# Dissipativity-Based Synchronization of Mode-Dependent Complex Dynamical Networks with Semi-Markov Jump Topology

Authors: Chao Ma; Wei Wu; Yidao Ji
Journal: Complexity
 (2018)

---

## Abstract

This paper considers the synchronization control for mode-dependent complex dynamical networks (CDNs) based on dissipativity theory. Particularly, the network topologies of the CDNs are governed by the semi-Markov process. By applying the Lyapunov-Krasovskii method, mode-dependent sufficient synchronization criteria are established while satisfying the desired dissipativity performance. On the basis of matrix transformation, the synchronization controllers are further designed. The effectiveness of our obtained results is illustrated with simulation results.


---

## Body

## 1. Introduction

Over the last decade, complex dynamical networks (CDNs) have been significantly investigated because of their extensive applications in computer networks, social networks, biological networks [1–5], and other areas. As a ubiquitous and important feature, the synchronization of CDNs has become a hot research issue, and there are various methods on analysis and synthesis of synchronization problems [6–11]. Furthermore, investigations on the synchronization phenomenon can give the researchers an insight into the intrinsic properties and dynamic characteristics of CDNs. Note that, in practice, CDNs are always subject to disturbances, which would make the synchronization fail or degrade the synchronization performance to some extent. To make sure that the synchronization can be well achieved in the presence of disturbances, synchronization methods should be developed with the abilities to deal with the disturbances. Fortunately, it has been verified that H∞ technique is an effective and popular approach for disturbance attenuation. Therefore, fruitful results of H∞ synchronization of CDNs have been reported in the literature and the references therein [12–14].It is noteworthy that dissipativity theory can provide a powerful framework in system theory and engineering from the energy-related perspective. Dissipativity theory introduces the system input and output descriptions and claims that it is a more general case of theH∞ and the passivity performances [15, 16]. Particularly, it has been proven that dissipativity is with great capability of disturbance attenuation. Under this circumstance, more flexible synchronization designs for CDNs can be obtained by utilizing the dissipativity. As a result, several attempts have been carried out on dissipativity-based synchronization issues, and significant achievements have been made [17–21].On the other hand, increasing attention has been paid to CDNs with a switching network topology, since the topology would change due to environmental abrupt or node failures [22, 23]. In particular, researchers have found that certain switching dynamics composed of a set of topologies can be described by Markov chains, which give rise to the research topics on CDNs with Markov topology [24, 25]. Recently, there has been some initial concern with the semi-Markov topology. Note the fact that the transition rates could be time varying and sojourn time cannot be exponentially distributed [26–28]. Semi-Markov topology can be considered as a general case of Markov topology in the real world. Naturally, it is meaningful to study the synchronization problem of CDNs with semi-Markov topology from the practical point of view. Furthermore, it is noted that the node dynamics of CDNs would be switching according to the mode of topology, such that mode-dependent CDNs models should be taken into account. However, so far, investigations on dissipativity-based synchronization of mode-dependent CDNs with semi-Markov jump topology have not been considered until now, which motivates this study.In this paper, we develop the synchronization model of CDNs with semi-Markov jump topology and mode-dependent nodes. More precisely, the topology is modulated by a continuous-time discrete-state semi-Markov process chain. Compared with the previous results, the main contributions of our paper are mainly threefold. Firstly, the mode-dependent model with semi-Markov jump topology and external disturbances is proposed to describe more realistic dynamics of CDNs in practical applications. Secondly, theQ,S,R dissipativity performance is adopted for dealing with the corresponding synchronization problem. Thirdly, a proper mode-dependent stochastic Lyapunov-Krasovskii functional is constructed, and sufficient synchronization conditions are derived. Synchronization controllers are further designed by LMIs such that the synchronization can be ensured with the desired dissipativity performance.The outline of our paper is stated by the following. In Section2, some preliminaries on the CDNs model are presented, and the dissipativity-based synchronization problem is formulated. The main theoretical results are given in Section 3. In Section 4, two numerical examples are provided for validating our synchronization method, and Section 5 concludes the paper with further remarks.Notation:
The notations throughout this paper are standard.ℝn and ℝm×n denote n dimensional Euclidean space and the set of all m×n matrices, respectively. ℒ20,∞ denotes the space of square-integrable vector functions over 0,∞. Ω,ℱ,P is a probability space, Ω is the sample space, ℱ is the σ-algebra of subsets of the sample space, and P is the probability measure on ℱ. E⋅ denotes the mathematical expectation of the stochastic process or vector. A−B≻0A−B≺0 denotes that A−B is positive definite (negative definite). A⊗B stands for the Kronecker product. ∗ is used as an ellipsis for the symmetry terms in symmetric block matrices, and diag⋯ denotes a block-diagonal matrix.


## 2. Preliminaries and Problem Formulation

Given a probability spaceΩ,ℱ,P, consider the following class of directed CDNs consisting of N identical nodes:
(1)ẋit=Aσtxit+Bσtxit−τt+Cσtfxit+∑j=1NGijσtΓσtxjt+uit+Dσtwit,i=1,2,…,N,where xit=xi1t,xi2t,…,xintT∈ℝn denotes the state vector of the ith node; ui∈ℝn and wi∈ℝq denote the control input and the disturbance input on the ith node, respectively; fxit∈ℝn is a smooth nonlinear function; τt is the time-varying delay; Aσt is a diagonal matrix; Bσt, Cσt, and Dσt are weight matrices; Γσt is the inner coupling matrix; and Gσt=Gijσt∈ℝN×N is the outer coupling matrix representing the directed network topology. If there is a directed coupling from node i to node ji≠j, then the coupling Gijσt≠0; otherwise, Gijσt=0. Moreover, Giiσt is defined by Giiσt=−∑j=1,j≠iNGijσt, i=1,2,…,N. Without loss of generality, the initial conditions are xit=ϕit, t∈−τ¯,0.σt,t≥0 denotes a continuous-time discrete-state semi-Markov process on Ω,ℱ,P taking values in a finite set ℐ=1,…,N. The transition probability matrix Θ≔παβh, h>0, ∀α,β∈ℐ is defined by
(2)Prσt+h=βσt=α=παβhh+oh,α≠β,1+πααhh+oh,α=β,with limoh/h=0. παβh≥0, α≠β, is the transition rate from mode α at time t to mode β at time t+h, satisfying πααh=−∑β=1,β≠αNπαβh, ∀α∈ℐ.Assumption 1.
For eachσt∈ℐ, the network topology keeps constant, and Aσt, Bσt, Cσt, Dσt, Gσt, and Γσt are known real constant matrices.Assumption 2.
The nonlinear functionfxit satisfies
(3)fxit−fxjt−F1xit−xjtTfxit−fxjt−F2xit−xjt≤0,∀xit,xjt∈ℝn, where F1 and F2 are constant matrices with F2−F1≥0.Assumption 3.
The time-varying delayτt satisfies 0<τt≤τ¯, where τ¯ is a positive constant.In this paper, the synchronization errors are defined as(4)eit≔xit−st,i=1,2,…,N,where st∈ℝn is the state trajectory of the unforced isolated node ṡt=Aσtst+Bσtst−τt+Cσtfst. Then, the synchronization error dynamics of the CDNs can be obtained as follows:
(5)ėit=Aσteit+Bσteit−τt+Cσtf~eit+∑j=1NGijσtΓσtejt+uit+Dσtwit,i=1,2,…,N,where f~eit≔fxit−fst.The following mode-dependent synchronization controller is designed:(6)uit=Kσteit,where Kσt is the controller gain matrix.Consequently, the closed-loop synchronization error dynamics can be derived by(7)ėit=Aσt+Kσteit+Bσteit−τt+Cσtf~eit+∑j=1NGijσtΓσtejt+Dσtwit,i=1,2,…,N,which can be further rewritten as
(8)ėt=IN⊗Aσt+Kσtet+IN⊗Bσtet−τt+IN⊗CσtFet+Gσt⊗Γσtet+IN⊗Dσtwt,where
(9)et≔e1Tt,e2Tt,…,eNTtT,Fet≔f~Te1t,f~Te2t,…,f~TeNtT,wt≔w1Tt,w2Tt,…,wNTtT.For notational simplicity, denoteσt by the index α, α∈ℐ. Then, one has
(10)ėt=IN⊗Aα+Kαet+IN⊗Bαet−τt+IN⊗CαFet+Gα⊗Γαet+IN⊗Dαwt.The following definition is given.Definition 1.
The mean-square stochasticallyQ,S,ℛ dissipative synchronization of the CDNs (1) is said to be achieved if there exist real symmetric matrices Q, ℛ, matrix S, and a scalar γ>0, such that for any t>0, the following condition holds with zero initial condition:
(11)Eet,Qett+2et,Swtt+wt,ℛwtt≥Eγwt,wtt,where et,Qett denotes ∫0teTsQesds, and the other symbols are similarly defined. Moreover, it is assumed that there exists a constant matrix Q−≻0, such that Q≤0 with −Q=Q−TQ−.Remark 1.
It can be found that theQ,S,ℛ dissipative synchronization problem is a more general case of H∞ and passivity synchronization by adjusting Q,S,ℛ matrices.Before proceeding further, the following lemma is introduced for subsequent analysis.Lemma 1.
[29] For any matrix ℳ≻0, scalars τ¯>0, τt satisfying 0≤τt≤τ¯, vector function ẋt:−τ¯,0⟶ℝn such that the concerned integrations are well defined, then
(12)−τ¯∫t−τ¯tẋTsℳẋsds≤ζTtΩζt,where
(13)ζt=xTt,xTt−τt,xTt−τ¯T,Ω=−ℳℳ0∗−2ℳℳ∗∗−ℳ.We aim to design a proper mode-dependent synchronization controller (6) for CDNs (1) to guarantee that mean-square stochastically Q,S,ℛ dissipative synchronization can be achieved.


## 3. Main Results

In this section, sufficient synchronization conditions are first derived. Then, the synchronization controllers are developed based on the established criteria.Theorem 1.
For given scalarsτ¯ and γ, the Q,S,ℛ dissipative synchronization of the CDNs (1) can be achieved with given mode-dependent synchronization controller gain Kα, if there exist mode-dependent matrix Pα≻0, matrices W1≻0 and W2≻0, such that Πα≺0 for each α∈ℐ, where
(14)Πα≔Πα1Πα2∗Πα3,with
(15)Πα1≔Πα11IN⊗PαBα+IN⊗W20∗−2IN⊗W2IN⊗W2∗∗−IN⊗W1−IN⊗W2,Πα11≔2IN⊗PαAα+IN⊗PαKα+Gα⊗PαΓα+IN⊗W1−IN⊗W2+∑β=1NπαβhIN⊗Pβ−F^1+Q−TQ−,Πα2≔IN⊗PαCα−F^2IN⊗PαDα−SΠα2200τ¯IN⊗BαTPα000,Πα22≔τ¯IN⊗AαTPα+IN⊗KαTPα+GαT⊗ΓαTPα,Πα3≔−I0τ¯IN⊗CαTPα∗−ℛ+γIτ¯IN⊗DαTPα∗∗IN⊗W2−2IN⊗Pα,and
(16)F^1≔F1ΛTF2Λ+F2ΛTF1Λ2,F^2≔−F1ΛT+F2ΛT2,F1Λ≔IN⊗F1,F2Λ≔IN⊗F2.Proof.
For eachα∈ℐ, define et=et+s, −τ¯≤s≤0. Choose the Markovian switched Lyapunov-Krasovskii functional as follows:
(17)Vet,α,t=∑l=13Vlet,α,t,where
(18)V1et,α,t≔eTtIN⊗Pαet,V2et,α,t≔∫t−τ¯teTφIN⊗W1eφdφ,V3et,α,t≔τ¯∫−τ¯0∫t+ηtėTφIN⊗W2ėφdφdη.
The weak infinitesimal operatorℒ of Vet,α,t is defined by
(19)ℒVet,α,t=limΔ⟶01ΔEVet+Δ,σt+Δ,t+Δσt=α−Vet,α,t.
Then, it can be deduced that(20)limΔ⟶01ΔGαh+Δ−Gαh1−Gαh=0,limΔ⟶01Δ1−Gαh+Δ1−Gαh=1,limΔ⟶01ΔqαβGαh−Gαh+ΔΔ1−Gαh=παβh,where h is the elapsed time at mode α, Gαh represents the cumulative distribution function of the sojourn time, and qαβ denotes the probability intensity jumping from mode α to mode β. As a result, one has
(21)ℒV1et,α,t=limΔ⟶01Δ∑β=1,β≠αNqαβGαh+Δ−Gαh1−GαheTt+ΔPβet+Δ=+1−Gαh+Δ1−GαheTt+ΔPαet+Δ−eTtPαet=ėTtIN⊗Pαet+eTtIN⊗Pαėt+eTt∑β=1NπαβhIN⊗Pβet=2eTtIN⊗PαIN⊗Aα+Kαet+IN⊗Bαet−τt+IN⊗CαFet+Gα⊗Γαet+IN⊗Dαwt+eTt∑β=1NπαβhIN⊗Pβet=2eTtIN⊗PαAα+IN⊗PαKαet+2eTtIN⊗PαBα×et−τt+2eTtIN⊗PαCαFet+2eTtGα⊗PαΓαet+2eTtIN⊗PαDαwt+eTt∑β=1NπαβhIN⊗Pβet.
Moreover, it can be obtained that(22)ℒV2et,α,t=eTtIN⊗W1et−eTt−τ¯IN⊗W1et−τ¯,and
(23)ℒV3et,α,t=τ¯2ėTtIN⊗W2ėt−τ¯∫t−τ¯tėTφIN⊗W2ėφdφ.
By Lemma1, it holds that
(24)−τ¯∫t−τ¯tėTφIN⊗W2ėφdφ≤etet−τtet−τ¯T−IN⊗W2IN⊗W20∗−2IN⊗W2IN⊗W2∗∗−IN⊗W2×etet−τtet−τ¯.
According to Assumption2, it can be verified that
(25)−etFetTF^1F^2∗IetFet≥0,where F^1 and F^2 are defined in (14).
Note that(26)τ¯2ėTtIN⊗W2ėt=τ¯2ζTtΠ~TIN⊗W2Π~ζt,where
(27)Π~≔IN⊗Aα+IN⊗Kα+Gα⊗Γα,IN⊗Bα,0,IN⊗Cα,IN⊗Dα,ζt≔eTt,eTt−τt,eTt−τ¯,FTet,wTtT.
Thus, one can obtain(28)ℒVet,α,t−EeTtQet+2eTtSwt+wTtℛ−γIwt≤∑l=13ℒVlet,α,t−EeTtQet+2eTtSwt+wTtℛ−γIwt−etFetTF^1F^2∗IetFet≤ζTtΠζt.
It follows from Schur complement lemma thatΠα≺0, α∈ℐ can guarantee ℒVet,α,t−EeTtQet−2eTtSwt−wTtℛ−γIwt<0. Therefore, by integrating both sides of the above inequality from 0 to t under zero initial condition and taking the expectation, it yields that Ee,Qet+2e,Swt+w,ℛwt≥Eγw,wt can hold, which implies that the Q,S,ℛ dissipative synchronization of the CDNs (1) can be well achieved according to Definition 1 and completes the proof.Remark 2.
It is worth mentioning that the derived sufficient synchronization conditions are not in the form of strict LMIs due to the time-varying dwell timeh. As a result, one reasonable assumption can be given as π¯αβ≤παβh≤π¯αβ, where π¯αβ and π¯αβ denote the upper and lower bounds of the transition rates, respectively. Then, we can obtain the following sufficient conditions with the strict LMIs.Theorem 2.
For given scalarsτ¯ and γ, the Q,S,ℛ dissipative synchronization of the CDNs (1) can be achieved with given mode-dependent synchronization controller gain Kα, if there exist mode-dependent matrix Pα≻0, matrices W1≻0 and W2≻0, such that Π¯α≺0 and Π¯α≺0 for each α∈ℐ, where
(29)Π¯α≔Π¯α1Π¯α2∗Π¯α3,with
(30)Π¯α1≔Π¯α11IN⊗PαBα+IN⊗W20∗−2IN⊗W2IN⊗W2∗∗−IN⊗W1−IN⊗W2,Π¯α11≔2IN⊗PαAα+IN⊗PαKα+Gα⊗PαΓα+IN⊗W1−IN⊗W2+∑β=1Nπ¯αβIN⊗Pβ−F^1+Q−TQ−,Π¯α2≔IN⊗PαCα−F^2IN⊗PαDα−SΠ¯α2200τ¯IN⊗BαTPα000,Π¯α22≔τ¯IN⊗AαTPα+IN⊗KαTPα+GαT⊗ΓαTPα,Π¯α3≔−I0τ¯IN⊗CαTPα∗−ℛ+γIτ¯IN⊗DαTPα∗∗IN⊗W2−2IN⊗Pα,and
(31)Π¯α≔Π¯α1Π¯α2∗Π¯α3,with
(32)Π¯α1≔Π¯α11IN⊗PαBα+IN⊗W20∗−2IN⊗W2IN⊗W2∗∗−IN⊗W1−IN⊗W2,Π¯α11≔2IN⊗PαAα+IN⊗PαKα+Gα⊗PαΓα+IN⊗W1−IN⊗W2+∑β=1Nπ¯αβIN⊗Pβ−F^1+Q−TQ−,Π¯α2≔IN⊗PαCα−F^2IN⊗PαDα−SΠ¯α2200τ¯IN⊗BαTPα000,Π¯α22≔τ¯IN⊗AαTPα+IN⊗KαTPα+GαT⊗ΓαTPα,Π¯α3≔−I0τ¯IN⊗CαTPα∗−ℛ+γIτ¯IN⊗DαTPα∗∗IN⊗W2−2IN⊗Pα.Remark 3.
Note that the above synchronization criteria are with certain conservativeness since the number of matrices in the LMIs is increased. One method to deal with this issue can be partly measurable transition rates with the linear combination technic, which follows thatπαβh=∑κ=1Kϰκπαβ,κ, ∑κ=1Kϰκ=1,ϰκ≥0 [30].Theorem 3.
For given scalarsτ¯ and γ, the Q,S,ℛ dissipative synchronization of the CDNs (1) can be achieved with given mode-dependent synchronization controller gain Kα, if there exist mode-dependent matrix Pα≻0, matrices W1≻0 and W2≻0, such that Π~α≺0 for each α∈ℐ and κ=1,2,…,K, where
(33)Π~α,κ≔Π~α,κ1Π~α,κ2∗Π~α,κ3,with
(34)Π~α,κ1≔Π~α,κ11IN⊗PαBα+IN⊗W20∗−2IN⊗W2IN⊗W2∗∗−IN⊗W1−IN⊗W2,Π~α,κ11≔2IN⊗PαAα+IN⊗PαKα+Gα⊗PαΓα+IN⊗W1−IN⊗W2+∑β=1Nπαβ,κIN⊗Pβ−F^1+Q−TQ−,Π~α,κ2≔IN⊗PαCα−F^2IN⊗PαDα−SΠ~α,κ2200τ¯IN⊗BαTPα000,Π~α,κ22≔τ¯IN⊗AαTPα+IN⊗KαTPα+GαT⊗ΓαTPα,Π~α,κ3≔−I0τ¯IN⊗CαTPα∗−ℛ+γIτ¯IN⊗DαTPα∗∗IN⊗W2−2IN⊗Pα.
Based on Theorem3, the following theorem can be given for the synchronization controller design problem.Theorem 4.
For given scalarsτ¯ and γ, the Q,S,ℛ dissipative synchronization of the CDNs (1) can be achieved with the mode-dependent synchronization controller gain Kα, if there exist mode-dependent matrices Pα≻0 and Vα, matrices W1≻0 and W2≻0, such that Ξα≺0 for each α∈ℐ and κ=1,2,…,K, where
(35)Ξα,κ≔Ξα,κ1Ξα,κ2∗Ξα,κ3,with
(36)Ξα,κ1≔Ξα,κ11IN⊗PαBα+IN⊗W20∗−2IN⊗W2IN⊗W2∗∗−IN⊗W1−IN⊗W2,Ξα,κ11≔2IN⊗PαAα+IN⊗Vα+Gα⊗PαΓα+IN⊗W1−IN⊗W2+∑β=1Nπαβ,κIN⊗Pβ−F^1+Q−TQ−,Ξα,κ2≔IN⊗PαCα−F^2IN⊗PαDα−SΞα,κ2200τ¯IN⊗BαTPα000,Ξα,κ22≔τ¯IN⊗AαTPα+IN⊗VαT+GαT⊗ΓαTPα,Ξα,κ3≔−I0τ¯IN⊗CαTPα∗−ℛ+γIτ¯IN⊗DαTPα∗∗IN⊗W2−2IN⊗Pα.
Furthermore, the mode-dependent synchronization controller gainKα can be obtained by Kα=Pα−1Vα.Proof.
LettingVα=PαKα, the rest of the proof can follow readily from Theorem 3.


## 4. Illustrative Examples

In the following section, we will give the following simulation examples to verify our developed theoretical results.Example 1.
Consider the CDNs (1) with four nodes (N=4), where each node is two dimensional (n=2). For the case of two modes (σt=1,2), the parameters of the CDNs are given as
(37)A1=−100−1.5,B1=0.50.30.30.6,C1=0.8−1.2−11.1,D1=0.10.1,Γ1=1001,and
(38)A2=−1.500−2,B2=0.10.20.20.4,C2=1−1−0.50.6,D2=0.20.2,Γ2=2002.Moreover, the jumping network topologies are depicted in Figure1, where the outer coupling matrix Gσt=Gijσt can be accordingly obtained by
(39)G1=00001−10001−10100−1,G2=−1100000010−10001−1.Figure 1
The jumping network topologies of CDNs.
(a)
(b)The time-varying delay is set byτt=0.15+0.05sint, such that one has τ¯=0.2. The nonlinear functions fxit is taken as
(40)fxit=0.2xi1t−tanh0.1xi1t0.1xi2t,such that
(41)F1=diag0.1,0.1,F2=diag0.2,0.1.In the simulation, the dissipative matrices are given asQ=−2I, S=4I⊗1,1T, ℛ=4I, and the external disturbances are set as wit=0.5sin10t. The transition rates are chosen by π11h∈−2.2,−1.6 and π22h∈−2,−1.5, which implies that π11,1=−1.6, π11,2=−2.2, π22,1=−1.5, and π22,2=−2 with K=2.With the above parameters, it can be verified that (35) has a feasible solution, and the desired mode-dependent synchronization controller gains can be calculated as follows:
(42)K1=−6.62005.10854.8080−6.8411,K2=−6.77202.93583.4072−2.0640.Consequently, the state responses of the resulting closed-loop CDNs can be shown in Figure2 with the obtained synchronization controllers and random initial conditions. In addition, the corresponding synchronization errors can be seen in Figure 3. Therefore, it can be observed that the synchronization can be well achieved by our designed mode-dependent synchronization controllers, which demonstrates our theoretical results.Example 2.
Consider the following Chua’s circuit as the unforced isolated node with two modes, which can be depicted in Figure4.
(43)ṡ1t=r1i−s1t+s2t−ϵs1t,ṡ2t=s1t−s2t+s3t,ṡ3t=−r2is2t,where ϵs1t=−0.68s1t+0.5−1.27+0.68s1t+1−s1t−1 and r11=10, r12=10.5, r21=14.87, r22=14.85.Figure 2
The state responses of the resulting closed-loop CDNs.Figure 3
The synchronization errors of the resulting closed-loop CDNs.Figure 4
The state responses of the unforced isolated node.
(a)
Mode 1
(b)
Mode 2The controlled CDNs are with four nodes, where the inner coupling matrices are given as(44)Γ1=100010001,Γ2=200020002,and the outer coupling matrices are the same as in Example 1.Moreover, the disturbance matrices are considered to be(45)D1=0.010.010.01,D2=0.020.020.02.With the same dissipative parameters and transition rates in Example1, the desired mode-dependent synchronization controller gains can be obtained as follows:
(46)K1=−28.8144−8.70460.0318−1.1806−24.8527−0.63120.085212.6531−28.9280,K2=−27.8000−8.64390.1146−0.9641−24.0207−0.64700.198912.7048−28.0110.As a result, the synchronization errors of the resulting closed-loop CDNs can be shown in Figure5, which also supports our theoretical results.Figure 5
The synchronization errors of the resulting closed-loop CDNs.


## 5. Conclusion

This paper deals with the dissipativity-based synchronization for mode-dependent CDNs with semi-Markov jump topology. Based on model transformation and stochastic analysis, sufficient conditions are given for guaranteeing the synchronization with the prescribed dissipativity performance. Then, the mode-dependent synchronization controllers are developed accordingly by LMIs. Finally, we provide the simulations that validate the usefulness of our developed synchronization scheme. Our future work encompasses investigating the synchronization problems of the CDNs with semi-Markov topology and communication network constraints.



---

*Source: 2348705-2018-10-18.xml*