Event-triggered sampled-data synchronization of complex networks with time-varying coupling delays

This paper investigates the event-triggered sampled-data synchronization problem of complex networks with time-varying coupling delays. The sampled-data controller is designed with event-triggered mechanisms. Some results in terms of a linear matrix inequality are obtained to guarantee the asymptotical synchronization of complex networks with time-varying coupling delays. Lastly, we test the effectiveness of the proposed method via some numerical examples.


Introduction
as: (1) event-triggered nonuniform sampling is considered in complex networks with timevarying coupling delays; (2) the nonlinear part of the node system is also handled.

Problem formulation
Consider the following complex networks with time-varying coupling delays consisting of N nodes via an event-triggered control approach: where i = 1, 2, . . . , N , N is the number of nodes, k = 0, 1, . . . , ∞, x i (t) ∈ R n denotes the state vector associated with the ith node, t i k denotes the event-triggered instant and it is determined by the transmission error and the state error, u i (t i k ) ∈ R n denotes the designed control law where the transmitted data packets are utilized along with the event-triggered control happening, τ (t) denotes the time-varying delay and satisfies the conditionτ (t) < ν ≤ 1, A ∈ R n×n , B ∈ R n×m are known constant matrices, f : R n → R m is a continuous vectorvalued function, c is a constant coupling strength, Γ ∈ R n×n denotes the inner coupling matrix, G = (g ij ) N×N is the coupling configuration matrix: if nodes i and j (i = j) are connected, then g ij > 0, otherwise g ij = 0, the diagonal elements of matrix G are defined by g ii = -N j=1,i =j g ij , i = 1, 2, . . . , N . t k denotes the whole sampling instant and it is irregular. Moreover, the serial number of transmitted data packets, denoted as i k , may be discontinuous for the existence of event-triggered mechanism where the transmitted data packets refer to the ones successfully arriving at the plant. i k ∈ N denotes the serial number of the transmitted data packet such that {i 0 , i 1 , i 2 , . . .} ⊆ {0, 1, 2, 3, . . .}.
Let e i (t) = x i (t)s(t) be the error vectors, where s(t) ∈ R n is a solution of a target node satisfyingṡ(t) = As(t) + Bf (s(t)). The sampled-data synchronization feedback controller is designed as where K i is the feedback gain matrix with appropriate dimensions. Then, the error dynamics of (1) can be obtained as follows: where i = 1, 2, . . . , N and g(e i (t)) = f (x i (t))f (s(t)). It is clear that (3) can be rewritten in a vector-matrix form, where e(t) = [e T 1 (t), . . . , e T N (t)] T ,ḡ(e(t)) = [g T (e 1 (t)), . . . , g T (e N (t))] T , K = diag{K 1 , . . . , K N }, The error between the current sampling instant and the latest transmission instant can be calculated asē(t i k +l ) = e(t i k +l )e(t i k ) where l = 1, 2, . . . , d, d = i k+1i k . The eventtriggered condition can be provided as where δ > 0 is a given scalar parameter, Φ is a positive-definite weighting matrix with appropriate dimensions. If the event-triggered condition is satisfied, then the transmitted signals will be sent. Otherwise the sampling signals will not be sent.
Assumption 1 There exists a constant h > 0 such that where h denotes the upper bound of the interval between two consecutive sampling instants.

Assumption 2 ([19])
There exists a diagonal matrix Λ = diag{λ 1 , . . . , λ N } > 0 such that the nonlinear part of the node system satisfies the following condition in the domain of definition: Combining (4) and (5), the final error dynamics can be described aṡ where (6). The control objective is to design the controller gain matrix K such that the error dynamics (8) is asymptotically stable, i.e., e(t) → 0 as t → ∞.

Main results
In this section, the event-triggered sampled-data synchronization scheme will be given.

Theorem 1
Given the scalars h > 0, δ > 0, the error dynamics (8) is global asymptotically stable concerning with the event-triggered condition (5), if there exist matrices P = P T > 0, with appropriate dimensions, and the controller gain matrix K , such that the following condition holds: where Proof Construct the following Lyapunov-Krasovskii functional:
Note that the condition (9) Moreover, the sampled-data controller gain is given by K = J -1 V .

Numerical examples
In this section, two numerical examples are given to demonstrate the effectiveness of the proposed event-triggered sampled-data synchronization scheme.
Example 1 ( [22]) A complex network with five coupled identical nodes is described as follows: Setting 1 = 2 = 1, and δ = 0.1, the condition (12) in Corollary 1 is feasible and the following result is obtained: Through the above two numerical examples, we can find that the event-triggered sampled-data synchronization is guaranteed for complex networks with time-varying coupling delays.

Conclusion
In the present work, we deal with the event-triggered sampled-data synchronization problem of complex networks with time-varying coupling delays. In future work, more performance requirements for the event-triggered sampled-data synchronization of complex networks will be considered in a uniform network topological structure.