Synchronization Approach for Drive-Response Complex Dynamical Networks with Randomly Occurring Controller Gain Fluctuations and Uncertainties

This paper investigates the nonfragile H ∞ synchronization problem of complex dynamical networks with randomly occurring controller gain fluctuations and uncertainties. These randomly occurring phenomena are described by independent stochastic variables satisfying Bernoulli distributions, which are adopted to model more realistic dynamical behaviors of the complex networks. By applying the Lyapunov-Krasovskiimethod, delay-dependent criteria are established to ensure that the synchronization can be achieved with the prescribed H ∞ disturbance attenuation. Moreover, the obtained results do not rely on the derivatives of time-varying delays. A set of nonfragile controllers are further designed in terms of linearmatrix inequality (LMI) approach. Finally, a numerical example is given to illustrate the effectiveness of our theoretical results.


Introduction
In the past decade, there have been intensive researches on dynamical properties of complex networks.This increasing interest in complex dynamical networks is mainly due to their wide ranging implications and applications in the real world, such as computer networks, communication networks, biological networks, and social networks [1][2][3][4][5][6][7][8].Roughly speaking, complex dynamical networks consisted of a large set of interconnected nodes displaying collective behaviors.Specially, the synchronization problem of complex dynamical networks is becoming a hot research area in recent years, which is a universal phenomenon in nature.As a result, many synchronization methods have been developed for complex dynamical networks [9][10][11].
It is worth mentioning that there are inevitable disturbances in complex dynamical networks caused by environmental circumstances, which gives rise to the  ∞ synchronization problem.Following this research line, several  ∞ synchronization results have been reported in the literature; see, for example, [12][13][14][15].In addition, it is generally known that time delays exist ubiquitously in almost all practical applications, which may lead to system performance degradation [16][17][18][19].Due to their impact, there have been some efficient synchronization results of complex dynamical networks with various time delays [20,21].Another important issue in the complex dynamical networks is the uncertainty.Since explicit descriptions of the network parameters are always hardly obtained, robust synchronization approaches against uncertainties are proposed.Some initial attention has been focused on this problem [22].
On the other hand, there is a class of complex dynamical networks that cannot achieve synchronization by themselves, such that synchronization controllers should be designed.It should be pointed out that, due to A/D conversion, D/A conversion, and finite word length in computations, it is always very expensive or even impossible to implement the designed controller gains precisely [23][24][25][26][27][28].Thus, nonfragile controllers are needed to tolerate the uncertainty in practical situations.Very recently, the issue of randomly occurring uncertainties has been introduced for neural networks and chaotic systems.However, to the best of the authors' knowledge, the nonfragile  ∞ synchronization problem for complex dynamical networks with randomly occurring controller 2 Mathematical Problems in Engineering gain fluctuations and uncertainties has never been tackled in the previous literatures and remains challenging, which motivates us to conduct this study.
In this paper, by introducing the complex dynamical networks subject to time-varying delays, external disturbances, and randomly occurring uncertainties, the  ∞ synchronization problem is investigated.In particular, nonfragile controllers are designed, in which the randomly occurring controller gain fluctuations are taken into consideration.The parameter uncertainties and coupling delays are assumed to be time-varying and norm bounded.Compared with the existing results, the main contributions of this paper can be summarized as threefold.Firstly, the complex dynamical networks model adopted in this paper is more general, which can better simulate the networks in the applications.Both self-time-varying delays and coupling time-varying delays are considered in the model.Secondly, it is the first time to establish the nonfragile  ∞ synchronization for complex dynamical networks with randomly occurring controller gain fluctuations and uncertainties.By utilizing the Lyapunov-Krasovskii method, novel  ∞ synchronization criteria are derived in terms of LMIs.Finally, the obtained delaydependent results impose no constraints on the derivative of time-varying delays, which are more applicable in the real world.
The remainder of this paper is organized as follows.Section 2 introduces the complex dynamical networks model and presents several assumptions and lemmas.The main results based on the LMI approach are established in Section 3. Section 4 gives a numerical example to demonstrate the effectiveness of our derived criteria.Finally, the paper is concluded in Section 5.
Notation.The notation in the paper is standard.R  denotes the  dimensional Euclidean space and R × represents the set of all  ×  real matrices. and 0 represent identity matrix and zero matrix with appropriate dimensions, respectively.The notation  > 0 means  is real symmetric and positive definite, and the superscript "" denotes matrix transposition.Pr{} means the occurrence probability of the event .E{} stands for the mathematical expectation of the stochastic variable . ⊗  denotes the Kronecker product of matrices  and .Moreover, in symmetric block matrices, * is used as an ellipsis for the terms that are introduced by symmetry and diag{⋅ ⋅ ⋅ } denotes a block-diagonal matrix.Finally, if not explicitly stated, all matrices are assumed to have compatible dimensions.
where   () = ( 1 (),  2 (), . . .,   ())  ∈ R  represents the response state vector of the th node,   ∈ R  denotes the control input to be designed for the th node, and   () is the disturbance on the th node.
Remark 3. Compared with the existing results, our proposed drive-response complex dynamical networks model is more general.When  = 1, the model can be utilized to describe synchronization problem of chaotic systems or neural networks.Moreover, the random appearance or disappearance of the parameter uncertainties is introduced in the model to reflect more complex behaviors of the real networks.
Define the synchronization errors as follows: such that the following synchronization error dynamics can be obtained: where f(,   ()) fl (,   ()) − (,   ()).Then, in order to deal with the real world controller implementation, the following nonfragile state feedback controllers with controller gain fluctuations are designed where   ∈ R × is the controller gain to be determined and the real valued matrix Δ  () denotes the controller gain fluctuation.Moreover, Δ  is assumed to have the following structure: where Υ  () ∈ R × and  = 1, 2, . . .,  is an unknown timevarying matrix satisfying and   ,   are known constant matrices.The stochastic variable () ∈ R is a Bernoulli distributed sequence, which describes the phenomena of randomly occurring controller gain fluctuation.() is defined by 1, controller gain fluctuations happen, 0, controller gain fluctuations do not happen, with where  ∈ [0, 1] is a known constant.Moreover, it is assumed that the stochastic variables () and () are mutually independent.Consequently, the closed-loop synchronization error dynamics can be obtained as which can be further rewritten as where Before proceeding, the following definitions and lemmas are given for subsequent analysis.Definition 4. The  ∞ synchronization is said to be reached if there exists a constant  > 0 such that for any nonzero () ∈ L 2 .
Then,  +  +       < 0, if and only if there exists a scalar  > 0 such that  +  −1   +    < 0, or equivalently The main purpose of the paper is to design a set of nonfragile controllers (10) for the response complex dynamical networks such that, for admissible external bounded disturbances, the  ∞ synchronization is achieved with a prescribed attenuation level .

Main Results
In this section, delay-dependent criteria will be established to solve the  ∞ synchronization problem of complex dynamical networks in the presence of randomly occurring controller gain fluctuations and uncertainties.Then, a design procedure is proposed for the nonfragile controllers in terms of LMIs.Theorem 7.For given scalars ,  ∈ [0, 1], and  and the upper bounds of time-varying delays  and , the  ∞ synchronization of the complex dynamical networks can be achieved with the given controller gains   ,  = 1, 2, . . ., , if there exist symmetric positive definite matrices , ,  ∈ R × , and constants  1 ,  2 > 0 such that Π < 0, where with Proof.Choose the Lyapunov-Krasovskii function candidate as follows: Note that (16) where Ã is defined in (21).
The infinitesimal operator L of () is defined as follows: Taking the derivative of ( 23) along the solution of system ( 24) yields By Lemma 5, it holds that Moreover, it can be obtained by Assumption 1 that where F1 and F2 are defined in (21).
Combining inequalities ( 27)-( 28 In order to show the  ∞ performance , the following function is defined: Then, it can be obtained from ( 27)-( 28) that   < 0, if E{  ()Π() +  ė  () ė () +  ė  () ė ()} < 0 holds, where Π (32) Then, according to Schur complement lemma [31], it can be verified that   < 0 if Π < 0 holds, where Note that − −1  ≤ −2 holds.By substituting ( 2) and ( 11) into (33) and performing a congruence transformation of diag{, , , , , , , , , } to (33), it follows that Π < 0 if the following inequality holds: where Therefore, it can be obtained by Lemma 6 that if Π < 0 holds, then   < 0. When () ̸ = 0, under zero initial conditions, it can be verified that the condition Π < 0 can guarantee that E{∫ ∞ 0 [  ()() −  2   ()()]} < 0, which implies that the  ∞ synchronization is achieved in the sense of Definition 4 and completes the proof.Remark 8.The two stochastic variables satisfying Bernoulli random binary distributions are adopted to character the randomly occurring phenomena in the synchronization problem for complex dynamical networks.It can be observed that when  = 1, the controllers (11) are degenerated to the general nonfragile controllers.By utilizing the stochastic information, a better synchronization performance can be obtained with less conservatism.Remark 9.It is worth mentioning that it is a very expensive or even impossible task to obtain the real values or the boundedness of the derivatives of time delays in many applications.Regarding this aspect, our established synchronization conditions impose no constraint on the derivatives of the time-varying delays () and (), which is applicable to more general case in the real world.Moreover, the derived results in this paper can be directly applied to deal with the cases of constant time delays.
Based on Theorem 7, the following theorem is presented to solve the  ∞ synchronization controller design problem in terms of LMIs.
Proof.The proof follows directly from Theorem 7 by letting  := .

Numerical Example
In this section, an illustrative example is presented to show the effectiveness and usefulness of our proposed control strategy.
Consider two complex dynamical networks (1) and ( 7) with three nodes and the state vector of each node being two-dimensional, such that  = 3 and  = 2.The network matrices are given as The inner coupling matrix is assumed to be Γ = diag{1, 1} and the outer coupling matrix  is given as The time-varying delays are chosen as () = 0.2 + 0.05 sin  and () = 0.4 + 0.1 cos .Accordingly, one has  = 0.25 and  = 0.5.The nonlinear function (,   ()) is taken as such that (,   ()) can satisfy Assumption 1 with The parameter uncertainties satisfy condition (11) with Set the initial states of the two complex dynamical networks with random values.Figure 1 shows the state trajectories of the synchronization errors without the controllers and the state trajectories of the controlled synchronization errors are shown in Figure 2. It can be clearly seen that the two complex dynamical networks can achieve the  ∞ synchronization with the designed controllers, which demonstrates our proposed method.

Conclusion
This paper is concerned with the problem of nonfragile  ∞ synchronization of complex dynamical networks with randomly occurring controller gain fluctuations and uncertainties.In particular, two independent stochastic variables satisfying Bernoulli distributions are adopted to model these randomly varying phenomena.By constructing appropriate Lyapunov-Krasovskii functionals and using stochastic analysis, delay-dependent sufficient conditions are established to guarantee the  ∞ synchronization of the drive-response complex dynamical networks.Moreover, the nonfragile controllers are further developed based on the obtained results in terms of LMIs.In the end, the effectiveness of our proposed strategy is demonstrated through a simulation example.Our further research would be to extend the current results to the cases with time-varying topologies and pinning control strategy, which are more comprehensive in the real world.