Exponential H∞ synchronization of non-fragile sampled-data controlled complex dynamical networks with random coupling and time varying delay

ABSTRACT This paper investigates the problem of non-fragile sampled-data control for synchronization of complex dynamical networks with randomly coupling and time varying delay under exponential approach. By adopting an appropriate Lyapunov Krasovskii functional (LKF) and taking into consideration full information on the sampling pattern, free-matrix based integral and Wirtinger inequalities are explored leading to the establishment of sufficient conditions to guarantee the exponential synchronization stability and disturbance attenuation of the closed loop network, with a designed non-fragile controller under all randomly admissible gain variations. The results are presented in terms of Linear matrix inequalities (LMIs), which can effectively be solved by some available softwares. Finally, two simulated results are demonstrated to show the effectiveness and less conservativeness of our proposed scheme.


Introduction
In the past few decades, a lot of attention have been devoted to the study of complex dynamical networks (CDNs), which have massive applications in areas of science and engineering such as Internet, biological networks, World Wide Web, epidemic spreading networks, social networks and electric power grids (Dorato, 1998;Hu, Wang, Gao, & Stergioulas, 2012;Kwon, Park, & Lee, 2008;Lakshmanan, Mathiyalagan, Park, Sakthivel, & Rihan, 2014;Li & Chen, 2004;Li, Zhang, Hu, & Nie, 2011;Liu, Guo, Park, & Lee, 2018;Seuret & Gouaisbaut, 2013;Strogstz, 2001;Su & Shen, 2015;Wu, Park, Su, Song, & Chu, 2012;Zeng, He, Wu, & She, 2015;Zhang, He, & Wu, 2010;Zhang, Wu, She, & He, 2005). Many of these networks have shown some complexities with regard to their topologies and dynamical characteristics concerning the network nodes and their coupled units. CDNs are represented by a large interconnecting of nodes, which comprise of nonlinear dynamical systems. There are some significant and important phenomena in CDNs, which can be described by coupled ordinary differential equations such as synchronization, spatiotemporal chaos, and selforganization . Synchronization as one of the fundamental and significant phenomena of CDNs, have in recent times received much attention amongst most researchers, because of its broad applications in CONTACT Chun Yin yinchun.86416@163.com, chunyin@uestc.edu.cn the fields of mathematics, engineering, traffic systems and computer science (Rakkiyappan, Latha, & Sivaranjani, 2017;Shi, Yang, Wang, Zhong, & Wang, 2018;Yin et al., 2016). The essence of synchronization phenomenon in CDNs, is to allow all coupled nodes or some selected nodes in the network to approach the trajectory of a target node. To ensure the synchronization of CDNs, many control schemes have been deeply studied and proposed, where adaptive pinning control was studied in Guo, Pan, and Nian (2016) and Astrom and Wittenmark (1994), whereby the authors utilized pinning adaptive control strategy to ensure the control of cluster synchronization problems of their proposed network. In Yang, Cao, and Lu (2012), the authors studied synchronization of complex networks with the hybrid adaptive and impulsive control methods. With recent advancement in digital measurement, high quality computers, communication networks and intelligent instruments, continuous-time controllers are been replaced with digital controllers (Lee, Park, Kwon, & Sakthivel, 2017). Furthermore, these have resulted in improved reliability, accuracy and better stability performance. The need for sampled-data control happens to be another important control strategy which have received much attention in the study of CDNs (Lee, Wu, & Park, 2012;Li et al., 2011;Liu et al., 2018;Shen, Wang, & Liu, 2012;Wang, Zhang, & Wang, 2015;Yucel, Ali Syed, Gunasekaran, & Arik, 2017;Zhang et al., 2010;Zheng, Zhang, Zhong, Wang, & Shi, 2017). The sampleddata control system is a continuous plant connected to discrete -time controller where digital -to-analogy (D/A) and analogy-to-digital(A/D) components are used. The sampled-data control schemes have many advantages, amongst them are but not limited to easy implementation, reduction of overall size, low cost of maintenances, amount of transmitted information greatly reduced and above all, increased efficiency in bandwidth usage (Li et al., 2011;Rakkiyappan et al., 2017;Yucel et al., 2017). Additionally, choosing a suitable sampling period is an important factor to be considered in sampled-data control.
Interestingly, it is an established fact that, a larger sampling period will lead to lower communication channel occupying, fewer actuation of controller and less frequent signal transmission which are very desirable (Li et al., 2011;Wu et al., 2012). The input delay approach is one of the most widely studied methods adopted in the analysis and synthesis of sampled data systems where the sampling holder is modeled as delay control inputs (Fridman, 2010). In fact, it is of great essence to ascertain how well will such systems be influenced by a designed sampled-data controller and also, to what extent will the desired performance be achieved? In real world applications, time delays are inevitable in most physical systems (Li, Dong, Han, Hou, & Li, 2017;Lu & Chen, 2004;Park, Kwon, Park, Lee, & Cha, 2012;Yang, Dong, Wang, Ren, & Alsaadi, 2016), therefore it is important to be considered in the investigation of synchronization of CDNs under the exponential H ∞ approach. The existence of time delays which might occur as a result of, limited information channels and large-scale interconnected complex networks could lead to undesired oscillation, instability and poor performance of the CDNs (Lakshmanan et al., 2014;Liang, Wu, & Chen, 2016;Zeng et al., 2015).
That notwithstanding, modeling errors, disturbance inputs and parameter perturbations or uncertainties do occur in CDNs, which contribute to the degradation and poor performance of the network. It is important to investigate such randomly coupling and time-varying delay CDNs under exponential H ∞ synchronization with nonfragile sampled-data controller, which is very critical and important in theory and practical applications . In the past few years, many researchers devoted quality time and resources to the study of nonfragile control and filter implementation design problems with some applications in control and communication field such as altitude control of satellites, missile control, chemical process control and macroeconomic system control (Li et al., 2017;Li, Deng, & Peng, 2012). The aim was to design controllers for some given systems such that, they become insensitive to some amount of errors and deviations with regards to their gains, environmental effects, modeling errors and uncertainties. It is therefore significant for designed controllers to tolerate some level of uncertainties in their parameters. In Hou, Dong, Wang, Ren, and Alsaadi (2016), a non-fragile state estimator is designed for admissible gain variations. In addition, the study of sampled-data control, which implies the successful updating of signal transmitted from the sampler to the controller and then to the zero-order hold (ZOH) at instant of t k can experience some form of a constant signal transmission delay as investigated in Liu et al. (2018), Lee et al. (2012) and Liu and Fridman (2012). Based on the reasons above, and as our first motivation in this paper, it is prudent and beneficial to investigate the exponential H ∞ synchronization of non-fragile sampled data control for randomly coupling and time varying delay of CDNs. Furthermore, in practical implementation of controller design, there are some uncertainties which might show up resulting from analogy-digital and digital-analogy conversions, round off errors in numerical computations and deterioration of system components. These occurrences have the tendency of deteriorating the performance or causing instability of the closed-loop systems hence, our second motivation to study and design a non-fragile sampled-data controller with norm bounded uncertainties which incorporates a constant signal transmission delay for the CDNs under exponential H ∞ synchronization Wu et al., 2012). The main contributions of this work are: 1) dealing with the CDNs in terms of the randomly occurring coupling , time varying delay, external disturbances and nonlinearities. 2) addressing the fragilities of the sampled-data controller in the presence of constant signal transmission delay. 3) designing and implementing the proposed controller to guarantee synchronization of the CDNs under an unstable circumstance. 4) Two solved numerical examples which indicated less conservativeness in our results compared with some existing simulation results (Li et al., 2011;Liu et al., 2018;Su & Shen, 2015;Wu et al., 2012;Yang, Shu, Zhong, & Wang, 2016).
The rest of the paper is organized as follows: In Section 2, the exponential H ∞ synchronization problem of CDNs with randomly coupling and time-varying delays is formulated. A synchronization criterion for the exponential H ∞ problem is derived with non-fragile sampleddata control in Section 3. Section 4 gives the simulation results of two numerical examples and the effectiveness of our proposed approach. Finally, conclusions are given in Section 5.
Notation: Throughout this paper the notations are standard. 'I' stands for identity matrix with appropriate dimensions, R n denotes the n-dimensional Euclidean space; R m×n is the set of all m × n real matrices; * indicates the symmetric elements of the symmetric matrix; The symmetric matrices X and Y, the notation X > 0,Y < 0 means that the matrix X is symmetric positive definite and Y negative definite matrices. The symbol '⊗' represents Kronecker product. Any matrix not having explicit dimension is assumed to be of appropriate dimension for algebraic operation.

Problem formulation and preliminaries
Consider the following CDNs comprising of N identical coupled nodes with each node being an n-dimensional dynamical system: (1) where x i (t) ∈ R n ,p i (t) ∈ R l and u i (t) ∈ R n are the state vectors, the corresponding outputs and the control input . . , f n (x i (t))] T is a nonlinear vector valued function describing the non-linear dynamics of nodes. 1 , 2 ∈ R n×n are constant inner coupling matrices , F = (F ij ) N×N is the outercoupling matrix which also represent the network topology, C ∈ R n×n , and D ∈ R n×k . If there is a connection between node i and node j (i = j), then F ij = 1, otherwise F ij = 0 (i = j). i (t) ∈ R k is the external disturbance which belongs to L 2 [0, ∞). The diagonal elements of matrix F are defined by The scalar γ (t) is taken as the time-varying delay which satisfies where γ > 0 and υ > 0 are known constants. δ 1 (t) ∈ R denotes a stochastic variable, which in the form of a Bernoulli's distribution sequences and defined by Below indicates the probability of stochastic variable δ 1 (t): with δ 1 ∈ [0, 1] being a known constant. Also, E{δ 1 (t)} is known to be the expectation of δ 1 (t), therefore we have Assumption 2.1: Let f (•) : R n → R n be a continuous vector -valued function which satisfies the following sectorbounded condition: where Q 1 and Q 2 are known constant matrices of appropriate dimensions.
The nonlinear function in (7) is very general which entails the well known Lipschitz condition as a special case. It is assumed that s(t) ∈ R n is the state trajectory of the unforced isolated node which satisfiesṡ = f (s(t)) withq = Cs(t) as the output of the unforced isolated node s(t). Then, the error vector becomes ϑ i (t) = x i (t) − s(t), whereby the synchronization error dynamics of (1) is obtained as follows: . Then the output error betweenp i (t) andq(t) has the following form: The control signal for the synchronization is generated by utilizing the Zero-Order-Hold (ZOH) function with a sequence holding times with the sampling interval defined as for all k ≥ 0, d > 0 is the largest sampling interval. The sampling is non-periodic, although it is assumed to be bounded. Considering the error dynamics (8), we adopt the design of a memory set of sampled-data state feedback controller in the form where u id (•) is a discrete-time control signal, t k represent the sampling instant of ith node and K 1i is the appropriate feedback control gain matrix to be estimated, and ι been the constant signal transmission delay. α(t) is the stochastic variable used to indicate the randomly occurring controller fluctuations, which follows the Bernoulli distributed sequence taking on values as stated in Hu et al. (2012): 1 : controller fluctuation occurs, 0 : controller fluctuation does not occur (13) Below indicates the probability and expectation of the stochastic variable α(t): with α 1 ∈ [0, 1] being a known constant. Let E{α(t)} be the expectation of α(t), therefore we can have The perturbation phenomenon is considered in the controller design as uncertainties in the form K 1i (t) been the controller gain fluctuations. K 1i (t) has the following representations: where B i and L 1i are known appropriately dimensioned constant matrices and (t) being an unknown matrix function which satisfies the condition Now, consider the state feedback controller (12) in the error dynamics (8), which becomes: It is obvious that, (18) can be represented in the kronecker form as: Remark 2.1: The occurrence of controller gain fluctuations might result from actuator degradations, hence, the need for readjustment of the controller gains during implementations (Dorato, 1998). These parameter uncertainties are unavoidable which would eventually affect the stability and performance of the network if not handled well.The non-fragile sampled-data control is considered to ensure synchronization of the CDNs with such parameter perturbations or uncertainties.
We now present the following definitions and lemmas to help derive the main results in this paper.  (1) is exponentially synchronized if the closed-loop error dynamics (19) is exponentially stable, when there exist two constants α > 0 and β > 0, with (t) = 0 such that, the following condition holds: where a non-zero (t) ∈ L[0, ∞) and the parameter is known as the H ∞ -norm bound or the disturbance attenuation level.
Lemma 2.1 (Free-matrix based inequality Zeng et al., For a given symmetric positive matricesR ∈ R n×n with Z 1 , Z 3 ∈ R 3n×3n and any matrices Z 2 ∈ R 3n×3n , and the following inequality holds:

Remark 2.2:
It is important to note that, the free matrices in the inequality of Lemma 2.1 gives a better relaxation in deriving the stability criteria which ensures the achievement of tight bound. It is easy to arrive at some well established integral inequalities as special cases of Lemma 2.1. An instance is lettingN 1 = [Y T , 0] T ,N 2 = 0, Z 1 = diag{X, 0}, Z 2 = 0, and Z 3 = 0, this reduces the above lemma to the case in Zhang et al. (2005, Lemma 2). In another situation, 2 , the integral inequality becomes the well known Wirtinger integral inequality (Seuret & Gouaisbaut, 2013).

Main results
In this section, we first establish sufficient conditions, which allow the error dynamic system (19) of the CDNs to be exponentially synchronized with i (t) = 0 under the designed non-fragile sampled data control for the CDN(1). For the sake brevity in our presentation of the main results, we first denotẽ To ensure simplicity of matrix representation, we use e T i = . . , 15) to represent block entry matrices.
Moreover, the desired control gain matrix is given by Proof: Consider the following Lyapunov-Krasovskii function (LKF) for the error system (19): where Now considering the infinitesimal operator L of V(t), which is defined as follows: Hence, calculating the time derivative of V(t) along the trajectory of the system (19), we have From Lemma 2.1, we shall have the integral part of (29) as whereÑ 1 ,Ñ 2 ∈ R 3n×n ,M 1 ,M 2 ∈ R 3n×n Combination of (29) and (30) Consider Lemma 2.2 and the integral term of (33) For appropriately dimensioned matrices F i , (i = 1, 2), it is easy to derive the following where the following inequalities hold Equation (33) is rewritten as For a matrixF with appropriate dimensions and any given scalars σ 1 and σ 2 , we derive from (19): Based on (7), with a scalarτ , it follows that (42) also, it is equivalently represented as Thus, adding the left-hand side of (41)-(43) to E{LV(t)}, where, Hence, is a convex combination of t − t k and d − (t − t k ) , therefore < 0, if and only if (23) with the following inequalities hold By considering the Schur complement, (45) and (46) are equal to (23), and (24) respectively.
It would be realized that, the terms containing t − t k and t k+1 − t vanishes before t k and after t k , that is V i (0) = 0(i = 6, 7), and by considering the generalized Itô's formula, Moreover, we can derive the following Likewise, from V 1 (t), it can be established that Therefore, considering Equation (48) and (50), it follows that: where According to Definition (2.1), we can infer from (51) that, the error dynamics (19) is exponentially synchronized with the decay rate α. This completes the proof.
Remark 3.1: Considering an appropriate LKF which is very important in the derivation of less conservative results. Based on the time-dependent LKF approach, the involvement of V 6 (t) and V 7 (t) which in our case contain t k and t k+1 terms ensured the full use of the available information pertaining to the actual sampling formation. Hence an improvement in the conservativeness of our proposed results. In the cases of Zhang et al. (2010) and Wu et al. (2012), such terms were not included, hence our Theorem 3.1 is deemed more efficient and practical.
Remark 3.2: Our Theorem 3.1, provide a new synchronization criterion for (1). It is obvious that, our given results been formulated by LMIs, can be readily verified using already existing tools such as Matlab LMI tools box. The LMIs in our case, also took into consideration transmission delays in the non-fragile sampled-data controller design and exponential decay rate α.

Remark 3.3:
When δ 1 (t) = 1 is considered in (53) with no disturbances ( (t) = 0) and transmission delays as well as absence of randomness in the controller gain uncertainties, with similar system considered in Wu et al. (2012), Liu et al. (2018) and Su and Shen (2015).

Numerical examples
This section presents two numerical examples to illustrate the validity and effectiveness of the proposed method in this paper.
Case (1): Given that, the inner coupling matrices 1 = 0 and 2 = 0.5 0 0 0.5 with the random coupling probability δ 1 = 1. The nonlinear function f (·) is taken as which can be found that, f (·) satisfies (7) with Choosing σ 1 = 0.3, σ 2 = 0.4, γ (t) = m(t) = 0.2 + 0.05 (10 sin(t)), accordingly by simple calculations, we have γ = 0.25,m = 0.25, υ = 0.5, ω = 0.5, α 1 = 0, the transmission delay in the controller ι = 0. By applying Theorem 3.2, the system under consideration is similar to that in Li et al. (2011), Liu et al. (2018 and Wu et al. (2012), when c = 1, it is considered as a special case in our paper. In this example, our maximum sampling period d = 1.2151 which is larger and less conservative as compared with the others in Table 1, where our results showed 5.04% improvement of the sampling period over that of Yang, Shu, et al., (2016), 16.50% in   the case of Liu et al. (2018), 118.03% increment in the case of Wu et al. (2012) and 124.64% improvement in the case of Li et al. (2011) having the exponential decay rate(α) set at 0.3 in our simulations. If one considers a sampling period d ∈ (1.1564,1.2151], then with exception of our proposed scheme, all the others as stated will not be able to achieve the desired error synchronization. Therefore, our results is less conservative and effective. Now, utilizing the Matlab LMI toolbox with α 1 = 0 , ι = 0 and the above given uncertainties parameters, the LMIs (54) and (55) Figure 2 shows the error synchronization of the CDNs (19) with our control design inputs. The results shows that, the error synchronization converges to the target trajectory (zero). The trajectories for the error dynamic system without the control inputs and the control inputs into the system are shown in Figures 1 and 3 respectively. with the random coupling probability δ 1 = 0.5, external disturbance of (t) = 2/(1 + t 2 ) into the system. We also set the following parameters as σ 1 = 0.3, σ 2 = 0.4, γ = 0.25,m = 0.25, υ = 0.5, ω = 0.5, α 1 = 1, the delay in the controller, ι = 0.05. According to Theorem 3.1 with sampling interval d = 0.465 and setting = 0.3., we achieved the following controller gain parameters: Chua's circuit is considered in this example as an unforced isolated node of CDNs (Hu et al., 2012), which is represented by the following equations: ϑ 3 (t) = −σ 2 ϑ 2 (t), whereσ 1 = 10,σ 2 = 14.87, ρ(ϑ 1 (t)) = 0.68ϑ 1 + 1 2 (−1.27 − 0.68)(|ϑ 1 + 1| − |ϑ 1 − 1|). It can be calculated from Equation (7) Q 1 =  The outer coupling matrix F = (F) N×N is taken as We assume the non existence of uncertainties in the controller design (α 1 = 0), with other given parameters as α = 0.3,τ = 0.07, σ 1 = 2, σ 2 = 3 and the time-varying delay chosen as γ (t) = 0.03 + 0.01 sin(t) which implies γ = 0.04 and υ = 0.01 by some calculations. By solving Theorem 3.2, the maximum sampling period d is 0.1858, which showed less conservativeness as compared with other results in Table 2. The resulting nonfragile sampled data controller gains are shown in Figure 4:      Figures 5, 6, and 7 which indicate the error synchronization with control inputs, the control inputs response and finally, the error system without control inputs respectively. From the above simulations, it can be concluded that our designed nonfragile sampleddata controller design is effective, less conservative and will also guarantee the synchronization CDNs with randomly occurring perturbations in the controller.

Conclusion
The exponential H ∞ synchronization of randomly coupling CDNs with time-varying delay, under non-fragile sampled-data control have been investigated. In this work, the time-varying Lyapunov Krasovskii functional is formulated in which the well known free-matrix based and Wirtinger inequalities have been used in the derivations of less conservativeness of our results as compared with other existing ones. Finally, two examples have been simulated to illustrate the feasibilities and effectiveness of our proposed scheme.