1. Introduction
It is well known that many practical systems can be described as complex networks such as Internet networks, biological networks, epidemic spreading networks, collaborative networks, social networks and neural networks [1–4]. Thus, during the past decades, the research on the dynamics of complex dynamical networks (CDNs) has attracted extensive attention of scientific and engineering researchers in all fields domestic and overseas since the pioneering work of Watts and Strogatz [5]. As one of the most significant and important collective behaviors in CDNs, synchronization has received more attention; please refer to [6–10] and references therein for more details.
Since time delay inevitably exists and has become an important issue in studying the CDNs, synchronization problems for complex networks with time delays have gained increasing research attention, and considerable progress has been made; see, for example, [6–17] and references therein for more details. However, in some practical applications of communication networks, signal transmission channels often involve network-induced delays, packet dropouts, bit errors, environment disturbances, and so on, which will cause the error of transmission from one system to another. Then past change rate of the state variables affects the dynamics of nodes in the networks. This kind of complex dynamical network is termed as neutral complex dynamical network (NCDN), which contains delays both in its states and the derivatives of its states. There are some results about the synchronization design problem for neutral systems [18–23]. In these works, [19, 20] had studied the synchronization control for a kind of master-response setup and were further extended to the case of neutral-type neural networks with stochastic perturbation. The authors of [18, 22] had researched the synchronization problem for a class of complex networks with neutral-type coupling delays. The authors in [21] had studied the global asymptotic stability of neural networks of neutral type with mixed delays, which include constant delay in the leakage term, time-varying delays, and continuously distributed delays. The authors in [23] had investigated the robust global exponential synchronization problem for an array of neutral-type neural networks. However, much fewer results have been proposed for neutral complex dynamical networks (NCDNs) compared with the rich results for CDNs with only discrete delays.
On the other hand, network mode switching is also a universal phenomenon in CDNs of the actual systems, and sometimes the network has finite modes that switch from one to another with certain transition rate; then such switching can be governed by a Markovian chain. The stability and synchronization problems of complex networks and neural networks with Markovian jump parameters and delays are investigated in [16, 24–30], and references therein. The authors in [24] have established sufficient global exponential stability conditions on Markovian jump neural networks with impulse control and time varying delays. The authors in [27] have studied synchronization in an array of coupled neural networks with Markovian jumping and random coupling strength. Particularly, Ma et al. [25] have considered the stability and synchronization problems for Markovian jump delayed neural networks with partly unknown transition probabilities, which have not been fully investigated and need to propose more good results. Besides, sampled-data systems have attracted great attention because the digital signal processing methods require better reliability, accuracy, and stable performance with the rapid development in digital measurement and intelligent instrument. There are some important and essential results which have been reported in the literature [31–35]. What is worth mentioning is that the sampled-data synchronization control problem has been investigated for a class of general complex networks with time-varying coupling delays in [34, 35], where conditions have been presented to ensure the exponential stability of the closed-loop error system, and the desired sampled data feedback controllers have been designed. However, few results are available for neutral complex dynamical networks (NCDNs) with sampled data. To the best of the authors' knowledge, the NCDNs are difficult to treat and they are very challenging, especially in the presence of Markovian jump parameters, mode-dependent time-varying delays, and sampled data. Motivated by the above analysis, the exponential synchronization and sampled-data controller problem for a class of NCDNs with Markovian jump parameters and mode-dependent time-varying delay is investigated in this paper. The addressed NCDNs consist of M modes, and the networks switch from one mode to another according to a Markovian chain with partially known transition rate.
In this paper, the synchronization and sampled-data controller problem is studied for NCDNs with Markovian jump parameters and partially known transition rates. The sampling period considered here is assumed to be time varying and bounded, while the neutral and discrete delays are interval mode-dependent and time varying. Firstly, by constructing a new augmented stochastic Lyapunov functional, exponential stability conditions are derived based on the Lyapunov stability theory and reciprocally convex lemma. Then the design method of the desired sampled-data controllers is solved on the basis of the obtained conditions. Moreover, all the derived results are in terms of LMIs that can be solved numerically, which are proved to be less conservative than the existing results.
The remainder of the paper is organized as follows. Section 2 presents the problem and preliminaries. Section 3 gives the main results, which are then verified by numerical examples in Section 4. Section 5 concludes the paper.
Notations. The following notations are used throughout the paper. ℝn denotes the n dimensional Euclidean space and ℝm×n is the set of all m×n matrices. X<Y (X>Y), where X and Y are both symmetric matrices, which means that X-Y is negative (positive) definite. I is the identity matrix with proper dimensions. For a symmetric block matrix, we use * to denote the terms introduced by symmetry. ℰ stands for the mathematical expectation; ∥v∥ is the Euclidean norm of vector v, ∥v∥=(vTv)1/2, while ∥A∥ is spectral norm of matrix A, ∥A∥=[λmax(ATA)]1/2. λmax(min)(A) is the eigenvalue of matrix A with maximum (minimum) real part. The Kronecker product of matrices P∈ℝm×n and Q∈ℝp×q is a matrix in ℝmp×nq which is denoted as P⊗Q. Let ς>0 and C([-ς,0],ℝn) denote the family of continuous function φ, from [-ς,0] to ℝn with the norm |φ|=sup-ς≤θ≤0∥φ(θ)∥. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.
2. Problem Statement and Preliminaries
Given a complete probability space {Ω,ℱ,{ℱt}t≥0,P} with a natural filtration {ℱt}t≥0 satisfying the usual conditions, where Ω is the sample space, ℱ is the algebra of events and P is the probability measure defined on ℱ. Let {r(t),t≥0} be a right-continuous Markov chain taking values in a finite state space S={1,2,3,…,M} with a generator Υ=(γij)M×M, i,j∈S, which is given by
(1)P(r(t+Δt)=j∣r(t)=i)={γijΔt+o(Δt)i≠j,1+γiiΔt+o(Δt)i=j,
where Δt>0, limΔt→0(o(Δt)/Δt)=0, and γij≥0 (i,j∈S,i≠j) is the transition rate from mode i to j, and for any state or mode i∈S, it satisfies
(2)γii=-∑j=1,j≠iNγij.
It is assumed that r(t) is irreducible and available at time t, but the transition rates of the Markov chain are partially known in this paper, which means that some elements in matrix Υ=(γij)M×M are inaccessible. For instance, in the system with six operation modes, the jump rates matrix Υ may be viewed as
(3)[γ110γ12γ130γ140γ15γ160γ21γ220γ230γ24γ250γ26γ310γ320γ33γ34γ350γ360γ410γ42γ430γ440γ450γ46γ51γ520γ530γ540γ55γ560γ61γ620γ630γ64γ65γ660],
where γij0,i,j∈S represents the unknown element. Furthermore, let γim, γiM be lower and upper bound for the diagonal element γii or γii0, for all i∈S. For notation clarity, we denote that 𝒮i=𝒮ki⋃𝒮uki, for all i∈S, and
(4)𝒮ki≜{j:πij is known for j∈S},𝒮uki≜{j:πij is unknown for j∈S}.
If 𝒮ki≠⌀, than it is further described as
(5)𝒮ki={k1i,k2i,…,kmi}, 1≤m≤M,
where kji,(j=1,2,…,m) represent the jth known element of the set 𝒮ki in the ith row of the transition rate matrix Υ. It should be noted that if 𝒮ki=∅, 𝒮i=𝒮uki which means that any information between the ith mode and the other M-1 modes is not accessible, then MJSs with M modes can be regarded as ones with M-1 modes.
The following neutral complex dynamical network (NCDN) consisting of N identical nodes with Markovian jump parameters and interval time-varying delays over the space {Ω,ℱ,{ℱt}t≥0,P} is investigated in this paper:
(6)x˙k(t)=∑l=1Ngkl(1)(r(t))A(r(t))xl(t)+∑l=1Ngkl(2)(r(t))B(r(t))xl(t-d(t,r(t)))+∑l=1Ngkl(3)(r(t))C(r(t))x˙l(t-τ(t,r(t)))+D(r(t))f(xk(t))+uk(t),
where xk(t)∈ℝn, uk(t)∈ℝn are state variable and the control input of the node k∈{1,2,…,N}, respectively. r(t) describes the evolution of the mode. A(r(t))∈ℝn×n, B(r(t))∈ℝn×n, and C(r(t))∈ℝn×n represent the inner-coupling matrices linking between the subsystems in mode r(t). G(1)(r(t))=[gkl(1)]N×N, G(2)(r(t))=[gkl(2)]N×N, and G(3)(r(t))=[gkl(3)]N×N are the coupling configuration matrices of the networks representing the coupling strength and the topological structure of the NCDNs in mode r(t), in which gkl(m) is defined as follows: if there exists a connection between kth and lth (k≠l) nodes, then gkl(m)(r(t))=glk(m)(r(t))=1; otherwise gkl(m)(r(t))=glk(m)(r(t))=0, and
(7)gkk(m)(r(t))=-∑l=1,l≠kNgkl(m)(r(t)) =-∑l=1,l≠kNglk(m)(r(t)), m=1,2,3; k=1,2,…,N.τ(t,r(t)) and d(t,r(t)) denote the mode-dependent time-varying neutral delay and retarded delay, respectively. They are assumed to satisfy
(8)0≤τi(t)≤τi¯, τ˙i(t)≤νi<1,0≤d1i≤di(t)≤d2i, maxi∈S{d1i}≤minj∈S{d2j}, d˙i(t)≤μi, when r(t)=i,
where τi¯, 0≤νi<1, d1i, and d2i, 0≤μi, are real constant scalars.
D
(
r
(
t
)
)
∈
ℝ
n
×
n
is a parametric matrix with real values in mode r(t). f: ℝn→ℝn is a continuous vector-valued nonlinear function. It is assumed to satisfy the following sector-bounded condition [36]:(9)[f(x)-f(y)-F1(x-y)]T ×[f(x) -f(y)-F2(x-y)] ≤0, ∀x,y∈ℝn,
where F1, F2 are two constant matrices. Such a description of nonlinear functions has been exploited in [37–39] and is more general than the commonly used Lipschitz conditions. The sector-bounded condition would be possible to reduce the conservatism of the main results caused by quantifying the nonlinear functions via a matrix inequality technique.
Let s(t)∈ℝn be the state trajectory of the unforced node s˙(t)=D(r(t))f(s(t)); then the synchronization error is defined to be ek(t)=xk(t)-s(t). So the error dynamics of NCDN (6) can be derived as follows:
(10)e˙k(t)=∑l=1Ngkl(1)(r(t))A(r(t))el(t)+∑l=1Ngkl(2)(r(t))B(r(t))el(t-d(t,r(t)))+∑l=1Ngkl(3)(r(t))C(r(t))e˙l(t-τ(t,r(t)))+D(r(t))fe(ek(t))+uk(t),
where fe(ek(t))=f(xk(t))-f(s(t)), k=1,2,…,N.
The control signal is assumed to be generated by using a zero-order-hold (ZOH) function with a sequence of hold times 0=t0<t1<⋯<ts<⋯. Thus the state feedback controller takes the following form:
(11)uk(t)=Kkek(ts), k=1,2,…,N, ts≤t<ts+1,
where Kk is sampled-data feedback controller gain matrix to be determined. ek(ts) is discrete measurement of ek(t) at the sampling instant ts satisfying lims→+∞ts=+∞. It is assumed that ts+1-ts=hs≤h¯ for any integer s≥0, where h¯>0 represents the largest sampling interval.
With (10) and (11), the error dynamics of NCDN (6) can be replaced as
(12)e˙k(t)=∑l=1Ngkl(1)(r(t))A(r(t))el(t)+∑l=1Ngkl(2)(r(t))B(r(t))el(t-d(t,r(t)))+∑l=1Ngkl(3)(r(t))C(r(t))e˙l(t-τ(t,r(t)))+D(r(t))fe(ek(t))+Kkek(t-h(t)),
where h(t)=t-ts and 0≤h(t)≤h¯.
For simplicity of notations, we denote A(r(t)), B(r(t)), C(r(t)), D(r(t)), G(m)(r(t)), (m=1,2,3) by Ai, Bi, Ci, Di, Gi(m), (m=1,2,3) for rt=i∈S. By utilizing the Kronecker product of the matrices, (12) can be written in a more compact form as
(13)e˙(t)=𝔸ie(t)+𝔹ie(t-di(t))+ℂie˙(t-τi(t))+𝔻iFe(e(t))+𝕂e(t-h(t)),
where 𝔸i=Gi(1)⊗Ai, 𝔹i=Gi(2)⊗Bi, ℂi=Gi(3)⊗Ci, 𝔻i=IN⊗Di, and
(14)e(t)=col{e1(t),e2(t),…,eN(t)},e(t-di(t))=col{e1(t-di(t)),e2(t-di(t)), …,eN(t-di(t))},e˙(t-τi(t))=col{e˙1(t-τi(t)),e˙2(t-τi(t)), …,e˙N(t-τi(t))},Fe(e(t))=col{{(eN(t))}fe(e1(t)), fe(e2(t)),…,fe(eN(t))},e(t-h(t))=col{e1(t-h(t)),e2(t-h(t)), …,eN(t-h(t))},𝕂=diag{K1,K2,…,KN}.
The purpose of this paper is to design a serial of sampled-data controllers (11) to ensure the exponential synchronization of NCDN (6). Before proceeding with the main results, we present the following definitions and lemmas.
Definition 1 (see [<xref ref-type="bibr" rid="B16">40</xref>]).
Define operator 𝔇: C([-ρ,0],ℝn)→ℝn as 𝔇(xt)=x(t)-Cx(t-τ). 𝔇 is said to be stable if the homogeneous difference equation
(15)𝔇(xt)=0, t≥0,x0=ψ∈{ϕ∈C([-ρ,0],ℝn):𝔇ϕ=0}
is uniformly asymptotically stable. In this paper, that is, ∥Gi(3)⊗Ci∥<1.
Definition 2 (see [<xref ref-type="bibr" rid="B31">41</xref>]).
Define the stochastic Lyapunov-Krasovskii function of the error system (13) as V(e(t),r(t)=i, t>0)=V(e(t),i,t) where its infinitesimal generator is defined as
(16)ΓV(e(t),i,t) =limΔt→01Δt[ℰ{V(e(t+Δt),r(t+Δt),t+Δt)∣e(t) =e,r(t)=i}-V(e(t),i,t)] =∂∂tV(e(t),i,t)+∂∂eV(e(t),i,t)e˙(t) +∑j=1NπijV(e(t),j,t).
Definition 3.
The NCDN (6) is said to be exponentially synchronized if the error system (13) is exponentially stable; that is, [42] there exist two constants ε>0 and κ≥1 such that for all, e(t),
(17)∥e(t)∥≤κexp{-ε(t-t0)}supθ∈[-ς,t0]∥e(θ)∥,
where ε is the exponential decay rate; ς=max{h¯,maxi∈S{τi¯},maxi∈S{d2i}}.
Lemma 4 (see [<xref ref-type="bibr" rid="B1">43</xref>]).
Given matrices A, B, C, and D with appropriate dimensions and scalar α, by the definition of the Kronecker product, the following properties hold:
(18)(αA)⊗B=A⊗(αB),(A+B)⊗C=A⊗C+B⊗C,(A⊗B)(C⊗D)=(AC)⊗(BD),(A⊗B)T=AT⊗BT.
Lemma 5.
For any constant matrix Q=QT>0, continuous functions 0≤h1(t)≤h2(t), constant scalars 0≤τ1<τ2, and constant ε>0 such that the following integrations are well defined:
(19)(a) exp{2εh2(t)}-exp{2εh1(t)}2ε ×∫t-h2(t)t-h1(t)exp{2ε(s-t)}xT(s)Qx(s)ds ≥[∫t-h2(t)t-h1(t)xT(s)ds]Q[∫t-h2(t)t-h1(t)x(s)ds],(b) τ22-τ122∫-τ2-τ1∫t+θtexp{2ε(s-t)}xT(s)Qx(s)ds dθ ≥exp{-2ετ2}[∫-τ2-τ1∫t+θtxT(s)ds dθ]Q ×[∫-τ2-τ1∫t+θtx(s)ds dθ].
Proof.
(a) is directly obtained from [44]. In addition, from -τ2≤θ≤-τ1 and t+θ≤s≤t, it is held that -τ2≤θ≤s-t≤0. Then,
(20)∫-τ2-τ1∫t+θtexp{2ε(s-t)}xT(s)Qx(s)ds dθ ≥exp{-2ετ2}∫-τ2-τ1∫t+θtxT(s)Qx(s)ds dθ;
(b) is thus true by [45].
Lemma 6 (see [<xref ref-type="bibr" rid="B23">46</xref>]).
For functions λ1(t), λ2(t)∈[0,1], λ1(t)+λ2(t)=1, and η1=0 with λ1(t)=0 and η2=0 with λ2(t)=0 and matrices P>0, Q>0, then there exists matrix T such that
(21)[PTTTQ]>0,
and the following inequality holds:
(22)1λ1(t)η1Pη1T+1λ2(t)η2Qη2T≥[η1η2][PTTTQ][η1Tη2T].
According to Definition 3, the aim will be achieved if we obtain the gain matrices 𝕂 such that the error system (13) is exponentially stable. So we give the main results as follows.