Novel finite-time synchronization criteria for coupled network systems with time-varying delays via event-triggered control

This paper is concerned with the finite-time synchronization of coupled networks with time-varying delays. We work without applying the finite-time stability theorem, which is widely used in finite-time synchronization for complex networks or finite-time consensus problems for multi-agent systems. We construct a novel Lyapunov functional and apply some new analytical techniques. Sufficient conditions are obtained to ensure synchronization within a setting time with no Zeno behaviors. The obtained conditions do not contain any uncertain parameter. The controllers are presented based on event-driven strategies, which can significantly reduce the communication consumption and the frequency of the controller updates. And the setting time is related to initial values of the network. Finally, numerical examples are examined to illustrate the effectiveness of the analytical results.


Introduction
A coupled system is a suitable model for many distributed phenomena in communication engineering, economics and biological science [1][2][3]. And asymptotic synchronization of coupled complex network systems has been studied widely in the last few decades [4][5][6][7][8][9].
In practical engineering, it is often expected that one might realize synchronization as fast as possible. In terms of secure communication, it is an important application of synchronization. As is well known, the range of time during which the chaotic oscillators are not synchronized corresponds to the range of time during which the encoded message can unfortunately not be recovered [10]. That is, asymptotic synchronization is not optimal because machines' and human's life spans are limited [11]. Therefore, finite-time control technique has been introduced in the continuous systems [12,13]. And it has better robustness and disturbance rejection properties when the system is stabilized within a setting time. Thus, the finite-time control technique has been extensively used to synchronize networks [11,[14][15][16][17].
Moreover, almost all the aforementioned works assume that the Markovian switching (or chain) is homogeneous, that is, the transition probabilities are time-invariant. How-ever, to the best of the authors' knowledge, finite-time synchronization problems have not yet been addressed for the coupled complex networks in terms of time-varying delays by using ETC. And the important issue is how to design the ETR (event-trigger rule) for each node to achieve finite-time synchronization of the coupled networks and meanwhile to prevent Zeno behavior.
Motivated by the above discussion, in this paper, we study finite-time synchronization for coupled networks with time-varying delays by using event-triggered control. The main novelties of this note are: (1) by constructing the new ETR of each node, designing new Lyapunov functional and using matrix inequalities, sufficient conditions are derived to guarantee the finite-time synchronization without Zeno behavior; (2) we do not use the well-known finite-time stability theorem in [12], and the setting time can be easily obtained.
Notation Throughout this paper, a graph is defined as a pair G = (V, A) consisting of a set of nodes V = {1, 2, . . . , N} and a time-varying matrix is piecewise-constant, that is, a ij (t) ≥ 0 is piecewise-constant for all i, j ∈ V. R n and R n×m denote respectively, the set of n × 1 real vectors and the set of all n × m real matrices. D + stands for Dini derivative. The superscript "T" denotes the transpose and the notation X ≥ Y (respectively, X > Y ) where X and Y are symmetric matrices, means that X -Y is positive semi-definite (respectively, positive definite); I N is the identity matrix with compatible dimension. · 1 refers to the 1-norm of a row (column) vector or a matrix, i.e.,

System description
Considering the coupled network composed of N identical nodes, each node is an ndimensional delayed dynamical system described bẏ where . . , f n (x i (t))) T ∈ R n is a continuous vector-valued function, and u i (t) = [u i1 (t), u i2 (t), . . . , u in (t)] T ∈ R n are the control inputs. For simplicity, we take the network model (1) as the drive system and consider the response coupled network system described as follows: where y i (t) denote the states of the response system, u i (t) denote the control inputs to realize finite-time synchronization, and the rest variables and parameters are the same as those in the drive system (1).
Definition 1 The complex networked system (1) is said to be synchronized with the drive system (2) in finite time if there exists a constant 0 ≤ T ≤ +∞, which depends on the initial values for arbitrary solutions of system (1) and system (2), such that lim t→T e(t) 1 = 0 and e(t) 1 ≡ 0 for ∀t > T, where T is called the setting time.
We need the following assumptions to study the finite-time synchronization of the system (1).

Main results
It is well known that time delays are unavoidable for complex network modeling. Therefore, it is very important to consider the dynamics for the complex networks with time delays, and finite-time synchronization analysis is obviously one of the most important problems. In order to achieve the finite-time synchronization defined in Definition 1, the controllers should be designed and brought in cooperation to the nodes of system (1). The following controllers of the paper are considered for t ∈ [t i k , t i k+1 ): where the trigger functions for node i are designed asĝ )|, and 0 <ρ < 1, 0 <ρ < 1.
However, the converse is not necessarily true. Therefore, the trigger condition (5) in this paper is weaker and easier to implement.
Subtracting (1) from (2) based on the controller (4), the following error dynamical systems are obtained in the following form for i = 1, 2, . . . , N : where Based on the controller (4) and the error system (6), the following result is derived.
Through the following theorem, it is proved that the system under the controller (4) shows no Zeno behavior. Theorem 2 Consider the drive coupled network system (1) and the response system (2) under the event-triggered controller (4) with trigger functions (5). Assume that the condition (7) is satisfied in Theorem 1, then, for a node i with e i (t i k ) 1 = 0 and e i (t i kτ (t)) = 0, t ∈ [t i k , t i k+1 ).

Numerical examples
In this section, the example is provided to demonstrate the effectiveness of the proposed approach.
Example 1 In this example, the finite-time synchronization of the time-varying delayed coupled drive system (1) and the response system (2) with the controller (4) is investigated as follows: where x i (t) = (x i1 (t), x i2 (t)) T and y i (t) = (y i1 (t), y i2 (t)) T are the state variables of the ith node for the drive and response system, respectively, x i (0) = x i,0 and y i (0) = y i,0 are the initial state values, τ (t) = sin t 2 , and the outer coupling matrices are assumed to be The nonlinear function f (·) is given by [24] f x i (t) = -Cx i (t) + Dg x i (t) , (33) in which By simple computation, we have θ 1 = 0.5303, in view of (3), (4) and the parameters of network (32). From Assumption 1 and condition (7), letting ξ = 0.5,ρ = 0.2,ρ = 0.4, we obtain k 1 ≥ 5.6828, k 3 ≥ 1.6667, and k 2 can be any positive constant. Choosing k 2 = 2, and the initial value arbitrarily chosen from (-2, 2) by uniform distribution and we get 4 i=1 e i (0) 1 = 4.4744, ∀t ∈ [-1, 0], and e i (t) = 0 for t < -1, it is derived from (8) that the network is synchronized within T = 3.7230. Figure 1 shows that x 1 (t) x 2 (t) x 3 (t) x 4 (t) and y 1 (t) y 2 (t) y 3 (t) y 4 (t) evolve with the above initial values. Figure 2 describes the time evolution of the synchronization errors with the controller. When the controllers are added to the addressed network, one can see that the synchronization can be realized within T = 3.7230. Figure 3 shows the control updates for each of the node in 2.6 seconds of the simulation, while Table 1 shows the average interevent time exhibited by each node during the simulation. It may be observed that the minimum of these values is above 0.1 s, which means  that the node that updates its control input more often performs less than 10 updates/s on average.

Conclusions
In this paper, we have dealt with the finite-time synchronization problem of time-delayed coupled networks with event-trigger controller. Sufficient conditions have been established in terms of inequalities. And the finite-time synchronization problem is solved for the addressed networked systems with time-varying delays without Zeno behavior. Without using the finite-time stability theorem, the synchronization conditions are obtained for the systems. The numerical example has been presented to illustrate the usefulness and effectiveness of the main results obtained. Furthermore, the occurrence of time delay in the nonlinear function f (x i (t)) in (1) will be the subject of further study.