Improved on Extended Dissipative Analysis for Sampled-Data Synchronization of Complex Dynamical Networks With Coupling Delays

This paper concerns the issue of extended dissipative analysis for complex dynamical networks with coupling delays under a sampled-data control scheme. Firstly, we derive the input delay method and combine it with an appropriate Lyapunov functional, which can make full use of the information of the sampling period. Secondly, novel sufficient synchronization criteria are established by applying Jensen’s inequality, Wirtinger’s integral inequality, a new integral inequality, free-weighting matrix technique, and convex combination method. Moreover, we focus on the extended dissipative analysis issue, which includes <inline-formula> <tex-math notation="LaTeX">$\mathcal {L}_{2}-\mathcal {L}_{\infty} $ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$\mathcal {H}_{\infty} $ </tex-math></inline-formula>, passivity, and dissipativity performance in a unified formulation. These conditions can express in Linear matrix inequalities (LMIs) restrictions, which can solve with readily accessible software. Finally, two numerical examples illustrate the effectiveness and reduced conservatism of our developed method.


I. INTRODUCTION
During the last few decades, complex dynamical networks (CDNs) have been applied in real-world systems such as social networks, the internet, brain networks, food chains, and disease-spreading networks [1], [2], [3]. There are many connected nodes in this system, each of which represents a dynamic system. Depending on the network topology, some of these nodes are typically connected. A graph is a mathematical concept that describes these networks. In such diagrams, vertices represent system membership, whereas edges represent their interaction. In addition, the time delay is unavoidable in many scenarios, including neural networks, electrical engineering systems, chemical or process control systems, the internet, transportation systems, etc. It can cause instability, oscillations, and poor system performance. Because coupling delay is a prevalent time delay in CDNs, it is critical to think about how time delay affects CDNs.
The associate editor coordinating the review of this manuscript and approving it for publication was Wonhee Kim . Furthermore, synchronization is vital in CDNs because it can explain various phenomena, such as synchronous data exchange on the World Wide Web and the internet. As a result, it has become a hot topic that has been thoroughly researched. Li and Chen [4] presented CDNs with coupling delay for continuous-and discrete-time systems in the last decade. Gao et al. [5] studied synchronization for a general class of CDNs with coupling delays to discover a novel criterion for ensuring that the system is asymptotically stable. In the case that all nodes of networks cannot achieve synchronization by themselves. So, it is challenging to design a practical controller that can synchronize the networks. Various control schemes can be applied in CDNs, such as the pinning control [6], [7], [8], the impulsive control [9], [10], [11], the intermittent control [12], [13], [14], and the adaptive control [15], [16], [17]. Recently, communication networks and computers have advanced to a high-speed version, and practically digital signal techniques have replaced analog signal methods, which provide a more consistent result. Thus, the sampled-data system has received much attention [18], [19], [20], [21]. VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ As is known, a vital point in sampled-data systems is to place importance on the desired controller under a more considerable sampling period. Li et al. [22] investigated the synchronization of regular CDNs for the first time, including coupling time-varying delay, where sampled-data controllers were designed. An adequate condition was obtained using the Jensen inequality to achieve the exponential stability of the synchronization error system. Wu et al. [23] presented the exponential synchronization of CDNs with coupling timevarying delay. The above authors reduced a synchronization criterion using the same Lyapunov functional and a new integral inequality that takes full merit of the detail on delays. In [24], such a problem was further addressed by applying a reciprocally convex technique with an improved inequality that can yield a tighter upper bound than Jensen inequality. The previously reported results [22], [23], [24] were enhanced further in [25] by using a new time-dependent Lyapunov functional method and Wirtinger-based integral inequality [26]. Chen et al. [27] introduced Wirtinger-based double integral inequalities [28] to reduce the conservatism of the synchronization criterion by incorporating the new Lyapunov functional in triple integral inequalities. Obtaining a more sampling period has proven to be a significant issue in earlier studies [22], [23], [24], [25], [27]. As a result, implementing less conservative synchronization requirements for CDNs with sampled-data control has become a hot topic, and designing CDNs with sampled-data control is worthwhile. However, it should be noted that while these studies focus on using the inequality methodology to improve results, the information on the sawtooth structure on the input delay is not fully utilized, leaving room for improvement. In addition, the works mentioned above take no account of external disturbances, which is unworkable in practice. Thus, to implement sampled-data synchronization in CDNs, it is required to account for external disturbances and fully benefit the sawtooth function class of sampling period, which is the inspiration for our paper.
Furthermore, the topic of network performance analysis, which is based on the relationship between input and output, is essential in practical systems. Over the last few decades, analyzing performance has been essential in engineering and science applications [30], [31], [32], [33], [34], [35]. Zhang et al. [35] introduced a generic performance called extended dissipativity, which effectively encompasses the passivity, L 2 − L ∞ , H ∞ , and dissipativity index and it has received additional research in the last few years. In [30], the extended dissipative performance was utilized in neural network systems with continuous time-varying delay. For the first time, Yang et al. [36] studied the extended dissipative for synchronization of CDNs with coupling delay. However, The vital information of t − t k and t k+1 − t, ∀t ∈ [t k , t k+1 ) is not fully maximum potential. As a result, it is only reasonable to seek a different perspective to construct a less conservative condition for synchronizing sampled-data CDNs with coupling delays. From those mentioned above, we dedicate ourselves to overcoming the issue of extended dissipative analysis for sampled-data synchronization for CDNs with coupling delays. The main contribution of this article highlights as follows • The proposed CDNs with coupling time-varying delays  are comprehensive models for the other existing CDNs  [22], [23], [25], [29], [37], [38]. We take the external disturbances into each node which is not considered in [22], [23], [25], [27], [29], [37], and [38].
• We construct a time-dependent Lyapunov functional different from the references in [22], [23], [24], [25], and [27], and the advantage information on discrete sample point t k is fully used.
• It is worth mentioning that the positive definitiveness of the proposed Lyapunov functional requires only sampling times that are not necessarily throughout the sampling periods. So, we derive a new inequality that applies to prove the exponential synchronization of CDNs.
• A less conservative criterion is obtained under the novel Lyapunov functional using Jensen's inequality, Wirtinger's integral inequality, a new integral inequality, the free-weighting matrix technique, and the convex combination approach. Moreover, our results can verify Chua's circuit system. This paper is separated in the following way. The problem formulation, definitions, assumptions, and lemmas provide in Section 2. The main results are present in Section 3, which considers the exponential synchronization via sampled-data control and the extended index analysis. Two numerical examples manifest the effectiveness and reduced conservatism of the existing results in section 4. Finally, we sum up this letter in Section 5.
Notation: This paper contains the following notations, R n stands for n-dimensional Euclidean space and R m×n is the set of m×n real matrices. The symmetric matrix P > 0 is that the matrix P is positive definite. λ min (P) and λ max (P) mean the minimum and maximum eigenvalues of P, sym{P} defines P + P T The superscript T is the transpose. The symbol * represents the symmetric entries of the symmetric matrix. The symbol ⊗ denotes the Kronecker product, and diag{. . . } is the block diagonal matrix.

II. PRELIMINARIES
Consider the complex dynamical networks, which consist of N identical linked nodes, each of which is an n-dimensional dynamical system: where x i (t) and u i (t) are, respectively the state variable and the control input of each node. f (x i (t)) ∈ R n is a nonlinear vector valued function defining the dynamics of ith nodes.
c > 0 is the coupling strength. The internal-coupling matrices are (1) , (2) ∈ R n×n and the matrix G = G ij N ×N ∈ R N ×N is the outer-coupling configuration matrix, in which G ij is satisfied as follows: if there exists a string from node i to node j (i = j), G ij > 0; otherwise, G ij = 0. While, the diagonal elements of G is defined as G ii = − N j=1,j =i G ij . ω i (t) ∈ R n is the external perturbation, which belongs to L 2 [0, ∞), z i (t) ∈ R n , is the output, and J is a given matrix with proper dimensions. The function τ (t) represents the coupling time-varying delay that satisfies: Letν(t) = f (ν(t)), where ν(t) ∈ R n is the state response of the unforced isolate node. Then the error vector is Let η i (t) be the output of isolate node, which defines as η i (t) = J ν i (t). Then, the output errorẑ i (t) is described as followŝ At the same time, a zero-order-hold (ZOH) function with a sequence of hold times of 0 = t 0 < t 1 < · · · t k < · · · , lim k→+∞ t k = +∞ is assumed to be used to create the control signal. Then, the sampled-data feedback controller constructs in the form as: where K i denotes a set of feedback controller gain matrices that can be designed. r i (t k ) is a discrete mearsurement of r i (t) at the sampling time t k . Here, we assume that t k+1 − t k = d k ≤ h u for all integer k ≥ 0, where h u > 0 stands for the largest sampling period. Then, we have the following error closed-loop system: Then, error dynamical (5) can be transformed as ways: Here, we give some assumptions, definitions, and lemmas that are required to obtain the new synchronization criterion. Assumption 1 [39] The continuous function f : R n → R n satisfies the following condition: where U and V are known constant matrices of compatible dimensions.
Lemma 1: (Reciprocally Convex Lemma [40]). For any vectors x 1 , x 2 , matrices M > 0, S, and scalars 1 > 0, 2 > 0, satisfying 1 + 2 = 1, the following inequality holds: Lemma 2: [26] For any matrix R > 0. Then for any continuous function x in [a, b] → R n the inequality VOLUME 10, 2022 [41]). Let X , Y , Z be given matrices such that Z > 0, then Lemma 4: Consider the dynamical system (6). Then there exist two scalars ϕ 1 and ϕ 2 satisfying where Proof: From (6), we have the following inequality for any t ∈ [t k , t k+1 ), By utilizing the Cauchy-Schwarz inequality, we obtain from that (13) Using the Cauchy-Schwarz inequality once more, we can deduce from (14) that Furthermore, it is clear from (8) that Therefore We can immediately obtain (12) by utilizing the Gronwall-Bellman Lemma to (16). The proof is now complete. [42]. However, it should be noted that the mentioned paper assumed that time delays are expected to be constant. In general, the case of time-varying delays is more practical than constant delays. This paper considers time-varying coupling delays, which are more general than constant time delays.

III. MAIN RESULTS
This section will show you how to solve the problem using the design method. Firstly, a less conservative synchronization assures the system (6) is exponentially stable with the sampled-data control (4). Secondly, the disturbance is input to satisfy extended dissipative analysis. For the sake of fluent presentation, we provide some notations as follows A. STABILITY ANALYSIS The first theorem gives an exponential synchronization condition for CDNs (6) with a non-disturbance.
Theorem 1: For given scalars h u , τ u and τ d , if there are matrices P > 0, where then, the error system (6) is exponentially synchronized and the feedback controller gain matrix can be obtained as Proof: Consider the following Lyapunov functional represent for system (6): e 2βs r T (s)T 2 r(s)ds VOLUME 10, 2022 V 6 (t) is obtained aṡ V 7 (t) can be presented aṡ r T (s)Sr(s)ds.
Utilizing Lemma 2, we obtain So,V 7 (t) is bounded bẏ An estimation ofV 8 (t) is gained bẏ Use Lemma 2 to obtain Note that the inequalities, An expression ofV 9 (t) is followed bẏ By using the free-weighting matrix method [43], [44], the below equations hold for any matrices N 1 and N 2 with compatible dimensions. and Furthermore, rely on the well-known inequality for any matrix N > 0, −2x T y ≤ x T N −1 x + y T N y, it is obvious to given that Combining (28), (36) and (38), we obtain the new estimation ofV 5 (t) as followṡ Combining (35), (37) and (39), we obtain all estimations oḟ V 9 (t) as followṡ An expression ofV 10 (t) is calculated bẏ Applying Lemmas 1 and 2, based on the inequality (21), a new calculating ofV 10 (t) transforms to Thus, we get all estimations ofV 10 (t) as followṡ An expression ofV 11 (t) is calculated bẏ Applying Lemmas 1 and Jensen's inequality, relied on the inequality (22), a new calculating ofV 11 (t) transforms to So,V 11 (t) can be gained aṡ Moreover, we can obtain from condition (7) for any ε > 0 For any proper dimension M , the following equation is clearly to be obtained = e 2βt ζ T (t)sym (e T 1 + e T 6 )M T (−e 6 + e 11 + c(G ⊗ (1) )e 1 + c(G ⊗ (2) )e 4 + Ke 2 ) ζ (t) = ζ T (t)e 2βt 12 ζ (t).
Adding Eq.(24)-(47) to the right-hand sides ofV (t) yieldṡ i and the other notations are provided in (17). From (48), it can be rearranged condition for system (6) as follows: where (1) . The inequality (49) presents a convex combination of d 1 (t) and d 2 (t). It is notably that (1) Applying Schur complement to (19) and from condi- Now, from (51), we havė Therefore, it can be readily presented that, for t ∈ [t k , t k+1 ) We can conclude from Lemma 4 and (53) that for t k ≤ t < t k+1 On the other hand, it should be pointed that V 4 (0)−V 9 (0) = 0 and hence where . Using (54) and (55), we can get From Definition 1, the error system (6) is exponentially stable. The proof is now complete. Remark 4: It is vital to seek an appropriate Lyapunov functional, which used to derive a less conservative condition. Thus, we introduce novel six (t k , t k+1 )-dependent terms VOLUME 10, 2022 V 4 (t) − V 9 (t), which is different from the construction of the Lyapunov functional in [22], [23], [24], [25], and [27].
We not only consider the information on the upper bound of the time-varying delay but also take the full merit of the available information about the actual sampling instant t k .
Remark 5: To compare the designed controller in [24] and [27], we employed the one free-matrix parameter M in Eq. (47) instead of two parameters in the zero equation in [24] and [27]. By means that two weighting parameters β 1 and β 2 are required to obtain the sampled-data feedback controller gain matrix K in [24] and [27]. However, we can get it without introducing more variables.

B. PERFORMANCE ANALYSIS
The below theorem introduces the extended dissipative performance criterion for CDNs (6) with nonzero ω(t) ∈ L 2 [0, ∞), under the zero initial condition.
Then, for t ∈ [0, t f ], (68) result in Thus, from (60), we obtain By combining (69) and (70), the system (6) is described as extended dissipative. The proof is now complete. Remark 6: The novelty of our method is that we take the external disturbances into each node which is not considered in [22], [23], [25], [27], [29], [37], and [38]. Moreover, we obtain newly exponential synchronization with extended dissipative containing, passive, H ∞ , L 2 −L ∞ and dissipative performance. The conditions are more general than those in [22], [23], [25], [27], [29], [37], and [38]. Therefore, we can notice that their conditions cannot be simulated to our examples. Moreover, we construct a new time-dependent Lyapunov functional, which is different from the proposed in [22], [23], [24], [25], and [27], and the advantage information on discrete sample point t k is fully used. Moreover, the positive definitiveness of the proposed Lyapunov functional is required only at sampling times that are not necessarily throughout the sampling periods. So, we derive a new inequality that applies to prove the exponential synchronization of CDNs.
Remark 7: Compared to the findings in [22], [23], [24], [25], [27], [29], [36], and [38], the result in this study is less conservative by constructing the appropriate Lyapunov functional and the technique for estimating the upper bound of its derivative. In contrast to the Lyapunov function in [40], [41], [42], and [43], we fully consider the critical information t − t k and t k+1 − t, ∀t ∈ [t k , t k+1 ). Additionally, compared to the convex combination technique and Jensen inequality, the inverse of the first-order approach, mixed convex combination and some effective integral inequalities can offer a more precise upper bound.
Remark 8: In this study, we address the sampled-data synchronization problem of CDNs with fixed coupling and time-varying delay in each dynamical system and obtained less conservative results. Because the sampled-data controller gain matrices must be obtained using system parameters, the proposed method is unsuitable for CDNs with time-varying coupling and multiple time delays. As a result, there is still much room for further investigation into obtaining the sampled-data synchronization criteria of CDNs with time-varying coupling. Some more effective methods, such as the proposed in [45] and [46], inspire us to conduct additional research.
Remark 9: Some free matrices are introduced in this paper using a time-dependent Lyapunov functional with complete information on the actual sampling instant t k , a new inequality, and a convex combination approach. As a result, the construction and computation technique of the Lyapunov functional are the primary keys to improving the outcomes of this work. All of this leads to a reduction in our results's conservatism compared to recent works and, in particular, numerical examples.

IV. NUMERICAL EXAMPLES
This part focuses on using two numerical examples to manifest the effectiveness of the suggested method in the above theorems.

V. CONCLUSION
This study uses a sampled-data controller to handle the problem of extended dissipative exponential synchronization of CDNs with time-varying coupling delays. An aperiodic sampled-data control scheme has been devised to solve this challenge, with the sampling period specified as time-varying but bounded. Then, using an enhanced Lyapunov function, integral inequalities, the free-weighting matrix technique, and the convex combination approach, a new adequate condition for strict LMIs was discovered. These conditions can also be used to investigate the extended dissipativity analysis issue, which includes the passivity, L 2 −L ∞ , H ∞ , and dissipativity performance in a unified formulation. Finally, two numerical examples indicate that our finding is better than the previous references, highlighting how this paper has developed. It is worthwhile to mention that the method in this article can be investigated further and applied to more complicated systems such as neutral-type CDNs [51], stochastic CDNs [52], T-S fuzzy CDNs [53].