfati-Saber, Fax, and Murray (2007), Yu et al. (2009), Yu, Chen, and Lü (2009), Yu, Chen, Lü, and Kurths (2013), Wang, Wang, and Liu (2010), Wang, Ho, Dong, and Gao (2010), Liang, Wang, Liu, and Liu (2008), Liang, Wang, and Liu (2009), Zhang, Tang, Fang, and Wu (2012), Lu and Ho (2010), Lu, Ho, and Cao (2010), Mahdavi, Menhaj, Kurths, and Lu (2013), Shen, Wang, and Liu (2011), Shen, Wang, and Liu (2012), Gan (2012), Lü and Chen (2005), Wen, Bao, Zeng, and Huang (2013), Huang, Li, Yu, and Chen (2009), Wang and Chen (2002b), Chen, Liu, and Lu (2007), Liu, Lu, and Chen (2011), Guan, Wu, and Feng (2012), Yang, Cao, and Lu (2012), Liu, Lu, and Chen (2013), Hu, Yu, Jiang, and Teng (2012), Zhao, Hill, and Liu (2011) and references therein. Theoretically, synchronization of a network is mainly contributed by the nodes’ dynamical behaviors connections among the nodes. Many results have been devoted to the structural characterization and evolution of complex networks. In Wang and Chen (2002a), the authors studied robustness and fragility of synchronization of scale-free networks through the spectral properties of the underlying structure. In terms of graph based theoretical bounds to synchronizability, Wu (2003) mainly focused on the bounds of its extreme eigenvalues with graph. Global and local synchronization of coupled networks were discussed in Wen et al. (2013) and Yu et al. (2009) where different techniques were employed to address time delays in the system level. The authors
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C. Li et al. / Neural Networks 54 (2014) 85–94
of Wang, Ho et al. (2010) and Wang, Wang et al. (2010) studied synchronization of discrete complex networks when the case involves randomly occurred nonlinearities and mixed time-delays. The bounded H∞ state estimation problem was studied in Shen et al. (2011) via RLMI method. In addition, synchronizability for general dynamical networks and fuzzy complex dynamical networks were discussed in Lu and Ho (2010) and Mahdavi et al. (2013). Meanwhile, to design a control law for synchronization of complex network, many efforts have been made for this purpose. In the sense of continuous control law, Lü and Chen (2005) described a remarkable idea of a time-varying complex dynamical network model and designed its controlled synchronization criteria. Stability schemes of delayed neural networks were studied by Zhang et al. (2012) in which time-varying impulses have been modeled. A set of sampled-data synchronization controllers is designed by utilizing both the Gronwall’s inequality and the Jenson integral inequality in Shen et al. (2012). Taking account into the consideration of discontinuous control approach, periodic intermittent controller (Gan, 2012) and intermittent controller (Hu et al., 2012) for synchronization of networks with time delays were investigated through different methods. Lu et al. (2010) first established a novel concept, namely average impulsive interval, for the synchronizability analysis of complex networks. By introducing intermittent linear state feedback, Huang et al. (2009) studied problems of cluster synchronization of linearly coupled complex networks and synchronization of delayed chaotic systems with parameter mismatches, respectively. Hybrid adaptive and impulsive control (Yang et al., 2012) was exploited for stochastic synchronization of complex networks. In addition, as known, certain complex networks display a scale-free feature, of which the connectivity distributions have the power-law form. The pinning scheme of the most highly connected nodes induced that a significant reduction in required controllers as compared to the traditional control scheme, see Wang and Chen (2002b). More general cases of pining control can be found in Yu et al. (2009), Yu et al. (2013). A single controller of pinning synchronization was designed, for example, without assuming symmetry, irreducibility, or linearity of the couplings, the authors (Chen et al., 2007) proved that a single controller can pin a coupled complex network to a homogeneous solution. By considering the average impulsive interval, some generic mean square stability criteria of synchronization control were derived in Lu et al. (2010). Obviously, these methods are effective for the synchronization control of different complex networks. In the practical network environment, information exchange of each node through communication channels often experiences link failures and transmission delay, which can be modeled by a sequence of switching interconnections with time delays as shown in Liu et al. (2011) and Olfati-Saber et al. (2007). Synchronization problems with respect to switched systems were studied by typical techniques, for instance, the multiple v -Lyapunov function method (Zhao et al., 2011). As aforementioned in Lu et al. (2010) and Yang et al. (2012), by average impulsive interval, it can be derived that a unified synchronization criterion of complex networks is less conservative no matter desynchronizing or synchronizing impulses. Likewise, by average dwell time (Vu & Morgansen, 2010; Zhang & Shi, 2009), the stability criterion of switched system is less conservative. If we can obtain the result by virtue of the relationship between average impulsive interval and average dwell time, it would be much less conservative than the previous results. The research undertaken so far in order to understand how synchronization phenomena are affected by impulsive control law and the topological substrate of interactions, especially, when complex networks are involving noisy disturbances. The main goal of this paper is to examine the effect of time delays in switching topologies exerting on synchronization through Lyapunov–Razumikhin
technique. Correspondingly, the major contributions in this paper are twofold. First, the relationship between average impulsive interval(AII) of control law and average dwell time(ADT) of switched topologies is established. Second, the average time approach presented in this paper offers attractive features that are potentially useful for controlling different categories of complex networks. The paper is organized as follows. We first introduce the basic mathematical descriptions of discrete complex networks that will be used henceforth. Next, we focus on the synchronization analysis of discrete complex networks through impulsive control in the sense of average dwell time. The effect of time delays is tackled by Lyapunov–Razumikhin technique. Section 4 is devoted to the analysis of the conditions for the synchronization of complex networks in the sense of average interval time. The relationship is established by means of the results in Section 3. In Section 5, we present a numerical example and the discussion on the tradeoff between AII and ADT is given. Finally, the last section rounds off the paper by giving our conclusions. 2. Preliminary Let Rd denote the d-dimensional Euclidean space and ∥ • ∥ be the Euclidean norm in Rd . Denote Z+ = {1, 2, . . .}, Zτ = {−τ , −τ + 1, −τ + 2, . . . , −1, 0}, the finite set H = {1, 2, . . . , H } with finite positive integer H, N = {0, 1, 2, . . .}, I be the identity matrix, and matrix X > 0 ( 0 is the inner coupling matrix between two connected nodes. τ (k) is a nonnegative constant which represents the time-delay of the signal transmitted from the network to the ith node, where the coupling time-delay τ (k) satisfies 0 ≤ τ (k) ≤ τ for a constant τ > 0. gi : Z × Rd → Rd is the noise intensity function vector. ω(k) is a scale Wiener process (Brownian motion) defined on (Ω , F , P ) with T
E{ω(k)} = 0, E{ω(i)ω(j)} = 0,
Nd
E{ω(k)2 } = 1,
(i ̸= j).
(2)
C. Li et al. / Neural Networks 54 (2014) 85–94
87
Let G = ({1, 2, . . . , N }, E ) be an undirected graph with N vertices, describing the topology of the network. The Graph Laplacian matrix LG associated with G is defined as LG = ∆G − AG , where ∆G is the degree matrix and AG is the adjacency matrix of the graph. The Laplacian matrix is symmetric, positive semi-definite and has 0 as an eigenvalue with eigenvector 1. More specifically, G is connected if and only if rank(LG ) = N − 1 and λ2 (LG ) serves the function as the determinant contributor to the algebraic connectivity of G. For networks with switching connected topologies, let G be the finite set corresponding to the connected undirected graphs with vertices {1, 2, . . . , N }, such that
G = {Gh = ({1, 2, . . . , N }, Eh )|h ∈ H }. Each Gh ∈ G is connected, undirected with different connections of edge. Let IG ⊂ N be an index set associated with the elements of G. A switching signal σ is a map σ : Z+ → IG . For each time k ∈ Z+ , the switching signal σ establishes the network graph Gσ (k) ∈ G utilized by the network agents. The logical rules that generate the switching signals constitute the switching logic, and the index σ (k) is called the active mode at the time instant k. Meanwhile, let km,x (m ∈ N ) be the switching instants to label each switching phenomenon, i.e., when σ (km,x − 1) = 1, the active mode is Mode 1; when σ (km,x ) = 2, the active mode is replaced by Mode 2. Let k0,x = 0 be the starting mode. Note that there is no-chattering requirement in the discrete-time switched system due to the finite state changes in each impulsive interval, see Zhang and Shi (2009). Lσ (k) = (lij,σ (k) )N ×N is the coupling configuration matrix representing the coupling topology of the dynamical network at the active mode σ (k). The element lij,σ (k) is defined as the connectivity of the whole complex network, in which if there is a connection between subnetwork i and subnetwork j (j ̸= i) then lij,σ (k) = lji,σ (k) = 1; otherwise, lij,σ (k) = lji,σ (k) = 0 (j ̸= i). Let the diagonal element lii,σ (k) be defined as lii,σ (k) = −
N
lij,σ (k) = −
N
lji,σ (k) ,
(3)
i=1,i̸=j
j=1,j̸=i
where 1 ≤ i, j ≤ N. The initial conditions associated with model (1) are given by xi (θ) = φi (θ ),
θ ∈ Zτ ,
(4)
where φ(θ ) ∈ L2F0 ([−τ , 0], Rd ) and the norm of φ(θ ) is defined by ∥φ(θ)∥τ = supθ∈Zτ {∥φ(θ )∥}. In order to design an impulsive control scheme to synchronize system (1), we consider the evolutionary state abruptly jumps at every impulsive instant of time kn,u from its open-loop state, which can be formularized by xi (kn,u ) = Jn xi (kn,u − 1),
n∈N
(5)
where xi (kn,u − 1) stands for the primal state at time instant kn,u without impulsive jump (see Zhang, Sun, & Feng, 2009). As usual, every impulsive instant of time kn,u satisfies 0 = k0,u < k1,u < k2,u < · · · < kn,u < kn+1,u < · · · and limn→∞ kn,u = ∞; Jn ∈ Rd×d represents the impulsive jump strength at time instant kn,u . Therefore, at each impulsive instant of time kn,u , the coupled states xi (k) and xj (k) can be described by xi (kn,u ) − xj (kn,u ) = Jn [xi (kn,u − 1) − xj (kn,u − 1)].
(6)
∞
δ(k − kn,u )Jn (xi (kn,u − 1)),
n ∈ N,
As shown in Fig. 1, the top picture represents the state evolution of discrete system and the bottom one denotes the switching signals among three different modes. In addition, the state switching occurs at each time instant ki,x (i = 1, 2, 3, . . . , 19) while impulsive control is deployed at time instant kj,u (j = 1, 2, 3, . . . , 8). Eventually, by means of impulsive control, system will converge to its equilibrium point. Remark 1. It is worthwhile mentioning here that every time instant of impulsive control is unnecessarily equal to the time instant of switching signal, i.e., kn,u ̸= km,x . Due to the existence of time delays in the dynamical system, it is difficult to design feedback controller for the switched systems in the presence of both state delays and switching delays (see Vu & Morgansen, 2010). When the information available to the controller is a delayed version of the desired one in a network environment, the closed loop will feature both switching signals and delayed states. However, the impulsive control law is asynchronous to switching signals, which is consummate even the existence of large time delays. Therefore, impulsive control is a better choice for this circumstance, in which information exchange experiences both link failures and time delays. The equivalent matrix form of impulsive controlled system is given by x(k + 1) = F (k, x(k)) + (Lσ (k) ⊗ Γ )x(k − τ (k)) + G(k, x(k))ω(k); k ̸= kn,u , k ∈ Z+ ; x(kn,u ) = [IN ⊗ Jn ]x(kn,u − 1),
(8)
where Lσ (k) is a configuration matrix defining the topology of each sub-state. The compact forms of each element are x(k) = [xT1 (k)
xT2 (k)
...
xTN (k)]T ,
F (k, x(k)) = [f T (k, x1 (k))
. . . f T (k, xN (k))]T , x(k − τ (k)) = [ ( − τ (k)) . . . xTN (k − τ (k))]T , xT1 k g1T k x1
G(k, x(k)) = [
( , (k))
...
gNT (k, xN (k))]T .
Assumption 1. For each nonlinear function f (·, ·), suppose that there exist constant fˆ satisfies
[f (k, x1 ) − f (k, x2 )]T [f (k, x1 ) − f (k, x2 )] ≤ fˆ [x1 (k) − x2 (k)]T [x1 (k) − x2 (k)], where any x1 , x2 ∈ Rd .
Intuitively, a family of impulsive controller can be designed as Ui (k, xi (k)) =
Fig. 1. The trend in the evolution of controlled system with switching signals.
(7)
n=1
where Ui (k, xi (k)) represents a class of impulsive controller at each instant of time kn,u ; δ(•) denotes the Dirac discrete-time function.
Assumption 2. For each noisy intensity function g (·, ·), suppose that there exists a matrix G ∈ Rd×d such that g T (k, xi (k))g (k, xi (k)) ≤ ∥Gxi (k)∥2 , for any xi ∈ Rd .
i = 1, 2, . . . , N ,
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C. Li et al. / Neural Networks 54 (2014) 85–94
Assumption 3. Without loss of generality, let Mn be the subset of H (Mn ⊆ H ) and represent sub-states appearing on impulsive interval k ∈ [kn,u , kn+1,u − 1] (n ∈ N ). Different impulsive interval has its own subset Mn .
Theorem 1. Under Assumptions (A1) and (A2), the dynamical system (8) is globally exponentially synchronized in the mean square if there exist positive scalars εn > 0, αn > 0, βn > 1 and µ ≥ 1 on each time interval k ∈ [kn,u , kn+1,u − 1] satisfying:
For the simplicity of calculation, we denote F = diag{fˆ , fˆ , . . . ,
I. for each sub-state k ∈ [km,x , km+1,x − 1] and certain positive τ ⌉, there exist positive definite integer nτ = ⌈ inf {k −k }
fˆ } and G = diag{ˆg , gˆ , . . . , gˆ }. Let
m∈N
xij (k) = xi (k) − xj (k), xij (k − τ ) = xi (k − τ ) − xj (k − τ ), Fij (k, x(k)) = Fi (k, x(k)) − Fj (k, x(k)), (2)
Lσ (k) WLσ (k) = Nlij,σ (k) ,
WLσ (k) = Nlij,σ (k) .
Now, we recall some useful related definitions as follow. Definition 1 (see Hespanha & Morse, 1999). For switching signal σ and any K > k > k0 , let Nσ (K , k) be the switching numbers of σ over the interval [k, K ). If for any given K and k (K > k > k0 ), we have Nσ (K , k) ≤
K −k
Ta
Ta
m,x
n ∈ N,
i, j ∈ H ,
where
αn = max
σ (k)∈Mn
λmax (Πσ (k) ) + µβn λmax (Ωσ (k) ) , λmin (Pσ (k) )
Πσ (k) = F T Pσ (k) F + GT Pσ (k) G − Nlij,σ (k) F T Qσ−(1k) F , (2)
Ωσ (k) = −Nlij,σ (k) Γ T Pσ (k) Γ − Nlij,σ (k) Γ T PσT (k) Qσ (k) Pσ (k) Γ ; II. it is given that each impulsive controller requires )(kn+1,u −kn,u )
with the average dwell time ln µ , γ∗ − ε
where γ ∗ = maxn∈N
(Jn ) . − 2knln+1λ,max u −kn,u
(ε+ lnTaµ )(2kn+1,u −kn,u −kn−nτ +1,u )
III. βn ≥ e
, where ε = maxn∈N εn .
Proof. Consider the following Lyapunov function: (9)
for any k ∈ [kn,u , kn,u − 1], n ∈ N , where
Definition 3. The impulsive controlled discrete complex networks (8) are said to be globally exponentially synchronized in the mean square, if for any initial condition φ(θ ) ∈ L2F0 ([−τ , 0], Rd ), there exist two positive constants λ and M0 ≥ 1 such that
E ∥xi (k) − xj (k)∥2 ≤ M0 e−γ (k−k0 ) ,
1≤i≤j≤N
holds for all k > k0 .
−1 . . . −1 −1 N − 1 . . . −1 ... . . . . . . . . . . It is given that, for −1 −1 . . . N − 1
N − 1 W =
θ ∈ [−τ , −1], we
have V (θ ) = xT (θ )(W ⊗ Pσ (θ ) )x(θ )
= −
N −1 N
wij (φi (θ ) − φj (θ ))T
i =1 j =i +1
Pσ (θ ) (φi (θ ) − φj (θ ))
Lemma 1. Let W = (wij )N ×N , P ∈ Rd×d , x = (xT1 , xT2 , . . . , xTN )T and y = (yT1 , yT2 , . . . , yTN )T with xk , yk ∈ Rd (k = 1, 2, . . . , N). If W = W T and each row sum of W is zero, then N −1 N
Ta ≥ Ta∗ =
V (k) = xT (k)(W ⊗ Pσ (k) )x(k),
+ Nζ0 ,
where Ta is called average impulsive interval.
xT (W ⊗ P )y = −
Qi ≤ µQj ,
ln αn ≤ εn ,
ln µ
Definition 2 (see Lu et al., 2010). The average impulsive interval of the impulsive sequence ζ = {t1 , t2 , ···} is less than Ta , if there exist a positive integer Nζ0 and a positive number Nζ (K , k) associated with the number of impulsive times of impulsive sequence ζ over the interval [k, K ), such that K −k
P i ≤ µP j ,
λ2max (Jn ) ≤ e−(ε+ Ta
+ Nσ0 ,
then Ta and Nσ0 are called average dwell time and the chatter bound, respectively.
Nζ (K , k) ≤
m+1,x
matrices Pi ∈ Rd×d and Qi ∈ Rd×d , such that
wij (xi − xj )T P (yi − yj ).
i=1 j=i+1
≤ λmax (Pσ (θ ) )
N −1 N
(−wij )∥φi (θ ) − φj (θ )∥2 .
i=1 j=i+1
Let ϕ(∥φi (θ ) − φj (θ )∥2 ) = λmax (Pσ (θ ) )∥φi (θ ) − φj (θ )∥2 , thus V (θ ) ≤
N −1 N
ϕ(∥φi (θ ) − φj (θ )∥2 )
i=1 j=i+1
3. Main results
. = ϕ(∥φ∥2τ ).
(10)
Choose M ≥ 1 and γ > 0, such that
3.1. Global exponential synchronizability
1 ≤ Me−γ (k1,u −k0,u ) e−ε0 (k1,u −k0,u )−Nσ (k1,u ,k0,u ) ln µ .
In this section, we consider the global exponential synchronization problem of discrete delayed systems with switching topology by employing Lyapunov–Razumikhin Technique. Specifically, by considering Lyapunov-like functions allowed to increase with the bounded increase rate on each impulsive interval, the criteria of global exponential synchronization are developed in terms of average dwell time (ADT) which has been widely used for the stability analysis of the switched system.
We claim that E{V (k)} ≤ M ϕ(∥φ∥2τ )e−γ (kn,u −k0,u ) ,
k ∈ [kn,u , kn+1,u − 1], n ∈ N .
(11)
Mathematical induction will be utilized to prove the result in two steps. In the first step, the given statement (11) is to prove for the first interval which is on k ∈ [k0,u , k1,u − 1] for n = 1. The inductive step is to prove that (11) for any interval k ∈ [kn−1,u , kn,u − 1]
C. Li et al. / Neural Networks 54 (2014) 85–94
implies the given statement for the next interval. Meanwhile, proof by contradiction is also applied in each step where we assume that the given statement of each step is not a truly proposition and show that there is a contradiction to this assumption. We first show that E{V (k)} ≤ M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) ,
k ∈ [k0,u , k1,u − 1].
(12)
89
+ G(k, x(k))ω(k)] − xT (k)(W ⊗ Pσ (k) )x(k)} = E{F T (k, x(k))(W ⊗ Pσ (k+1) )F (k, x(k)) + xT (k − τ )(LTσ (k) ⊗ Γ T )(W ⊗ Pσ (k+1) ) × (Lσ (k) ⊗ Γ )x(k − τ ) + 2F T (k, x(k)) × (W ⊗ Pσ (k+1) )(Lσ (k) ⊗ Γ )x(k − τ )
Obviously, for k ∈ [k0,u − τ , k1,u − 1],
× GT (k, x(k))(W ⊗ Pσ (k+1) )G(k, x(k))
E{V (k)} ≤ ϕ(∥φ∥2τ )
− xT (k)(W ⊗ Pσ (k) )x(k)}. −γ (k1,u −k0,u ) −ε0 (k1,u −k0,u ) −Nσ (k1,u −k0,u ) ln µ
≤ M ϕ(∥φ∥τ )e
e
< M ϕ(∥φ∥τ )e
e
2
2
e
By further computation, (20) is given by
−γ (k1,u −k0,u ) −ε0 (k1,u −k0,u ) Nσ (k+1,k)
< M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) .
(13)
If (12) is not true, consider the case that no switching appears on this interval, then there exists an instant of time k¯ ∈ [k0,u , k1,u − 1],
E{V (k + 1)} ≤ µ
k¯ = min{k ∈ [km,x , km+1,x − 1]|E{V (k)}
E
xTij (k)F T
× Pσ (k) F xij (k) +
N −1 N
xTij (k)GT Pσ (k) Gxij (k)
i=1 j=i+1
−
N −1 N
(2)
xTij (k − τ )Nlij,σ (k) Γ T Pσ (k) Γ
i=1 j=i+1
× xij (k − τ ) −
N −1 N
xTij (k)Nlij,σ (k) F T
i=1 j=i+1
> M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) }
× Qσ−(1k) F xij (k) −
such that
N −1 N
xTij (k − τ )N
i=1 j=i+1
E{V (k¯ )} > M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u )
× lij,σ (k) Γ Pσ (k) Qσ (k) Pσ (k) Γ xij (k − τ )
> M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) e−ε0 (k1,u −k0,u )
T
> M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) e−ε0 (k1,u −k0,u ) e−Nσ (k1,u ,k0,u ) ln µ > ϕ(∥φ∥2τ )
(14)
≤ µNσ (k+1,k) E
T
N −1 N
xTij (k)[F T Pσ (k) F
i=1 j=i+1
and E{V (k)} ≤ M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) ,
k ∈ [k0,u − τ , k¯ − 1].
(15)
Consider
+ GT Pσ (k) G − Nlij,σ (k) F T Qσ−(1k) F ] × xij (k) + xTij (k − τ )[−Nl(ij2,σ) (k) Γ T Pσ (k) Γ
k = max{k ∈ [k0,u , k¯ ]|E{V (k)} ≤ M ϕ(∥φ∥2τ )
− Nlij,σ (k) Γ Pσ (k) Qσ (k) Pσ (k) Γ ]xij (k − τ )
∗
×e
N −1 N i=1 j=i+1
k¯ = min{k ∈ [k0,x , k1,u − 1]|E{V (k)} > M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) }. Or the case that certain state-switchings occur, then there exists an instant of time k¯ ∈ [km,x , km+1,x − 1] belonging to the time interval [k0,u , k1,u − 1] satisfying
(20)
−γ (k1,u −k0,u ) −ε0 (k1,u −k0,u ) −Nσ (k1,u ,k0,u ) ln µ
e
e
T
},
(16)
(17)
(18)
Obviously, state-switching would occur more than one time on each of impulsive intervals. Due to the fact that σ (k) ∈ M0 on the interval [k0,u , k1,u − 1] and Pσ (k) ≤ µPσ (k−τ ) , it is obtained that
from which we have E{V (k)} > M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) e−ε0 (k1,u −k0,u )
k ∈ [k∗ + 1, k¯ − 1].
Therefore, we can conclude that, for k ∈ [k∗ , k¯ ], one gives E{V (k∗ )} ≤ E{V (k)} ≤ E{V (k¯ )},
E{V (k + s)} ≤ M ϕ(∥φ∥τ )e
E{V (k + 1)} ≤ µNσ (k+1,k)
−γ (k1,u −k0,u )
≤ eε0 (k1,u −k0,u ) eNσ (k1,u ,k0,u ) ln µ E{V (k∗ )} ln µ
≤ e(ε+ Ta )(k1,u −k0,u ) E{V (k∗ )} ≤ β1 E{V (k)}.
(19)
Consider the increment of V (k) along the solution of discrete complex networks (1) in the interval k ∈ [k∗ , k¯ − 1]. For each active instant of sub-system k ∈ [k∗m,x , k∗m+1,x − 1], E{∆V (k)} = E{V (k + 1)} − E{V (k)}
= E{[F (k, x(k)) + (Lσ (k) ⊗ Γ )x(k − τ ) + G(k, x(k))ω(k)]T (W ⊗ Pσ (k+1) ) × [F (k, x(k))(Lσ (k) ⊗ Γ )x(k − τ )
λmax (Πσ (k) ) λmin (Pσ (k) ) λmax (Ωσ (k) ) + β0 −1 E{V (k)} µ λmin (Pσ (k) )
and, taking time delay s ∈ Zτ into account, one has, 2
λmax (Πσ (k) ) E{V (k)} λmin (Pσ (k) ) λmax (Ωσ (k) ) + E{V (k − τ )} . λmin (Pσ (k−τ (k)) )
≤ µNσ (k+1,k)
× e−Nσ (k1,u ,k0,u ) ln µ ,
T
≤ µNσ (k+1,k) α0 E{V (k)}. Therefore, we have ¯ ∗ (k¯ −k∗ ) E{V (k¯ )} ≤ µNσ (k,k ) α0 E{V (k∗ )} (k
−k0,u )
E{V (k∗ )}
(k
−k0,u )
M ϕ(∥φ∥2τ )
< µNσ (k1,u ,k0,u ) α0 1,u ≤ µNσ (k1,u ,k0,u ) α0 1,u
× e−γ (k1,u −k0,u ) e−ε0 (k1,u −k0,u ) e−Nσ (k1,u ,k0,u ) ln µ
(21)
90
C. Li et al. / Neural Networks 54 (2014) 85–94
= M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) < E{V (k¯ )},
=
which is a contradiction. Thus, (12) is true. Next, we claim (11) is true for n = 0, 1, 2, . . . , n¯ , such that k ∈ [kn−1,u , kn,u − 1].
k ∈ [kn¯ ,u , kn¯ +1,u − 1].
E{V (kn¯ ,u )} = E
×e
(k
{xTij (kn¯ ,u − 1)
< M ϕ(∥φ∥2τ )e−γ (kn¯ +1,u −k0,u ) < E{V (k˜ )},
≤ λ2max (Jn¯ )M ϕ(∥φ∥2τ )e−γ (kn¯ ,u −k0,u )
≤ M ϕ(∥φ∥2τ )e−γ (kn¯ +1,u −k0,u ) e−εn¯ (kn¯ +1,u −kn¯ ,u ) .
(25)
¯ > n¯ such that If (24) is not true, then there exists an integer m k˜ ∈ [km¯ ,x , km¯ +1,x − 1] on interval k ∈ [kn¯ ,u , kn¯ +1,u − 1] satisfying
min {λmin (Pσ (k) )}E
σ (k)∈H
≤E
k˜ = min{k ∈ [km¯ ,x , km¯ +1,x − 1]|E{V (k)} ≤ M ϕ(∥φ∥2τ )
E
k∗∗ = max{k ∈ [km∗∗ ,x , km∗∗ +1,x ]|E{V (k)} ≤ M ϕ(∥φ∥2τ )
N −1 N
(27)
(xi (k) − xj (k)) Pσ (k) (xi (k) − xj (k)) T
(33)
∥xi (k) − xj (k)∥
2 2
M
≤ min
For any k ∈ [k∗∗ + 1, k˜ ], we have
σ (k)∈H
E{V (k)} > M ϕ(∥φ∥2τ )e−γ (kn¯ +1,u −k0,u ) e−εn¯ (kn¯ +1,u −kn¯ ,u )
ϕ(∥φ∥2τ )e−γ (k−k0,u ) λmin (Pσ (k) )
(34)
which implies that
.
(28)
In addition, for any k ∈ [k∗∗ , k˜ ], we have E V (k∗∗ ) ≤ E{V (k)} ≤ E{V (k˜ )}.
(29)
Then, for any k ∈ [km¯ ,x , km¯ +1,x − 1], E {V (k + 1)} ≤ µNσ (k+1,k) E {V (k)}
λmin (Pσ (km¯ ,x ) ) (30)
In light of Condition (III), it observes that for any s ∈ Zτ , one has k + s ∈ [kn¯ −nτ ,u , k˜ − 1]. Then, for any k ∈ [k∗∗ , k˜ − 1], we have
≤ M ϕ(∥φ∥2τ )e−γ (kn¯ +1,u −k0,u )
N −1 N
i=1 j=i+1
× e−γ (kn¯ +1,u −k0,u ) e−εn¯ (kn¯ +1,u −kn¯ ,u ) e−Nσ (kn¯ +1,u ,kn¯ ,u ) ln µ }.
E{V (k + s)} ≤ M ϕ(∥φ∥2τ )e−γ (kn¯ −nτ +1,u −k0,u )
∥xi (k) − xj (k)∥
i=1 j=i+1
Therefore, for any k ∈ Z+ , we have
Denote
≤ µNσ (k+1,k) αn¯ E {V (k)} .
2 2
≤ M ϕ(∥φ∥2τ )e−γ (k−k0,u ) .
(26)
Consequently, we have k˜ > km¯ ,x and V (k˜ ) > V (kn¯ ,u ). Also there ¯ exists another integer m∗∗ such that m∗∗ ≤ m.
×
N −1 N
i=1 j=i+1
× e−γ (kn¯ +1,u −k0,u ) }.
λmax (Πσ (km¯ ,x ) ) + µβn¯ λmax (Ωσ (km¯ ,x ) )
(32)
which is a contradiction. Hence, (24) holds for k = k˜ + 1. And by virtue of mathematical induction method, the claim (11) is true for each k ∈ Z+ . In view of Lemma 1, it can be obtained that, for any k ∈ Z+ ,
= λ2max (Jn¯ )E{V (kn¯ ,u − 1)}
−kn¯ ,u )
× e−ε(kn¯ +1,u −kn¯ ,u ) e−Nσ (kn¯ +1,u ,kn¯ ,u ) ln µ × M ϕ(∥φ∥2τ )e−γ (kn¯ +1,u −k0,u )
× Pσ (kn¯ ,u −1) xij (kn¯ ,u − 1)}
(31)
< µNσ (kn¯ +1,u −kn¯ ,u ) ln µ αn¯ n¯ +1,u
×e
E{V (k∗∗ + 1)}
˜ ∗∗ (k˜ −k∗∗ ) E{V (k˜ )} ≤ µNσ (k,k ) pn¯ ,u E{V (k∗∗ )}
i=1 j=i+1
−Nσ (kn¯ +1,u ,kn¯ ,u ) ln µ
i=0
0, ε− < 0, αn > 0, βn > 1 and µ ≥ 1 on each time interval k ∈ [kn,u , kn+1,u − 1] satisfying I. For each sub-state k ∈ [km,x , km+1,x − 1] and certain positive τ ⌉, there exist matrices Pi ∈ Rd×d , integer nτ = ⌈ inf m∈N {km+1,x −km,x } and Qi ∈ Rd×d , such that P i ≤ µP j , Qi ≤ µQj , i, j ∈ H , ε+ = max{ln αn } if αn > 1, n∈N
≤ ϕ(∥φ∥2τ ).
E {V (k + 1)} ≤ eε− T− (k,km,x )+ε+ T+ (k,km,x ) E V (km,x ) = eε− T− (k,km,x )+ε− T+ (k,km,x ) × e−ε− T+ (k,km,x )+ε+ T+ (k,km,x ) E V (km,x ) ≤ e(ε+ −ε− )Tmax eε− (k−km,x ) E V (km,x ) ,
αn = max
σ (k)∈Mn
λmax (Πσ (k) ) + µβn λmax (Ωσ (k) ) , λmin (Pσ (k) )
Πσ (k) = F T Pσ (k) F + GT Pσ (k) G − Nlij,σ (k) F T Qσ−(1k) F , (2)
Ωσ (k) = −Nlij,σ (k) Γ T Pσ (k) Γ − Nlij,σ (k) Γ T PσT (k) Qσ (k) Pσ (k) Γ . II. The relationship between average impulsive interval and average dwell time requires
+ ε− +
ln µ + (ε+ − ε− )Tmax
2 max
Ta
Contents lists available at ScienceDirect
Neural Networks journal homepage: www.elsevier.com/locate/neunet
Impulsive synchronization schemes of stochastic complex networks with switching topology: Average time approach Chaojie Li a , Wenwu Yu b , Tingwen Huang c,∗ a
Platform Technologies Research Institute, RMIT University, Melbourne, VIC 3001, Australia
b
Department of Mathematics, Southeast University, Nanjing 210096, PR China
c
Texas A&M University at Qatar, Doha, P.O. Box 23874, Qatar
article
info
Article history: Received 1 November 2013 Received in revised form 29 January 2014 Accepted 27 February 2014 Available online 12 March 2014 Keywords: Impulsive synchronization Stochastic complex networks Switching topology Average dwell time Average impulsive interval
abstract In this paper, a novel impulsive control law is proposed for synchronization of stochastic discrete complex networks with time delays and switching topologies, where average dwell time and average impulsive interval are taken into account. The side effect of time delays is estimated by Lyapunov–Razumikhin technique, which quantitatively gives the upper bound to increase the rate of Lyapunov function. By considering the compensation of decreasing interval, a better impulsive control law is recast in terms of average dwell time and average impulsive interval. Detailed results from a numerical illustrative example are presented and discussed. Finally, some relevant conclusions are drawn. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Consensus has received increasing attention since the collective dynamical behaviors of complex networks have been a subject of intensive research with potential applications in physical, social, biological, and technological fields. In general, complex networks are modeled by a graph with non-trivial topological features where every node is an individual element of the whole system with certain pattern of connections, which are neither entirely regular nor entirely random (Barabási & Albert, 1999; Strogatz, 2001; Watts & Strogatz, 1998). These features do not occur in the mathematical models of the networks that have been studied in the past, such as lattices or random graphs, but they do truly exist in nature. However, the phenomenon of synchronization of large populations is a challenging problem and requires different hypothesis to be solved. Synchronization processes in populations of locally interacting elements are in the focus of intensive research. The analysis of synchronizability has benefited not only from the advance in the understanding of complex networks, but also has it contributed to the understanding of general emergent properties of networked systems, such as the Internet, Human and Robot interaction networks, human collaboration networks, etc. As
∗
Corresponding author. Tel.: +974 44230174. E-mail address: [email protected] (T. Huang).
http://dx.doi.org/10.1016/j.neunet.2014.02.013 0893-6080/© 2014 Elsevier Ltd. All rights reserved.
a consequence, many results for synchronization of different complex networks have been extensively studied in the physics and mathematics literature; e.g. Wang and Chen (2002a), Wu (2003), Olfati-Saber, Fax, and Murray (2007), Yu et al. (2009), Yu, Chen, and Lü (2009), Yu, Chen, Lü, and Kurths (2013), Wang, Wang, and Liu (2010), Wang, Ho, Dong, and Gao (2010), Liang, Wang, Liu, and Liu (2008), Liang, Wang, and Liu (2009), Zhang, Tang, Fang, and Wu (2012), Lu and Ho (2010), Lu, Ho, and Cao (2010), Mahdavi, Menhaj, Kurths, and Lu (2013), Shen, Wang, and Liu (2011), Shen, Wang, and Liu (2012), Gan (2012), Lü and Chen (2005), Wen, Bao, Zeng, and Huang (2013), Huang, Li, Yu, and Chen (2009), Wang and Chen (2002b), Chen, Liu, and Lu (2007), Liu, Lu, and Chen (2011), Guan, Wu, and Feng (2012), Yang, Cao, and Lu (2012), Liu, Lu, and Chen (2013), Hu, Yu, Jiang, and Teng (2012), Zhao, Hill, and Liu (2011) and references therein. Theoretically, synchronization of a network is mainly contributed by the nodes’ dynamical behaviors connections among the nodes. Many results have been devoted to the structural characterization and evolution of complex networks. In Wang and Chen (2002a), the authors studied robustness and fragility of synchronization of scale-free networks through the spectral properties of the underlying structure. In terms of graph based theoretical bounds to synchronizability, Wu (2003) mainly focused on the bounds of its extreme eigenvalues with graph. Global and local synchronization of coupled networks were discussed in Wen et al. (2013) and Yu et al. (2009) where different techniques were employed to address time delays in the system level. The authors
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C. Li et al. / Neural Networks 54 (2014) 85–94
of Wang, Ho et al. (2010) and Wang, Wang et al. (2010) studied synchronization of discrete complex networks when the case involves randomly occurred nonlinearities and mixed time-delays. The bounded H∞ state estimation problem was studied in Shen et al. (2011) via RLMI method. In addition, synchronizability for general dynamical networks and fuzzy complex dynamical networks were discussed in Lu and Ho (2010) and Mahdavi et al. (2013). Meanwhile, to design a control law for synchronization of complex network, many efforts have been made for this purpose. In the sense of continuous control law, Lü and Chen (2005) described a remarkable idea of a time-varying complex dynamical network model and designed its controlled synchronization criteria. Stability schemes of delayed neural networks were studied by Zhang et al. (2012) in which time-varying impulses have been modeled. A set of sampled-data synchronization controllers is designed by utilizing both the Gronwall’s inequality and the Jenson integral inequality in Shen et al. (2012). Taking account into the consideration of discontinuous control approach, periodic intermittent controller (Gan, 2012) and intermittent controller (Hu et al., 2012) for synchronization of networks with time delays were investigated through different methods. Lu et al. (2010) first established a novel concept, namely average impulsive interval, for the synchronizability analysis of complex networks. By introducing intermittent linear state feedback, Huang et al. (2009) studied problems of cluster synchronization of linearly coupled complex networks and synchronization of delayed chaotic systems with parameter mismatches, respectively. Hybrid adaptive and impulsive control (Yang et al., 2012) was exploited for stochastic synchronization of complex networks. In addition, as known, certain complex networks display a scale-free feature, of which the connectivity distributions have the power-law form. The pinning scheme of the most highly connected nodes induced that a significant reduction in required controllers as compared to the traditional control scheme, see Wang and Chen (2002b). More general cases of pining control can be found in Yu et al. (2009), Yu et al. (2013). A single controller of pinning synchronization was designed, for example, without assuming symmetry, irreducibility, or linearity of the couplings, the authors (Chen et al., 2007) proved that a single controller can pin a coupled complex network to a homogeneous solution. By considering the average impulsive interval, some generic mean square stability criteria of synchronization control were derived in Lu et al. (2010). Obviously, these methods are effective for the synchronization control of different complex networks. In the practical network environment, information exchange of each node through communication channels often experiences link failures and transmission delay, which can be modeled by a sequence of switching interconnections with time delays as shown in Liu et al. (2011) and Olfati-Saber et al. (2007). Synchronization problems with respect to switched systems were studied by typical techniques, for instance, the multiple v -Lyapunov function method (Zhao et al., 2011). As aforementioned in Lu et al. (2010) and Yang et al. (2012), by average impulsive interval, it can be derived that a unified synchronization criterion of complex networks is less conservative no matter desynchronizing or synchronizing impulses. Likewise, by average dwell time (Vu & Morgansen, 2010; Zhang & Shi, 2009), the stability criterion of switched system is less conservative. If we can obtain the result by virtue of the relationship between average impulsive interval and average dwell time, it would be much less conservative than the previous results. The research undertaken so far in order to understand how synchronization phenomena are affected by impulsive control law and the topological substrate of interactions, especially, when complex networks are involving noisy disturbances. The main goal of this paper is to examine the effect of time delays in switching topologies exerting on synchronization through Lyapunov–Razumikhin
technique. Correspondingly, the major contributions in this paper are twofold. First, the relationship between average impulsive interval(AII) of control law and average dwell time(ADT) of switched topologies is established. Second, the average time approach presented in this paper offers attractive features that are potentially useful for controlling different categories of complex networks. The paper is organized as follows. We first introduce the basic mathematical descriptions of discrete complex networks that will be used henceforth. Next, we focus on the synchronization analysis of discrete complex networks through impulsive control in the sense of average dwell time. The effect of time delays is tackled by Lyapunov–Razumikhin technique. Section 4 is devoted to the analysis of the conditions for the synchronization of complex networks in the sense of average interval time. The relationship is established by means of the results in Section 3. In Section 5, we present a numerical example and the discussion on the tradeoff between AII and ADT is given. Finally, the last section rounds off the paper by giving our conclusions. 2. Preliminary Let Rd denote the d-dimensional Euclidean space and ∥ • ∥ be the Euclidean norm in Rd . Denote Z+ = {1, 2, . . .}, Zτ = {−τ , −τ + 1, −τ + 2, . . . , −1, 0}, the finite set H = {1, 2, . . . , H } with finite positive integer H, N = {0, 1, 2, . . .}, I be the identity matrix, and matrix X > 0 ( 0 is the inner coupling matrix between two connected nodes. τ (k) is a nonnegative constant which represents the time-delay of the signal transmitted from the network to the ith node, where the coupling time-delay τ (k) satisfies 0 ≤ τ (k) ≤ τ for a constant τ > 0. gi : Z × Rd → Rd is the noise intensity function vector. ω(k) is a scale Wiener process (Brownian motion) defined on (Ω , F , P ) with T
E{ω(k)} = 0, E{ω(i)ω(j)} = 0,
Nd
E{ω(k)2 } = 1,
(i ̸= j).
(2)
C. Li et al. / Neural Networks 54 (2014) 85–94
87
Let G = ({1, 2, . . . , N }, E ) be an undirected graph with N vertices, describing the topology of the network. The Graph Laplacian matrix LG associated with G is defined as LG = ∆G − AG , where ∆G is the degree matrix and AG is the adjacency matrix of the graph. The Laplacian matrix is symmetric, positive semi-definite and has 0 as an eigenvalue with eigenvector 1. More specifically, G is connected if and only if rank(LG ) = N − 1 and λ2 (LG ) serves the function as the determinant contributor to the algebraic connectivity of G. For networks with switching connected topologies, let G be the finite set corresponding to the connected undirected graphs with vertices {1, 2, . . . , N }, such that
G = {Gh = ({1, 2, . . . , N }, Eh )|h ∈ H }. Each Gh ∈ G is connected, undirected with different connections of edge. Let IG ⊂ N be an index set associated with the elements of G. A switching signal σ is a map σ : Z+ → IG . For each time k ∈ Z+ , the switching signal σ establishes the network graph Gσ (k) ∈ G utilized by the network agents. The logical rules that generate the switching signals constitute the switching logic, and the index σ (k) is called the active mode at the time instant k. Meanwhile, let km,x (m ∈ N ) be the switching instants to label each switching phenomenon, i.e., when σ (km,x − 1) = 1, the active mode is Mode 1; when σ (km,x ) = 2, the active mode is replaced by Mode 2. Let k0,x = 0 be the starting mode. Note that there is no-chattering requirement in the discrete-time switched system due to the finite state changes in each impulsive interval, see Zhang and Shi (2009). Lσ (k) = (lij,σ (k) )N ×N is the coupling configuration matrix representing the coupling topology of the dynamical network at the active mode σ (k). The element lij,σ (k) is defined as the connectivity of the whole complex network, in which if there is a connection between subnetwork i and subnetwork j (j ̸= i) then lij,σ (k) = lji,σ (k) = 1; otherwise, lij,σ (k) = lji,σ (k) = 0 (j ̸= i). Let the diagonal element lii,σ (k) be defined as lii,σ (k) = −
N
lij,σ (k) = −
N
lji,σ (k) ,
(3)
i=1,i̸=j
j=1,j̸=i
where 1 ≤ i, j ≤ N. The initial conditions associated with model (1) are given by xi (θ) = φi (θ ),
θ ∈ Zτ ,
(4)
where φ(θ ) ∈ L2F0 ([−τ , 0], Rd ) and the norm of φ(θ ) is defined by ∥φ(θ)∥τ = supθ∈Zτ {∥φ(θ )∥}. In order to design an impulsive control scheme to synchronize system (1), we consider the evolutionary state abruptly jumps at every impulsive instant of time kn,u from its open-loop state, which can be formularized by xi (kn,u ) = Jn xi (kn,u − 1),
n∈N
(5)
where xi (kn,u − 1) stands for the primal state at time instant kn,u without impulsive jump (see Zhang, Sun, & Feng, 2009). As usual, every impulsive instant of time kn,u satisfies 0 = k0,u < k1,u < k2,u < · · · < kn,u < kn+1,u < · · · and limn→∞ kn,u = ∞; Jn ∈ Rd×d represents the impulsive jump strength at time instant kn,u . Therefore, at each impulsive instant of time kn,u , the coupled states xi (k) and xj (k) can be described by xi (kn,u ) − xj (kn,u ) = Jn [xi (kn,u − 1) − xj (kn,u − 1)].
(6)
∞
δ(k − kn,u )Jn (xi (kn,u − 1)),
n ∈ N,
As shown in Fig. 1, the top picture represents the state evolution of discrete system and the bottom one denotes the switching signals among three different modes. In addition, the state switching occurs at each time instant ki,x (i = 1, 2, 3, . . . , 19) while impulsive control is deployed at time instant kj,u (j = 1, 2, 3, . . . , 8). Eventually, by means of impulsive control, system will converge to its equilibrium point. Remark 1. It is worthwhile mentioning here that every time instant of impulsive control is unnecessarily equal to the time instant of switching signal, i.e., kn,u ̸= km,x . Due to the existence of time delays in the dynamical system, it is difficult to design feedback controller for the switched systems in the presence of both state delays and switching delays (see Vu & Morgansen, 2010). When the information available to the controller is a delayed version of the desired one in a network environment, the closed loop will feature both switching signals and delayed states. However, the impulsive control law is asynchronous to switching signals, which is consummate even the existence of large time delays. Therefore, impulsive control is a better choice for this circumstance, in which information exchange experiences both link failures and time delays. The equivalent matrix form of impulsive controlled system is given by x(k + 1) = F (k, x(k)) + (Lσ (k) ⊗ Γ )x(k − τ (k)) + G(k, x(k))ω(k); k ̸= kn,u , k ∈ Z+ ; x(kn,u ) = [IN ⊗ Jn ]x(kn,u − 1),
(8)
where Lσ (k) is a configuration matrix defining the topology of each sub-state. The compact forms of each element are x(k) = [xT1 (k)
xT2 (k)
...
xTN (k)]T ,
F (k, x(k)) = [f T (k, x1 (k))
. . . f T (k, xN (k))]T , x(k − τ (k)) = [ ( − τ (k)) . . . xTN (k − τ (k))]T , xT1 k g1T k x1
G(k, x(k)) = [
( , (k))
...
gNT (k, xN (k))]T .
Assumption 1. For each nonlinear function f (·, ·), suppose that there exist constant fˆ satisfies
[f (k, x1 ) − f (k, x2 )]T [f (k, x1 ) − f (k, x2 )] ≤ fˆ [x1 (k) − x2 (k)]T [x1 (k) − x2 (k)], where any x1 , x2 ∈ Rd .
Intuitively, a family of impulsive controller can be designed as Ui (k, xi (k)) =
Fig. 1. The trend in the evolution of controlled system with switching signals.
(7)
n=1
where Ui (k, xi (k)) represents a class of impulsive controller at each instant of time kn,u ; δ(•) denotes the Dirac discrete-time function.
Assumption 2. For each noisy intensity function g (·, ·), suppose that there exists a matrix G ∈ Rd×d such that g T (k, xi (k))g (k, xi (k)) ≤ ∥Gxi (k)∥2 , for any xi ∈ Rd .
i = 1, 2, . . . , N ,
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C. Li et al. / Neural Networks 54 (2014) 85–94
Assumption 3. Without loss of generality, let Mn be the subset of H (Mn ⊆ H ) and represent sub-states appearing on impulsive interval k ∈ [kn,u , kn+1,u − 1] (n ∈ N ). Different impulsive interval has its own subset Mn .
Theorem 1. Under Assumptions (A1) and (A2), the dynamical system (8) is globally exponentially synchronized in the mean square if there exist positive scalars εn > 0, αn > 0, βn > 1 and µ ≥ 1 on each time interval k ∈ [kn,u , kn+1,u − 1] satisfying:
For the simplicity of calculation, we denote F = diag{fˆ , fˆ , . . . ,
I. for each sub-state k ∈ [km,x , km+1,x − 1] and certain positive τ ⌉, there exist positive definite integer nτ = ⌈ inf {k −k }
fˆ } and G = diag{ˆg , gˆ , . . . , gˆ }. Let
m∈N
xij (k) = xi (k) − xj (k), xij (k − τ ) = xi (k − τ ) − xj (k − τ ), Fij (k, x(k)) = Fi (k, x(k)) − Fj (k, x(k)), (2)
Lσ (k) WLσ (k) = Nlij,σ (k) ,
WLσ (k) = Nlij,σ (k) .
Now, we recall some useful related definitions as follow. Definition 1 (see Hespanha & Morse, 1999). For switching signal σ and any K > k > k0 , let Nσ (K , k) be the switching numbers of σ over the interval [k, K ). If for any given K and k (K > k > k0 ), we have Nσ (K , k) ≤
K −k
Ta
Ta
m,x
n ∈ N,
i, j ∈ H ,
where
αn = max
σ (k)∈Mn
λmax (Πσ (k) ) + µβn λmax (Ωσ (k) ) , λmin (Pσ (k) )
Πσ (k) = F T Pσ (k) F + GT Pσ (k) G − Nlij,σ (k) F T Qσ−(1k) F , (2)
Ωσ (k) = −Nlij,σ (k) Γ T Pσ (k) Γ − Nlij,σ (k) Γ T PσT (k) Qσ (k) Pσ (k) Γ ; II. it is given that each impulsive controller requires )(kn+1,u −kn,u )
with the average dwell time ln µ , γ∗ − ε
where γ ∗ = maxn∈N
(Jn ) . − 2knln+1λ,max u −kn,u
(ε+ lnTaµ )(2kn+1,u −kn,u −kn−nτ +1,u )
III. βn ≥ e
, where ε = maxn∈N εn .
Proof. Consider the following Lyapunov function: (9)
for any k ∈ [kn,u , kn,u − 1], n ∈ N , where
Definition 3. The impulsive controlled discrete complex networks (8) are said to be globally exponentially synchronized in the mean square, if for any initial condition φ(θ ) ∈ L2F0 ([−τ , 0], Rd ), there exist two positive constants λ and M0 ≥ 1 such that
E ∥xi (k) − xj (k)∥2 ≤ M0 e−γ (k−k0 ) ,
1≤i≤j≤N
holds for all k > k0 .
−1 . . . −1 −1 N − 1 . . . −1 ... . . . . . . . . . . It is given that, for −1 −1 . . . N − 1
N − 1 W =
θ ∈ [−τ , −1], we
have V (θ ) = xT (θ )(W ⊗ Pσ (θ ) )x(θ )
= −
N −1 N
wij (φi (θ ) − φj (θ ))T
i =1 j =i +1
Pσ (θ ) (φi (θ ) − φj (θ ))
Lemma 1. Let W = (wij )N ×N , P ∈ Rd×d , x = (xT1 , xT2 , . . . , xTN )T and y = (yT1 , yT2 , . . . , yTN )T with xk , yk ∈ Rd (k = 1, 2, . . . , N). If W = W T and each row sum of W is zero, then N −1 N
Ta ≥ Ta∗ =
V (k) = xT (k)(W ⊗ Pσ (k) )x(k),
+ Nζ0 ,
where Ta is called average impulsive interval.
xT (W ⊗ P )y = −
Qi ≤ µQj ,
ln αn ≤ εn ,
ln µ
Definition 2 (see Lu et al., 2010). The average impulsive interval of the impulsive sequence ζ = {t1 , t2 , ···} is less than Ta , if there exist a positive integer Nζ0 and a positive number Nζ (K , k) associated with the number of impulsive times of impulsive sequence ζ over the interval [k, K ), such that K −k
P i ≤ µP j ,
λ2max (Jn ) ≤ e−(ε+ Ta
+ Nσ0 ,
then Ta and Nσ0 are called average dwell time and the chatter bound, respectively.
Nζ (K , k) ≤
m+1,x
matrices Pi ∈ Rd×d and Qi ∈ Rd×d , such that
wij (xi − xj )T P (yi − yj ).
i=1 j=i+1
≤ λmax (Pσ (θ ) )
N −1 N
(−wij )∥φi (θ ) − φj (θ )∥2 .
i=1 j=i+1
Let ϕ(∥φi (θ ) − φj (θ )∥2 ) = λmax (Pσ (θ ) )∥φi (θ ) − φj (θ )∥2 , thus V (θ ) ≤
N −1 N
ϕ(∥φi (θ ) − φj (θ )∥2 )
i=1 j=i+1
3. Main results
. = ϕ(∥φ∥2τ ).
(10)
Choose M ≥ 1 and γ > 0, such that
3.1. Global exponential synchronizability
1 ≤ Me−γ (k1,u −k0,u ) e−ε0 (k1,u −k0,u )−Nσ (k1,u ,k0,u ) ln µ .
In this section, we consider the global exponential synchronization problem of discrete delayed systems with switching topology by employing Lyapunov–Razumikhin Technique. Specifically, by considering Lyapunov-like functions allowed to increase with the bounded increase rate on each impulsive interval, the criteria of global exponential synchronization are developed in terms of average dwell time (ADT) which has been widely used for the stability analysis of the switched system.
We claim that E{V (k)} ≤ M ϕ(∥φ∥2τ )e−γ (kn,u −k0,u ) ,
k ∈ [kn,u , kn+1,u − 1], n ∈ N .
(11)
Mathematical induction will be utilized to prove the result in two steps. In the first step, the given statement (11) is to prove for the first interval which is on k ∈ [k0,u , k1,u − 1] for n = 1. The inductive step is to prove that (11) for any interval k ∈ [kn−1,u , kn,u − 1]
C. Li et al. / Neural Networks 54 (2014) 85–94
implies the given statement for the next interval. Meanwhile, proof by contradiction is also applied in each step where we assume that the given statement of each step is not a truly proposition and show that there is a contradiction to this assumption. We first show that E{V (k)} ≤ M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) ,
k ∈ [k0,u , k1,u − 1].
(12)
89
+ G(k, x(k))ω(k)] − xT (k)(W ⊗ Pσ (k) )x(k)} = E{F T (k, x(k))(W ⊗ Pσ (k+1) )F (k, x(k)) + xT (k − τ )(LTσ (k) ⊗ Γ T )(W ⊗ Pσ (k+1) ) × (Lσ (k) ⊗ Γ )x(k − τ ) + 2F T (k, x(k)) × (W ⊗ Pσ (k+1) )(Lσ (k) ⊗ Γ )x(k − τ )
Obviously, for k ∈ [k0,u − τ , k1,u − 1],
× GT (k, x(k))(W ⊗ Pσ (k+1) )G(k, x(k))
E{V (k)} ≤ ϕ(∥φ∥2τ )
− xT (k)(W ⊗ Pσ (k) )x(k)}. −γ (k1,u −k0,u ) −ε0 (k1,u −k0,u ) −Nσ (k1,u −k0,u ) ln µ
≤ M ϕ(∥φ∥τ )e
e
< M ϕ(∥φ∥τ )e
e
2
2
e
By further computation, (20) is given by
−γ (k1,u −k0,u ) −ε0 (k1,u −k0,u ) Nσ (k+1,k)
< M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) .
(13)
If (12) is not true, consider the case that no switching appears on this interval, then there exists an instant of time k¯ ∈ [k0,u , k1,u − 1],
E{V (k + 1)} ≤ µ
k¯ = min{k ∈ [km,x , km+1,x − 1]|E{V (k)}
E
xTij (k)F T
× Pσ (k) F xij (k) +
N −1 N
xTij (k)GT Pσ (k) Gxij (k)
i=1 j=i+1
−
N −1 N
(2)
xTij (k − τ )Nlij,σ (k) Γ T Pσ (k) Γ
i=1 j=i+1
× xij (k − τ ) −
N −1 N
xTij (k)Nlij,σ (k) F T
i=1 j=i+1
> M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) }
× Qσ−(1k) F xij (k) −
such that
N −1 N
xTij (k − τ )N
i=1 j=i+1
E{V (k¯ )} > M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u )
× lij,σ (k) Γ Pσ (k) Qσ (k) Pσ (k) Γ xij (k − τ )
> M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) e−ε0 (k1,u −k0,u )
T
> M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) e−ε0 (k1,u −k0,u ) e−Nσ (k1,u ,k0,u ) ln µ > ϕ(∥φ∥2τ )
(14)
≤ µNσ (k+1,k) E
T
N −1 N
xTij (k)[F T Pσ (k) F
i=1 j=i+1
and E{V (k)} ≤ M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) ,
k ∈ [k0,u − τ , k¯ − 1].
(15)
Consider
+ GT Pσ (k) G − Nlij,σ (k) F T Qσ−(1k) F ] × xij (k) + xTij (k − τ )[−Nl(ij2,σ) (k) Γ T Pσ (k) Γ
k = max{k ∈ [k0,u , k¯ ]|E{V (k)} ≤ M ϕ(∥φ∥2τ )
− Nlij,σ (k) Γ Pσ (k) Qσ (k) Pσ (k) Γ ]xij (k − τ )
∗
×e
N −1 N i=1 j=i+1
k¯ = min{k ∈ [k0,x , k1,u − 1]|E{V (k)} > M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) }. Or the case that certain state-switchings occur, then there exists an instant of time k¯ ∈ [km,x , km+1,x − 1] belonging to the time interval [k0,u , k1,u − 1] satisfying
(20)
−γ (k1,u −k0,u ) −ε0 (k1,u −k0,u ) −Nσ (k1,u ,k0,u ) ln µ
e
e
T
},
(16)
(17)
(18)
Obviously, state-switching would occur more than one time on each of impulsive intervals. Due to the fact that σ (k) ∈ M0 on the interval [k0,u , k1,u − 1] and Pσ (k) ≤ µPσ (k−τ ) , it is obtained that
from which we have E{V (k)} > M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) e−ε0 (k1,u −k0,u )
k ∈ [k∗ + 1, k¯ − 1].
Therefore, we can conclude that, for k ∈ [k∗ , k¯ ], one gives E{V (k∗ )} ≤ E{V (k)} ≤ E{V (k¯ )},
E{V (k + s)} ≤ M ϕ(∥φ∥τ )e
E{V (k + 1)} ≤ µNσ (k+1,k)
−γ (k1,u −k0,u )
≤ eε0 (k1,u −k0,u ) eNσ (k1,u ,k0,u ) ln µ E{V (k∗ )} ln µ
≤ e(ε+ Ta )(k1,u −k0,u ) E{V (k∗ )} ≤ β1 E{V (k)}.
(19)
Consider the increment of V (k) along the solution of discrete complex networks (1) in the interval k ∈ [k∗ , k¯ − 1]. For each active instant of sub-system k ∈ [k∗m,x , k∗m+1,x − 1], E{∆V (k)} = E{V (k + 1)} − E{V (k)}
= E{[F (k, x(k)) + (Lσ (k) ⊗ Γ )x(k − τ ) + G(k, x(k))ω(k)]T (W ⊗ Pσ (k+1) ) × [F (k, x(k))(Lσ (k) ⊗ Γ )x(k − τ )
λmax (Πσ (k) ) λmin (Pσ (k) ) λmax (Ωσ (k) ) + β0 −1 E{V (k)} µ λmin (Pσ (k) )
and, taking time delay s ∈ Zτ into account, one has, 2
λmax (Πσ (k) ) E{V (k)} λmin (Pσ (k) ) λmax (Ωσ (k) ) + E{V (k − τ )} . λmin (Pσ (k−τ (k)) )
≤ µNσ (k+1,k)
× e−Nσ (k1,u ,k0,u ) ln µ ,
T
≤ µNσ (k+1,k) α0 E{V (k)}. Therefore, we have ¯ ∗ (k¯ −k∗ ) E{V (k¯ )} ≤ µNσ (k,k ) α0 E{V (k∗ )} (k
−k0,u )
E{V (k∗ )}
(k
−k0,u )
M ϕ(∥φ∥2τ )
< µNσ (k1,u ,k0,u ) α0 1,u ≤ µNσ (k1,u ,k0,u ) α0 1,u
× e−γ (k1,u −k0,u ) e−ε0 (k1,u −k0,u ) e−Nσ (k1,u ,k0,u ) ln µ
(21)
90
C. Li et al. / Neural Networks 54 (2014) 85–94
= M ϕ(∥φ∥2τ )e−γ (k1,u −k0,u ) < E{V (k¯ )},
=
which is a contradiction. Thus, (12) is true. Next, we claim (11) is true for n = 0, 1, 2, . . . , n¯ , such that k ∈ [kn−1,u , kn,u − 1].
k ∈ [kn¯ ,u , kn¯ +1,u − 1].
E{V (kn¯ ,u )} = E
×e
(k
{xTij (kn¯ ,u − 1)
< M ϕ(∥φ∥2τ )e−γ (kn¯ +1,u −k0,u ) < E{V (k˜ )},
≤ λ2max (Jn¯ )M ϕ(∥φ∥2τ )e−γ (kn¯ ,u −k0,u )
≤ M ϕ(∥φ∥2τ )e−γ (kn¯ +1,u −k0,u ) e−εn¯ (kn¯ +1,u −kn¯ ,u ) .
(25)
¯ > n¯ such that If (24) is not true, then there exists an integer m k˜ ∈ [km¯ ,x , km¯ +1,x − 1] on interval k ∈ [kn¯ ,u , kn¯ +1,u − 1] satisfying
min {λmin (Pσ (k) )}E
σ (k)∈H
≤E
k˜ = min{k ∈ [km¯ ,x , km¯ +1,x − 1]|E{V (k)} ≤ M ϕ(∥φ∥2τ )
E
k∗∗ = max{k ∈ [km∗∗ ,x , km∗∗ +1,x ]|E{V (k)} ≤ M ϕ(∥φ∥2τ )
N −1 N
(27)
(xi (k) − xj (k)) Pσ (k) (xi (k) − xj (k)) T
(33)
∥xi (k) − xj (k)∥
2 2
M
≤ min
For any k ∈ [k∗∗ + 1, k˜ ], we have
σ (k)∈H
E{V (k)} > M ϕ(∥φ∥2τ )e−γ (kn¯ +1,u −k0,u ) e−εn¯ (kn¯ +1,u −kn¯ ,u )
ϕ(∥φ∥2τ )e−γ (k−k0,u ) λmin (Pσ (k) )
(34)
which implies that
.
(28)
In addition, for any k ∈ [k∗∗ , k˜ ], we have E V (k∗∗ ) ≤ E{V (k)} ≤ E{V (k˜ )}.
(29)
Then, for any k ∈ [km¯ ,x , km¯ +1,x − 1], E {V (k + 1)} ≤ µNσ (k+1,k) E {V (k)}
λmin (Pσ (km¯ ,x ) ) (30)
In light of Condition (III), it observes that for any s ∈ Zτ , one has k + s ∈ [kn¯ −nτ ,u , k˜ − 1]. Then, for any k ∈ [k∗∗ , k˜ − 1], we have
≤ M ϕ(∥φ∥2τ )e−γ (kn¯ +1,u −k0,u )
N −1 N
i=1 j=i+1
× e−γ (kn¯ +1,u −k0,u ) e−εn¯ (kn¯ +1,u −kn¯ ,u ) e−Nσ (kn¯ +1,u ,kn¯ ,u ) ln µ }.
E{V (k + s)} ≤ M ϕ(∥φ∥2τ )e−γ (kn¯ −nτ +1,u −k0,u )
∥xi (k) − xj (k)∥
i=1 j=i+1
Therefore, for any k ∈ Z+ , we have
Denote
≤ µNσ (k+1,k) αn¯ E {V (k)} .
2 2
≤ M ϕ(∥φ∥2τ )e−γ (k−k0,u ) .
(26)
Consequently, we have k˜ > km¯ ,x and V (k˜ ) > V (kn¯ ,u ). Also there ¯ exists another integer m∗∗ such that m∗∗ ≤ m.
×
N −1 N
i=1 j=i+1
× e−γ (kn¯ +1,u −k0,u ) }.
λmax (Πσ (km¯ ,x ) ) + µβn¯ λmax (Ωσ (km¯ ,x ) )
(32)
which is a contradiction. Hence, (24) holds for k = k˜ + 1. And by virtue of mathematical induction method, the claim (11) is true for each k ∈ Z+ . In view of Lemma 1, it can be obtained that, for any k ∈ Z+ ,
= λ2max (Jn¯ )E{V (kn¯ ,u − 1)}
−kn¯ ,u )
× e−ε(kn¯ +1,u −kn¯ ,u ) e−Nσ (kn¯ +1,u ,kn¯ ,u ) ln µ × M ϕ(∥φ∥2τ )e−γ (kn¯ +1,u −k0,u )
× Pσ (kn¯ ,u −1) xij (kn¯ ,u − 1)}
(31)
< µNσ (kn¯ +1,u −kn¯ ,u ) ln µ αn¯ n¯ +1,u
×e
E{V (k∗∗ + 1)}
˜ ∗∗ (k˜ −k∗∗ ) E{V (k˜ )} ≤ µNσ (k,k ) pn¯ ,u E{V (k∗∗ )}
i=1 j=i+1
−Nσ (kn¯ +1,u ,kn¯ ,u ) ln µ
i=0
0, ε− < 0, αn > 0, βn > 1 and µ ≥ 1 on each time interval k ∈ [kn,u , kn+1,u − 1] satisfying I. For each sub-state k ∈ [km,x , km+1,x − 1] and certain positive τ ⌉, there exist matrices Pi ∈ Rd×d , integer nτ = ⌈ inf m∈N {km+1,x −km,x } and Qi ∈ Rd×d , such that P i ≤ µP j , Qi ≤ µQj , i, j ∈ H , ε+ = max{ln αn } if αn > 1, n∈N
≤ ϕ(∥φ∥2τ ).
E {V (k + 1)} ≤ eε− T− (k,km,x )+ε+ T+ (k,km,x ) E V (km,x ) = eε− T− (k,km,x )+ε− T+ (k,km,x ) × e−ε− T+ (k,km,x )+ε+ T+ (k,km,x ) E V (km,x ) ≤ e(ε+ −ε− )Tmax eε− (k−km,x ) E V (km,x ) ,
αn = max
σ (k)∈Mn
λmax (Πσ (k) ) + µβn λmax (Ωσ (k) ) , λmin (Pσ (k) )
Πσ (k) = F T Pσ (k) F + GT Pσ (k) G − Nlij,σ (k) F T Qσ−(1k) F , (2)
Ωσ (k) = −Nlij,σ (k) Γ T Pσ (k) Γ − Nlij,σ (k) Γ T PσT (k) Qσ (k) Pσ (k) Γ . II. The relationship between average impulsive interval and average dwell time requires
+ ε− +
ln µ + (ε+ − ε− )Tmax
2 max
Ta
