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

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 LyapunovKrasovskii 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.


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][2][3][4][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][7][8][9][10][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][13][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 the H ∞ 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][18][19][20][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][27][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 modedependent 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, the Q, 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 Section 2, 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.ℝ n and ℝ m×n denote n dimensional Euclidean space and the set of all m × n matrices, respectively.ℒ 2 0, ∞ 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 ≻ 0 A − 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.

Preliminaries and Problem Formulation
Given a probability space Ω, ℱ, P , consider the following class of directed CDNs consisting of N identical nodes: where x i t = x i1 t , x i2 t , … , x in t T ∈ ℝ n denotes the state vector of the ith node; u i ∈ ℝ n and w i ∈ ℝ q denote the control input and the disturbance input on the ith node, respectively; f x i t ∈ ℝ n is a smooth nonlinear function; τ t is the time-varying delay; A σ t is a diagonal matrix; B σ t , C σ t , and D σ t are weight matrices; Γ σ t is the inner coupling matrix; and G σ t = G ij σ t ∈ ℝ N×N is the outer coupling matrix representing the directed network topology.If there is a directed coupling from node i to node j i ≠ j , then the coupling Assumption 1.For each σ t ∈ ℐ, the network topology keeps constant, and and Γ σ t are known real constant matrices.
Assumption 2. The nonlinear function f x i t satisfies Assumption 3. The time-varying delay τ t satisfies 0 < τ t ≤ τ, where τ is a positive constant.
In this paper, the synchronization errors are defined as where s t ∈ ℝ n is the state trajectory of the unforced isolated node s t = A σ t s t + B σ t s t − τ t + C σ t f s t .Then, the synchronization error dynamics of the CDNs can be obtained as follows: 2 Complexity where f e i t ≔ f x i t − f s t .
The following mode-dependent synchronization controller is designed: where K σ t is the controller gain matrix.Consequently, the closed-loop synchronization error dynamics can be derived by which can be further rewritten as For notational simplicity, denote σ t by the index α, α ∈ ℐ.Then, one has The following definition is given.
Definition 1.The mean-square stochastically Q, 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: Remark 1.It can be found that the Q, 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, τ t satisfying 0 ≤ τ t ≤ τ, vector function x t : −τ, 0 ⟶ ℝ n such that the concerned integrations are well defined, then where We aim to design a proper mode-dependent synchronization controller (6) for CDNs (1) to guarantee that meansquare stochastically Q, S, ℛ dissipative synchronization can be achieved.

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 scalars τ and γ, the Q, S, ℛ dissipative synchronization of the CDNs (1) can be achieved with given mode-dependent synchronization controller gain K α , if there exist mode-dependent matrix P α ≻ 0, matrices W 1 ≻ 0 and W 2 ≻ 0, such that Π α ≺ 0 for each α ∈ ℐ, where Proof.For each α ∈ ℐ, define e t = e t + s , −τ ≤ s ≤ 0.
Choose the Markovian switched Lyapunov-Krasovskii functional as follows: V l e t , α, t , 17 where The weak infinitesimal operator ℒ of V e t , α, t is defined by Then, it can be deduced that where h is the elapsed time at mode α, G α h represents the cumulative distribution function of the sojourn time, and q αβ denotes the probability intensity jumping from mode α to mode β.As a result, one has  where F1 and F2 are defined in (14).Note that where 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 time h.As a result, one reasonable assumption can be given as π αβ ≤ π αβ h ≤ 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 scalars τ and γ, the Q, S, ℛ dissipative synchronization of the CDNs (1) can be achieved with given mode-dependent synchronization controller gain K α , if there exist mode-dependent matrix P α ≻ 0, matrices W 1 ≻ 0 and W 2 ≻ 0, such that Π α ≺ 0 and Π α ≺ 0 for each α ∈ ℐ, where with with 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 that Theorem 3.For given scalars τ and γ, the Q, S, ℛ dissipative synchronization of the CDNs (1) can be achieved with given mode-dependent synchronization controller gain K α , if there exist mode-dependent matrix P α ≻ 0, matrices W 1 ≻ 0 and W 2 ≻ 0, such that Π α ≺ 0 each α ∈ ℐ and κ = 1, 2, … , K, where , 33 Based on Theorem 3, the following theorem can be given for the synchronization controller design problem.Theorem 4. For given scalars τ and γ, the Q, S, ℛ dissipative synchronization of the CDNs (1) can be achieved with the mode-dependent synchronization controller gain K α , if there exist mode-dependent matrices P α ≻ 0 and V α , matrices W 1 ≻ 0 and W 2 ≻ 0, such that Ξ α ≺ 0 for each α ∈ ℐ and κ = 1, 2, … , K, where Furthermore, the mode-dependent synchronization controller gain K α can be obtained by K α = P −1 α V α .
Proof.Letting V α = P α K α , the rest of the proof can follow readily from Theorem 3.

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 (σ t = 1, 2), the parameters of the CDNs are given as

37
and Moreover, the jumping network topologies are depicted in Figure 1, where the outer coupling matrix G σ t = G ij σ t can be accordingly obtained by The time-varying delay is set by τ t = 0 15 + 0 05 sin t, such that one has τ = 0 2. The nonlinear functions f x i t is taken as such that In the simulation, the dissipative matrices are given as With the above parameters, it can be verified that (35) has a feasible solution, and the desired modedependent synchronization controller gains can be calculated as follows:  7 Complexity 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 Figure 4.As a result, the synchronization errors of the resulting closed-loop CDNs can be shown in Figure 5, which also supports our theoretical results.

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  8 Complexity 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.
state responses of the resulting closed-loop CDNs can be shown in Figure2with the

Figure 1 :
Figure 1: The jumping network topologies of CDNs.

43 where ϵ s 1 t
= −0 68s 1 t + 0 5 −1 27 + 0 68 s 1 t + 1 − s 1 t − 1 and r 11 = 10, r 12 = 10 5, r 21 = 14 87, r 22 = 14 85.The controlled CDNs are with four nodes, where the inner coupling matrices are given coupling matrices are the same as in Example 1.Moreover, the disturbance matrices are considered to be dissipative parameters and transition rates in Example 1, the desired mode-dependent synchronization controller gains can be obtained as follows:

4 Figure 2 :
Figure 2: The state responses of the resulting closed-loop CDNs.

Figure 3 :
Figure 3: The synchronization errors of the resulting closed-loop CDNs.

Figure 4 :
Figure 4: The state responses of the unforced isolated node.

Figure 5 :
Figure 5: The synchronization errors of the resulting closed-loop CDNs.
Qe t t denotes t 0 e T s Qe s ds, and the other symbols are similarly defined.Moreover, it is assumed that there exists a constant matrix Q Qe t t + 2 e t , Sw t t + w t , ℛw t t ≥ E γ w t , w t t , 11 where e t , It follows from Schur complement lemma that Π α ≺ 0, α ∈ ℐ can guarantee ℒV e t , α, t − E e T t Qe t − 2e T t Sw t − w T t ℛ − γI w t < 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 E e, Qe t + 2 e, Sw t + w, ℛw t ≥ E γ w, w t 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.
, e T t − τ t , e T t − τ , F T e t , w T t T 27 Thus, one can obtain ℒV e t , α, t − E e T t Qe t + 2e T t Sw t + w T t ℛ − γI w t ≤ 〠 3 l=1 ℒV l e t , α, t − E e T t Qe t + 2e T t Sw t + w T t ℛ − γI w t −