Distributed periodic event-triggered synchronization for discrete-time complex dynamical networks with time-varying delay

In this paper, we study the synchronization control problem for a class of discrete-time complex dynamical networks (CDNs) under an event-triggered mechanism (ETM). First, a discrete-time distributed periodic ETM is developed based on communication measurement. Under this mechanism, the signal is sampled in a periodic manner, but whether the control signal is updated or not depends on the pre-designed triggering condition. The sampling signal is designed as the communication measurement calculated by both state feedback and inner communication among nodes. Compared with the existing discrete-time ETMs, this method considers the topology connection among network nodes and avoids the continuous measurement of signals. Therefore, this mechanism effectively saves communication resources and reduces the waste of computing resources. Second, in order to analyze the serrated constraint caused by the periodic sampling error, a piecewise Lyapunov function is constructed. Third, under the distributed periodic ETM, sufficient conditions are derived for bounded synchronization of discrete-time CDNs with time-varying delay. Finally, a simulation example verifies the effectiveness and accuracy of the conclusion.


Introduction
Complex dynamical networks (CDNs) are usually made up of a great number of intercommunicated nodes under a certain topology. The synchronization of CDNs has attracted extensive research interest, since it not only describes natural collective behavior but also has a wide range of applications in engineering fields, such as secure communication, information processing, control engineering, etc [1][2][3][4].
The consumption of communication resources has become a huge obstacle to the realization of synchronization control. In this regard, event-triggered mechanisms (ETMs) are proposed, see [5][6][7][8]. Under the framework, signals are transmitted only when meeting the set triggering condition. Therefore, the mechanisms can effectively save network communication resources and reduce energy consumption. In the early stage, many ETMs that perform signal detection in a continuous-time manner have been widely used in the study of CDNs synchronization control, see [9][10][11][12][13]. Although this effectively reduces signal transmission, the waste of computing resources is inevitable. To solve such problems, the periodic ETM is proposed by combining the advantages of traditional triggered mechanisms, see [14]. Under the framework, the sensor periodically samples the output error, but whether the sampling data is transmitted to the controller depends on the triggering condition. This not only effectively saves communication resources, but also reduces the waste of computing resources. To date, periodic ETM has been further studied and applied. For example, in [15], a predictor-based periodic ETM is proposed to study a class of nonlinear uncertain systems affected by input delay. In [16], the periodic event-triggered suboptimal control scheme is taken to analyze the stability and performance upper bound of linear systems. In [17], the author studies a robust control strategy based on periodic ETM, which realizes the global stability of the closed-loop hybrid system.
In the periodic sampling mechanism, the sampling error means the difference between the current instant and the last sampling instant, and the sampling error at the sampling instant is zero. This makes the sampling error discontinuous at the sampling instant. Due to the discontinuity of the sampling error signal, the serrated-like constraint will inevitably occur. It is worth noting that the great majority of the periodic ETMs are applied in continuous systems at present. From the perspective of system processing methods, there are great differences between discrete-time systems and continuous systems. In continuous systems, the serrated constraint is processed by constructing a piecewise continuous loop-functional, see [18,19]. But in discrete-time systems, it is difficult to develop a parallel function. Therefore, when applying periodic ETM to discretetime CDNs for synchronization analysis, how to deal with the serrated constraint caused by periodic sampling is a challenging problem.
In addition, in the current research on periodic ETM in the field of CDNs synchronization, the triggering condition only determines the triggering instant according to the node state. For example, in [20], an ETM is adopted in the system, which determines whether the sampling signal is transmitted by detecting the output state of the node. In [21], a periodic ETM adopted in the system determines whether the sampling signal is transmitted by detecting the output error. In this case, the overall connection of the system is ignored, which makes the control effect limited. Therefore, how to formulate a periodic ETM that considers the impact of inner communication among nodes is a fundamental problem.
In this paper, in order to deal with the above problems, a distributed periodic ETM is developed based on communication measurement to realize the synchronization of discrete-time CDNs. The main innovations are described as follows: (1) A distributed periodic ETM is proposed based on communication measurement. In this method, the sensor samples the state feedback information and the inner communication information among nodes in a periodic manner. Whether the sampling signal is transmitted or not is detected by the preset triggering condition. Compared with ETMs [9][10][11], this method not only avoids the measurement of signal point-to-point but also reduces the frequency of event triggering. This effectively saves computing resources and communication resources. Moreover, this method fully considers the influence of network communication on the state of nodes, so that the ETM makes more accurate judgments under the overall influence of the network, thereby optimizing the control effect and reducing unnecessary information transmission. (2) To deal with the serrated constraint caused by periodic communication measurement sampling error signals, the discrete-time Wirtinger inequality is introduced to construct the piecewise Lyapunov function. This function is designed in a discontinuous style, that is, the value of the function is 0 at the sampling instant and is nonuniform at the neighborhood of the sampling instant. It is worth mentioning that the application of the piecewise Lyapunov function not only expands the size of the sampling period but also reduces the complexity of the stability criterion.
The rest of this paper is arranged as follows: In Sect. 2, we describe the CDN model and propose a distributed periodic ETM. In Sect. 3, the synchronization criterion of the discrete-time CDNs with time-varying delay is given. In Sect. 4, a simulation example proves the effectiveness of the proposed method. Finally, the conclusion is given in Sect. 5.
Notations: R n represents the n-dimensional vector. R n×n denotes n × n real matrix. The symbol ⊗ is employed to indicate the Kronecker product. The superscript T is used to denote the matrix transpose. The symbol * is employed to say term caused by symmetry. The identity matrix is represented by I . P > 0 states a positive-definite matrix. The diagonal block matrix is expressed by diag n {A i } with diagonal elements A 1 , A 2 , . . . , A n . r states the Euclidean norm of a vector r . N[a, b] indicates all integers in the interval [a, b]. col{· · · } is used to represent column vectors.

Problem description and preliminaries
Consider a discrete-time CDN model: represents external disturbance and satisfies ν i (k) ≤ ν i0 . τ (k) indicates the time-varying delay satisfying 0 ≤ τ ≤ τ (k) ≤ τ , k ∈ N, and τ , τ are known integers. W = (ω i j ) N ×N ∈ R N ×N is the network coupling configuration matrix. The nodes of CDN (1) are connected through a directed network. ω i j represents the adjacency strength and satisfies the following constraints: (1) if i = j and information is transmitted directionally from node i to node j, then ω i j > 0, else ω i j = 0; (2) if i = j, then According to the constraint conditions for W , we get rank(W ) = N − 1.

Remark 1
The adjacency matrix W is employed to indicate the physical topology of system. Its selection affects the difficulty of synchronization. In the directed network, ω i j represents the adjacency strength from node i to node j. When the value of ω i j increases, the communication from node i to node j will enhance. Assumption 1 For any p, q ∈ R n , nonlinear functions f (·) and g(·) satisfy The initial conditions for the CDN (1) are given as follows: In this paper, it is assumed that the trajectories of all nodes in the CDN (1) are synchronized to the following average state: where denotes the left eigenvector of the zero eigenvalue of the coupling matrix W .
The initial value corresponding to the weight average state system (3) is expressed as Through the CDN (1) and the weight average state system (3), one has This paper presents a communication mode by event-triggered. Specifically, when the triggering con-dition is met, the communication information among nodes is updated; otherwise, the communication among nodes remains in the communication state of the previous triggering instant. Under this mechanism, the CDN (1) is re-described as where the event triggering time sequence . . for node i are generated by ETM. Let the communication measurement of CDN (1) and weight average state system (3) be In order to decrease the update frequency of the controller and reduce the consumption of communication resources, we next utilize communication measurement (7) to design a distributed periodic ETM in CDN (1). In this case, the communication measurement ϕ i (k) is sampled only at periodic instant jh, where h represents the sampling period, j ∈ N. Whether the sampling data ϕ i ( jh) is transmitted depends on the following distributed periodic ETM: where denotes a communication measurement error between the current instant and the last triggering instant, Obviously, in the case of h = 1, the ETM (8) will be transformed into a continuous ETM. Therefore, without explicit instructions, we assume h ≥ 2 to distinguish periodic and continuous ETM.

Remark 2
To understand the influence among nodes, this paper proposes a method to measure nodes' topological communication, called communication measurement. The method is a detection method that comprehensively considers the topology connection among  [9,10] nodes and their own state feedback. It comprehensively analyzes the correlation of system. Compared with ETMs in [6,13,15], and [16], the influence of the overall state of the system on the triggering instant is fully considered during the design of distributed periodic ETM based on communication measurement, which effectively reduces the limitation and independence of the control effect.

Remark 3
In this paper, the distributed periodic ETM (8) is developed based on communication measurement, which effectively reduces communication load and saves computing resources. Compared with ETM in [22], the proposed periodic ETM avoids continuous measurement and expands the lower bound of the interval between events. Compared with ETM in [23], the communication measurement error is utilized in the triggering condition. In this sense, the influence of inter-node communication is considered, and the triggering instant is determined separately for each node. This is more suitable for the characteristics of the actual system.

Remark 4
The proposed periodic ETM is more comprehensive than the existing ones, and the control effect is more advantageous. Table 1 shows the existing eventtriggered control methods, which can be regarded as special cases of the method in this paper.

Main results
In this section, the synchronization of discrete-time CDNs with time-varying delay is discussed. Under the distributed periodic ETM, a sufficient condition is obtained for the synchronization of CDN (1) and weight average state system (3).
Define an error function as e i (k) = x i (k) − x(k). From (5) and (6), we have the following error dynamical system: − τ (k))), and Σ = col{ξ, ξ, . . . , ξ}. By exploiting Kronecker product, the error dynamical system (9) is rewritten as e(k + 1) = I N ⊗ Ae(k) + MF(e(k)) + MG(e(k − τ (k))) where M = [(I N − Σ) ⊗ I n ], and According to the characteristics of distributed periodic ETM (8), one has Consider the following Lyapunov function: where where V i,h (k) is constructed to deal with the sampling signal of node i with the following form: and real matrices P > 0, Q > 0, R i > 0, parameter ∈ (0, 1), and By taking (13) and (14) into account, the error system (10) is transformed into where It is seen from Lyapunov function (11) that the expression near the sampling instants k = k i l +( j −1)h is nonuniform. Therefore, the key of synchronization is the calculation of ΔV (k) at sampling instants k = k i l + ( j − 1)h. Next, we will give the synchronization theorem under the distributed periodic ETM. ∈ (0, 1), c, h, σ i , ε i and the internal coupling matrix Γ be given. If there exist positive-definite symmetric matrices P ∈

Theorem 1 Let positive scalars
where

then the error system (13) is finally bounded, and the error bound is calculated by
Proof According to the Lyapunov function (11), one has Next, we calculate the items (1− )V a (k)+ΔV a (k), (a = 1, . . . , 4) as follows: V 2 (k) and V 3 (k) are used to cope with time-varying delay. The results are as follows: and Thus, from (18) and (19), we can conclude The calculation of V 1 (k), V 2 (k), and V 3 (k) in the Lyapunov function (11) is conventional, but the calculation of V 4 (k) needs to be discussed in different cases When Therefore, we obtain When k = k i l + jh − 1, according to the definition of function V i,h (k), it yields Based on Lemma 1, we have Therefore, for k = k i l + jh − 1, one has According to (21)-(23), it deduces, Based on (14), (24) can be converted into − τ (k))) + M F(e(k)) According to Assumption 1, we obtain According to the ETM (8), one has Combining with (26)-(28), it can be concluded from (17), (20), and (25) that where According to Schur complement lemma, it is obtained that (16) holds if and only if the following inequality holds it derives which means that where ∈ (0, 1). Subsequently, according to the definition of V (k), one has Finally, it is easy to obtain that The proof is completed.

Remark 5
In Theorem 1, the synchronization problem of discrete-time CDNs with time-varying delay is studied under distributed periodic ETM. In order to solve the serrated constraint caused by periodic sampling, a piecewise Lyapunov function is constructed with the aid of Lemma 1, so as to reduce the complexity of calculation and the conservativeness of stability criterion. In addition, the developed distributed periodic ETM (8) is based on communication measurement sampling signals for detection. It makes the control effect more advantageous and reflects the influence characteristics of global communication among nodes in the actual system. Moreover, in the distributed periodic ETM, when the error state is large, the triggering threshold σ i plays a major role, which effectively expands triggering intervals and reduces the number of event triggers. However, as the error gradually becomes smaller and close to stability, it becomes negligible. In contrast, at this point, the triggering threshold ε i plays a major role, which effectively expands triggering intervals and reduces the event update frequency. Therefore, the integrated use of triggering thresholds σ i and ε i effectively save communication resources.

Remark 6
In this paper, the distributed periodic ETM is adopted to reduce the update frequency of controller and the consumption of communication resources. It is worth noting that the predefined triggering conditions in the existing ETMs [9][10][11]23] only consider the state of node itself, and do not consider the communication among nodes. These methods ignore the influence of system correlation on nodes. In this case, these ETMs cannot make a more precise control based on how the nodes are really affected. To deal with the limitations of triggering judgments, this paper introduces the communication measurement into the ETM. In this case, the ETM can make a more suitable judgment under the overall influence of network. Thus, the control effect is further optimized and unnecessary waste of resources is reduced.
The matrices satisfying the Assumption 1 are calculated as follows:  Other parameters are selected as = 0.95, c = 0.5, ε i = 0.0001, σ i = 0.1, and h = 3. By solving LMI (16), we obtain a set of feasible solutions as follows: In order to explain the effectiveness of periodic ETM in this paper, we also verify the case of h = 1. Figure 3 shows the triggering instants of h = 3 and h = 1, respectively. Notably, when h = 1, the ETM (8) is a triggering mode combining continuous relative ETM and continuous absolute ETM. Through Fig. 3, it is clearly concluded that the proposed method is effective. Table 2 lists the comparison results between the traditional time-triggered mechanism and existing ETMs [9][10][11]22,23]. Compared with the traditional timetriggered mechanism, the distributed periodic ETM (8) reduces the total number of events of nodes by  [9][10][11]22,23], the distributed periodic ETM (8) reduces the total number of events of nodes by 83.72%, 61.23%, 56%, and 84.99%, respectively. Moreover, ETMs [9][10][11]22] require continuous measurement of the signal, resulting in a large consumption of computing resources. The difference is that the method only detects the ETM (8) with a period h, which effectively saves computing resources. Specifically, the computing resource cost required by this method is about 1/ h of [9][10][11]22]. The effectiveness of the proposed method is further illustrated by the above comparisons.
In addition, to show the effectiveness of the proposed method, Fig. 4 shows the comparisons for the trajectories of e(k) under different methods. Furthermore, we calculate the time to reach the same error norm bound under different methods. Through the simulation results, it is concluded that the times for error reaching bound e(k) = 0.05 are k = 36, k = 37, k = 42, and k = 35 for the ETMs in [9][10][11]22,23], respectively. However, it is k = 32 under the proposed method, which is faster than the existing results. Moreover, compared with ETMs [9][10][11]22,23], this method not only avoids continuous measurement of signals but also reduces the number of events. This effectively saves computing resources and communication resources. Meanwhile, this method comprehensively considers the influence of system communication on nodes, which enables ETM to make more accurate judgments and improves the control effect. The above comparisons further illustrate the effectiveness of the proposed method.

Conclusion
In this paper, a distributed periodic ETM based on communication measurement has been proposed for synchronization control of discrete-time CDNs. In this scheme, the sensor periodically samples the communication measurement signal, then whether the sampling signal is transmitted is determined through the pre-designed triggering condition. Compared with the existing discrete-time ETMs, this scheme considers the influence among node states and avoids point-to-point continuous measurement detection. The communication resources and calculation loss were saved effectively. Moreover, in order to analyze the serrated constraint caused by the sampling error of periodic communication measurement, a piecewise Lyapunov function was constructed. Thus, sufficient conditions were obtained for bounded synchronization of discrete-time CDNs with time-varying delay. Finally, a simulation example verifies the effectiveness and accuracy of the conclusion.