Finite-time dissipative synchronization of discrete-time semi-Markovian jump complex dynamical networks with actuator faults

This paper is concerned with the problem of finite-time synchronization for a class of discrete-time semi-Markovian jumping complex dynamical networks (CDNs) with actuator faults based on reliable control. The main aim of this paper is to design a state feedback controller such that the resulting closed-loop system is finite-time synchronized under a prescribed dissipativity performance level even in the presence of actuator failures. Moreover, a stochastic nature followed by Bernoulli distribution is described in the considered networks due to the occurrence of probabilistic nature in time-varying delays. By composing a suitable Lyapunov–Krasovskii functional containing triple summation terms with the aid of Kronecker product properties, Lyapunov stability theory and free weighting matrix approach, sufficient criteria are established in terms of linear matrix inequalities that assure finite-time synchronization and meet the dissipativity performance to the addressed CDNs. The usefulness of the presented design scheme is finally verified by numerical examples.


Introduction
Complex network is a large-scale network composed of a large number of nodes joined by edges, and it can mirror several important structure characteristics and dynamic performances of real networks (Zhang et al. 2019;Dong et al. 2017). In recent years, the study on CDNs serves as one of the most active research field in control theory that have found successful applications in real-world disciplines such as supply chain and manufacturing networks, electricity power grids, transportation systems, water distribution systems, gas transmission, world wide web, social interaction, biological networks, co-authorship and citation of networks of scientists. Due to these utilizations, the researchers have been devoting their interest enormously to investigate the control problems and dynamical behaviors of CDNs (Zhang et al. 2019;Dong et al. 2017;Hao et al. 2016).
On the other hand, in practice, synchronization is interesting and significant dynamic phenomena of CDNs compared to other dynamical performance such as auto-waves, spatiotemporal chaos, self-organization and spiral waves. Inspired by these advantages, the research communities have investigated synchronization of a CDNs presented in He et al. (2016), , and Li et al. (2018).  studied the synchronization problem for a class of discrete-time CDNs with partial mixed impulsive effects. But in practice, the networks might be expected to achieve synchronization in finite-time interval rather than asymptotically. To achieve faster convergence rate in time delay complex networks, an effective method called finite-time synchronization or finite-time boundedness can be used. In addition, finite-time control design has been proposed for the synchronization of different CDNs as authorized by various literatures (Han et al. 2017;Sheng et al. 2018;Li et al. 2019). The finite-horizon bounded H ∞ synchronization and state estimation problem for a class of discrete-time com-plex networks with missing measurements are reported in Han et al. (2017).
Moreover, few actuators may be faulty and uniform of inferior quality, the CDNs structure may be changed as well as the closed-loop system becomes unstable. To improve system reliability and security, it is most important to design a reliable controller so that the stability and the performance of the Takagi-Sugeno fuzzy closed-loop system can operate fine, even in the existence of some actuator faults in Shen et al. (2014). Consequently, the study on fault tolerant control or reliable control for dynamical systems has received many results, see for example (Sakthivel et al. 2014;Dong et al. 2015;Sakthivel et al. 2015). Naturally, the inclusion of time delays is inevitable phenomenon in all kinds of dynamical systems and it arises due to the finite propagation speed between nodes, traffic congestions and memory effects, and it has the capability to alter notoriously the dynamic behavior of the system such as stability. Therefore, much work has been done for synchronization of discrete-time networks with time delay (Qunjiao et al. 2019;Zhang et al. 2018;Park and Kwon 2017). The existence of time delay obeys stochastic fashion in some practical systems, it consequences the researchers have focused the synchronization analysis of CDNs with probabilistic time-varying delay (Yang et al. 2015;Cheng and Peng 2016;Cheng et al. 2019). The problem of cluster synchronization in finite-time complex networks with probabilistic coupling delays has been addressed in Yang et al. (2015).
In most of the practical networks, the abrupt variations often expose in system dynamics due to random failures, changes in the subsystem interconnections and information latching. Therefore, in order to tackle this issues, it is essential to describe CDNs with Markovian jumping parameters, where each state denotes a discrete-time system with finite discrete jumping mode governed by the Markovian process, and few interesting results regarding Markov jump systems have been addressed in Wang et al. (2016), Ren et al. (2018), Akbari et al. (2020. It is pointed that the above studies have dealt with transition rates of Markov process which are constants due to that the sojourn time between two jumps of the Markov chain is governed by exponential distribution. Also, it should be mentioned that the Markovian process might consist of time-varying transition rates when modeling practical systems, such kind of process is known as semi-Markovian process. Apparently, semi-Markovian jump CDNs are comparatively more general than the Markovian jump CDNs. Shen et al. (2015) obtained the finite-time H ∞ synchronization criterion for complex networks with time-varying delays and semi-Markovian jump topology. The existence of sufficient conditions in Liang et al. (2018) and convex optimization technique which ensure the L 2 − L ∞ synchronization for singularly perturbed complex networks subject to semi-Markovian jump topology. Furthermore, the dissipativity theory was introduced by Willems (1972), and it serves as a powerful tool in control applications such as robotics, active vibration damping, electromechanical system and circuit theory. Generally speaking, dissipativity which postulates the quantity of energy supplied from the external source is not less than the energy lost inside the dynamic system . The main idea of dissipative theory is the dissipative quality of a system can keep the system internally stable. Also, the dissipativity system theory is a more general criterion when compared with passivity and H ∞ performance. In Ma et al. (2020), a set of sufficient conditions is established by using Lyapunov functional methodology and completing square technique for dissipativity control of discrete nonlinear Markovian jump systems with discrete and distributed time delays. Wang et al. (2019) addresses the problem of generalized dissipativity-based synchronization for complex networks with semi-Markovian jump topology. In spite, up to now, the finite-time dissipative synchronization has not yet been reported for a class of discrete-time semi-Markovian jump CDNs.
Motivated by the above discussions, this paper analyzes finite-time synchronization problem for discrete-time semi-Markovian jumping CDNs with probabilistic time-varying delay components based on dissipative performance. The reliable control strategy and probability distribution of the time-varying delays are proposed. By constructing a new LKF, utilizing the Kronecker product technique and Lyapunov stability theory, delay-dependent sufficient conditions are derived under which the CDNs are synchronized in the given finite-time interval. The derived conditions depend not only on the size of the delay but also on the probability of the delay taking values in some intervals. Based on this derived condition, a design algorithm of the proposed state feedback controller which ensures the finite-time synchronization of the CDNs with actuator faults. Finally numerical examples are provided to illustrate the effectiveness of our theoretical results. Notation: Throughout this paper, superscripts "T " and "(−1)" stand for matrix transposition and matrix inverse, respectively. R n denotes the n-dimensional Euclidean space, and R n×n denotes the set of all n × n real matrices. P > 0 and P < 0 represent positive definite and negative definite, respectively. L 2 [0, ∞) stands for the space of n-dimensional square integrable function over [0, ∞). I and 0 represent identity matrix and zero matrix with compatible dimensions. The asterisk " * represents a term that is induced by symmetry. diag{...} stands for a block-diagonal matrix. The Kronecker product of matrices S ∈ R m×n and T ∈ R p×q is a matrix in R mp×nq and denoted as S ⊗ T .

Network model and preliminaries
Consider the following discrete-time semi-Markovian jumping CDNs consisting of N nonlinearly coupled nodes, with each node being of n-dimensional and having identical dynamic performance. It can be designated as where x l (k) ∈ R n denotes the state vector of the l th node; A(σ (k)), B(σ (k)), C(σ (k)), R(σ (k)) and J (σ (k)) are constant matrices with suitable dimensions at instant k; f (k, x l (k)) and f (k, x l (k − d(k)) represent the nonlinear vector-valued functions without and with time delays, respectively. d lm (σ (k)), g lm (σ (k)) are the elements of the outer coupling configuration matrices D(σ (k)) and G(σ (k)) which describes the coupling structure of the CDNs. The positive diagonal matrices 1 and 2 denote inner coupling matrices with suitable dimension; V f l (k) is the control input of the l th node. If there is a link from node l to node m, then d lm (σ (k)) > 0, g lm (σ (k)) > 0; otherwise d lm (σ (k)) = g lm (σ (k)) = 0.
Assumption 1 Considering the information of probability distribution of the time delays d(k) and τ (k), we define Therefore the stochastic variable α(k), β(k) can be defined as Assumption 2 β(k) and α(k) are the Bernoulli distributed sequences with where E{β(k)} and E{α(k)} are the expectations of β(k) and α(k), respectively.
From Assumption 1, it is easy to say that Next, the discrete delays d 1 (k), d 2 (k), τ 1 (k) and τ 2 (k) are introduced in the following manner: By using Assumptions 1 and 2, CDNs (1) can be rewritten as Assumption 3 For the nonlinear function f (k, , there exists the known real constant matrix K 1 ,K 2 and K 3 such that Let e l (k) = x l (k) − s(k) be the synchronization error, where s(k) be a solution of an isolated node which is described as Then the corresponding error dynamics of the CDNs (1) can be obtained as follows: . Now consider the actuator fault model, for the control input V f l (k). We designate V f l (k) to describe the signal sent from the actuator and satisfies where G is the actuator fault matrix defined as G = diag{g 1 , g 2 , . . . , g m }, 0 ≤ g i ≤ g i ≤ḡ i ≤ 1, where g i andḡ i , i = 1, 2, . . . , m are given constants. g i = 0 means that ith actuator fails, g i = 1 means that i th actuator is normal, when 0 < g i < 1 denotes the i th actuator meets partial failures. In this paper, we design a state feedback controller of the form: where K p is the state feedback controller gain matrix to be determined. Then, by substituting (5) in (4) and using Kronecker product properties, the resulting closed-loop system can be written as Further, we introduce the following lemmas and definitions, which will be important for the derivations of the main results.
Definition 1 (Cheng et al. 2015) The error dynamics (6) subject to an exogeneous disturbance w(k) satisfying Definition 2 (Zhang et al. 2014) The error dynamics (6) is said to be stochastically finite-time stable(SFTS) with respect to (c 1 , c 2 , S p , N), where S p is a positive definite Definition 3  The error dynamics (6) is for any nonzero satisfies N k=0 w T (k)w(k) < δ, here L, M and R are real-valued matrices of appropriate dimensions with L and R are symmetric. Without loss of generality, it is assumed that L ≤ 0, then we have −L = (L 1/2 ) 2 .

Remark 1
In the view points of the above definitions, it should be noted that the SFTS of the errror system (6) can guarantee the SFTB. That is, in the absence of exogenous disturbance inputs w(k) = 0, the concept of SFTB is reduced to SFTS. Thus, SFTB implies SFTS, but the converse is not true. Consequently, the system (1) is finite-time synchronized under the feedback control law (5).

Main results
In this section, to design a state feedback controller which ensures the stochastic synchronization in a finite-time interval of the CDNs (1). First we find the stochastic finite-time boundedness for the closed-loop system (6). Next, we find the stochastic finite-time boundedness with dissipativity performance index.

A. Stochastic finite-time boundedness
Theorem 1 Under Assumptions 1 and 2, given scalars μ ≥ 1, κ i > 0(i = 1, 2) and known actuator fault matrix G, 2,3,4) such that the following inequalities hold: where 1,1 = −μ(I ⊗ P p ) Proof Construct the following LKF for the error system (6) as where Then, taking mathematical expectation of the forward difference formula V(k) = V(k +1)−V(k) along the trajectories of the system (6), we have where Furthermore, we have If we take Z p = [α 1 (I ⊗ P p ) T α 2 (I ⊗ P p ) T ] T with α i > 0(i = 1, 2), the following equation holds: where From Assumption 3, we can obtain the following inequalities Combining (9)-(18), we have where Further, by using schur complement to the right hand side of (19), we get the required LMI in (7). If the matrix inequality in (7) holds, it is obvious that Further, if μ ≥ 1, it follows that Next, we define the following parameters: Then, from (8), we can have By using the inequalities (20) and (21), we can obtain From the above computations, we can conclude that the closed-loop error system (6) is stochastically finite-time bounded with respect to (c 1 , c 2 , S p , N, δ).

B. Stochastic finite-time dissipative
A sufficient condition is given to ensure the finite-time boundedness with dissipative performance of CDNs is analyzed in the following theorem: Theorem 2 Under Assumptions 1 and 2, for given scalar μ ≥ 1, the error system (6) is stochastic finite-time dissipative with respect to (c 1 , c 2 , S p , N, δ, L, M, R), if there exist symmetric matrices X p > 0 ( p ∈ ψ),Q bp > 0 (b = 1, 2, · · · , 6), 1, 2, 3, 4) such that the following conditions hold: . Moreover, the gain matrices are given by K p = Y p X −1 p .
Proof In order to describe the dissipative performance of the closed-loop system (6), we consider the energy function J as The proof follows from Theorem 1, and it follows that By simple modifications, it is quite easy to get that Under zero initial condition and the fact V (k) ≥ 0, ∀ k = 1, 2, · · · , N , we have Then, from (26), it is easily to get the inequality in the Definition 3. To complete the proof of this theorem, let X p = P −1 p ,Q bp = X p Q p X p (b = 1, 2, · · · , 6),T cp = X p T c X p ,Û cp = X p U c X p (c = 1, 2),V lp = X p V l X p (l = 1, 2, 3, 4), then performing the congruence transformations to (7) by diag{(I ⊗ X p ), · · · , (I ⊗ X p ) where U = −diag{(I ⊗ X 1 ), (I ⊗ X 2 ), · · · , (I ⊗ X N )} and Letting K p X p = Y p , we can easily to get the required LMI (23).
Remark 2 By using Definitions 2 and 3, it should be noted that if w(k) = 0, the concept of SFTB is reduced to SFTS. Also we can observe that if the error system (6) is stochastically finite-time stable, then the proposed CDNs (1) is finite-time synchronized under the control law (5). In CDNs (1), removing the nonlinear function and external disturbance and considering only one mode in operation, then it can be rewritten as follows: Then the closed-loop error system (6) becomes Corollary 1 For given matrices S > 0, and scalars μ ≥ 1, c 1 > 0, c 2 > 0, N and α i ≥ 0 (i = 1, 2) and under Assumptions 1 and 2, the error system (27) is stochastic finitetime synchronization with respect to (c 1 , c 2 , S, N), if there exist symmetric matrices X > 0,Q b > 0 (b = 1, 2, 3), T c > 0,Û c > 0 (c = 1, 2),V l > 0 (l = 1, 2) such that the following inequalities hold:ˆ Proof Consider the following LKF Remaining proof follows from Theorem 2, we can obtain the required result. Thus, the proof is completed.

Numerical simulations
This section provides a simulation examples to show the effectiveness and superiority of the established criteria for finite-time synchronization of the proposed CDNs. The schematic diagram of synchronization for the addressed system with actuator faults is represented in Fig. 1.
Example 1 Consider a class of CDNs in the form of (2) with three nodes, and dimension of the state vector of each node is two.
The nonlinear function f ( it can easily satisfy Assumption 3 with the matrices K 1 = K 2 = K 3 = diag{0.4, 0.4} and also the noise signal is chosen as w(k) = 0.05 exp(−0.1k) sin k. Further, the time delays are taken as d 1 (k) = 3.01 + 0.25 sin(0.1k), d 2 (k) = 3.5 + 0.25 sin(0.1k), τ 1 (k) = 1.05 + 0.25 sin(0.5k), τ 2 (k) = 2.01 + 0.25 sin(0.02k), and the actuator failure matrix is G = diag{0. For the simulation purposes, the initial conditions for the states of the nodes and the isolated node are taken as Based on the above values, simulation results are presented in Figs. 2, 3, 4, 5, 6, 7, and 8. Specifically, the state responses of the first, second and third nodes together with the isolated node are plotted in Fig. 2. It can be seen from this figure that the states of the nodes are exactly synchronized with the isolated node. T , he error responses with and without control are depicted in Fig. 3. It can be observed from Fig. 3a that the error state trajectories are synchronized  signal, respectively. In addition, the Bernoulli random variable α(k) with α 0 = 0.6 and β(k) with β 0 = 0.4 are plotted in Figs. 6 and 7. The time history of e T i S p e i (k)(i = 1, · · · 6) is depicted in Fig. 8. From these figures, it can be realized that the CDNs(2) is finite-time bounded with respect to (0.01, 13.8344, 0.2, 20, 0.1) even in the existence of the network-induced imperfections such as actuator fault and time-varying delays.
For the simulation purposes, the initial conditions for the states of the nodes and the isolated node are taken as x 1 (0) = Based on the initial condition and system parameter values, simulation results are presented in Figs. 9, 10, and 11. Specifically, the state responses of the first, second and third nodes together with the isolated node are plotted in Fig. 9. It can be seen from this figure that the states of the nodes are exactly synchronized with the isolated node. The error responses with control are depicted in Fig. 10, and the time history of e T i Se i (k)(i = 1, · · · 6) is depicted in Fig. 11. From these figures, it can be realized that the CDNs (27) are finitetime synchronization with respect to (1.5, 9.5379, 0.1, 20) even in the existence of the network-induced imperfections such as actuator fault.

Conclusion
In this paper, the dissipative-based finite-time synchronization problem has been investigated for a discrete-time CDNs subject to semi-Markovian jumping parameters, probabilistic time-varying delays and actuator faults through the reliable control. By constructing suitable Lyapunov functional method and Kronecker product properties, the required criteria for ensuring the finite-time synchronization with dissipativity performance index of the CDNs have been obtained in the form of LMIs. Finally, two numerical examples have been exploited to show the effectiveness of the established proposed results. Zhang Q, Chen G, Wan L (2018)  Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.