Robust synchronization of uncertain Markovian jumping complex interconnected neural networks via adaptive fault-tolerant control

Abstract

This article inspects the problem of robust synchronization for uncertain Markovian jumping complex interconnected neural networks with randomly occurring uncertainties and time delays. The uncertainties considered here occur randomly and are assumed to follow certain mutually uncorrelated Bernoulli distributed white noise sequences. The presence of sensor faults may cause degradation or even instability of the entire network. Therefore, control laws are designed with sensor faults to ensure the controlled synchronization of the complex interconnected neural networks. Three types of fault-tolerant controls are designed based on the Lyapunov stability theory and adaptive schemes which include passive and adaptive fault-tolerant control laws. By constructing a new Lyapunov-Krasovskii functional (LKF) and by using Jensen’s inequality with a free-weighting matrix approach, some new delay-dependent synchronization criteria are obtained in terms of linear matrix inequalities (LMIs). By using the Lyapunov stability theory, the existence condition for the adaptive controller that guarantees the robust mean-square synchronization of complex interconnected neural networks in terms of LMIs are derived. Finally, a numerical example is presented to demonstrate the performance of the developed approach.

Introduction

During the last few years, complex dynamical networks have received considerable attention from various scientific communities, due to their broad applications in many fields including secure communication, information science, sociology, biological science, etc. The set of interconnected nodes which interact in different ways form the complex dynamical networks. Complex networks aid in modeling many practical systems, such as World Wide Web (WWW), food webs, social networks, electric power grids and so on [1], [12], [45]. In particular, synchronization is the most intriguing phenomenon in complex networks. The phenomena of synchronization appear in many real systems, such as in an array composing of identical delayed neural networks, biological systems and information sciences. The synchronization problem for coupled networks has been receiving increasing research attention and the results have been proposed in the existing literature [4], [31], [32], [43], [47], [60]. In [4], global synchronization in arrays of delayed neural networks with constant and delayed coupling has been investigated. Synchronization of coupled connected neural networks with delays has been studied in [32]. In [31], authors have analyzed the self-synchronization of Hopfield neural networks with randomly switching connections. Synchronization for an array of coupled stochastic discrete-time neural networks with mixed delays has been established in [43]. The fixed-time synchronization-based secure communication issue with a two-layer hybrid coupled networks are discussed in [47], [60] which deals with both delay-free impulses and delayed impulses to the synchronization of coupled neural networks. Synchronization of complex interconnected neural networks has also been studied extensively and corresponding results have been reported in the existing literature [5], [17], [35].

At the same time, in the real-world applications of systems, there exist time delay naturally due to the finite information processing speed and the finite switching speed of amplifiers. It is well known that time delay often causes undesirable dynamic behaviors such as performance degradation and instability of the systems. Synchronization of coupled neural networks with time delay has been discussed in [2], [3], [24], [29], [41], [40]. For instance, Cao et al. [2] deals with the global synchronization in an array of delayed neural networks with hybrid coupling, Cao and Li [3] interprets the synchronization in an array of hybrid coupled neural networks with delay. Moreover, the exponential synchronization for arrays of coupled neural networks with delayed couplings has been established in [29]. The problem of synchronization of Lur’e systems for chaotic secure communication systems with interval time-varying delay feedback control has been proposed in [24] and quasi-synchronization problem for reaction-diffusion neural networks was discussed in [40] with hybrid coupling and parameter mismatches via sampled-data control technique. However, the dynamics of complex interconnected neural networks with delayed coupling have been analyzed in recent years [27], [28]. In [27], the synchronization stability result for a class of complex dynamical networks (CDNs) with coupling delays has been established. In [28], a class of dynamical networks with different coupling delays has been considered and the authors have established several theorems on the synchronization properties by using the definition of matrix measure. Furthermore, system modeling consists of two powerful complexities such as nonlinearities and uncertainties. In the real world, nonlinear disturbances and parameter uncertainties may be subject to random changes in environmental circumstances, for instance, network-induced random failures and repairs of components, sudden environmental disturbances, and so on. Therefore, both the nonlinearities and uncertainties may occur in a probabilistic way with certain types and intensity, which is especially true in a networked environment. Therefore, robust stability analysis of neural networks with randomly occurring uncertainties (ROUs) and time delays have been studied in the existing literature [18], [23], [25], [26], [33], [53]. New delay-dependent robust stability criterion for uncertain neural networks with time-varying delays has been analyzed in [23]. In [33], a robust adaptive control scheme based on neural networks has been proposed for a class of delayed uncertain nonlinear systems. In [18], robust sliding mode control problem for a class of uncertain nonlinear stochastic systems with mixed time delays has been investigated. Recently, the authors in [39], [42] interested to study the stability analysis of semi-Markov jump systems and hybrid neutral stochastic systems respectively.

Moreover, Markovian jump systems have the advantage of modeling dynamic systems due to random abrupt variation in their structures. These systems can model physical systems with abrupt structural changes accurately, such as fault-tolerant control systems, economic systems, communication systems and other practical systems. Many researchers have made a lot of progress in Markovian jump systems with time delay, see [11], [52], [55], [56], [61]. For instance, in [52] focus the event-triggered filter design problem for delayed Markovian jump systems. Robust adaptive stabilization of uncertain stochastic jumping dynamical networks with mode-dependent mixed delays has been investigated in [56]. In [11], authors have focused on the stochastic stabilization of a class of Markovian jump systems with time delay. A general model of coupled neural networks with Markovian jumping and random coupling strengths has been introduced in [55]. The authors in [48]-[46] investigated Sliding model control and fuzzy controller design for nonlinear singularly perturbed MJSs.

On the other hand, the synchronization of complex networks is generally achieved through transferring information among interconnected nodes via couplings. Moreover, the synchronization of complex networks and adaptive control methods have been described in [10], [16], [19], [20], [22], [34], [57], [58], [59]. In [20], adaptive lag synchronization for uncertain CDNs with delayed coupling has been analyzed. The adaptive synchronization of two nonlinearly coupled CDNs with delayed coupling has been investigated in [59]. Based on Lyapunov stability theory the performance of adaptive learning control scheme for complex dynamical networks was studied in [10] and adaptive event-triggered synchronization control problem for a class of complex networks is investigated [34]. In [58], authors pointed out the projective synchronization of drive-response delayed CDNs with time-varying coupling and adaptive approach. In [22], adaptive synchronization problem for a class of uncertain dynamical complex networks against network deterioration has been studied and an adaptive approach has been proposed to adjust unknown coupling factors for the deteriorated networks. In [57], authors have proposed robust adaptive synchronization of uncertain complex networks with multiple time-varying coupled delays. Adaptive synchronization of uncertain CDN using fuzzy disturbance observer has been established in [19].

In addition, faults generally occur in actuators and sensors. Actuator faults act on the system, resulting in the fluctuation of the process variables. Sensor faults act on the sensors that calculate the system variables and do not influence the process immediately. These sensor faults will affect the process if the measurement signals are used to cause the input control signal. fault-tolerant control for complex networks have received increasing research attention in recent years, for example [9], [13], [14], [21], [44], [49]. In [21], the authors studied the adaptive fault-tolerant tracking control of near-space vehicle using T-S fuzzy models. In [14], authors have considered the active fault-tolerant control design for reusable launch vehicle using adaptive sliding mode technique and in [44] studied sliding mode fault-tolerant control for networked control systems. In [49], the problem of fault-tolerant synchronization for a class of complex interconnected neural networks with constant delay has been analyzed and fault-tolerant control laws have been designed. Recently, some interesting works on stochastic systems, memristive neural networks and stochastic reaction-diffusion neural networks have been discussed in [6], [7], [8]. Very recently, the authors in [36], [37], [38] have prominently studied the coupled NNs with Markovian jumping parameters with time-varying delays via round-robin protocol and gain scheduling approach. To the best of the author’s knowledge, the adaptive fault-tolerant synchronization for complex interconnected neural networks with constant time delays under the case of sensor faults has not yet been studied in the literature.

Motivated by the above analysis, in this paper,

• Robust synchronization for uncertain complex interconnected neural networks with ROUs and time delay under the case of sensor faults is investigated.

• Also, passive fault-tolerant controller is designed to contract the network synchronization under the sensor faults and adaptive fault-tolerant control laws are designed based on Lyapunov stability theory and adaptive control theory.

• New LKF is proposed and some sufficient conditions for synchronization are derived in the form of LMIs by using Jensen’s inequality and free-weighting matrix approach.

• Finally, a numerical example is given to illustrate the effectiveness of the proposed methods.

Notations: ${\mathbb{R}}^n$ denotes the $n$-dimensional Euclidean space and ${\mathbb{R}}^{n\times m}$ be the set of all $n\times m$ real matrices. For real symmetric matrices $X$ and $Y$, $X\ge Y$ (respectively, $X>Y$) means that the matrix $X-Y$ is a positive-semi definite (respectively, positive definite) matrix. Let $A^T$ stands for the transpose of matrix $A$. The superscript $T$ denotes the transposition of the matrix of appropriate dimension. $I$ is the identity matrix. $diag(\cdots )$ stands for block diagonal matrix. For an arbitrary matrix $B$ and two symmetric matrices $A$ and $C$, $\left[\begin{array}{cc}A& B\\ *& C\end{array}\right]$ denotes a symmetric matrix, where $*$ denotes the symmetric terms in a symmetric matrix. Let $(\Omega ,\mathbb{F},P)$ be a complete probability space with filtration ${\left\{{\mathbb{F}}_t\right\}}_{t\ge 0}$ satisfying usual conditions, where $\Omega$ is a sample space, $\mathbb{F}$ is the $\sigma$-algebra of subset of the sample space and $P$ is the probability measure on $\mathbb{F}$. Denote by $L_{{\mathbb{F}}_0}^P\left([-\varsigma ,0]\right)$ the family of all ${\mathbb{F}}_0$-measurable $C([-\varsigma ,0];{\mathbb{R}}^n)$-valued random variables $\phi =\left\{\phi \right(s):-\varsigma \le s\le 0\}$ such that ${\sup }_{-\varsigma \le s\le 0}\mathbb{E}{\parallel \phi \left(s\right)\parallel }^P<\infty$. $\mathbb{E}\{.\}$ stands for the mathematical expectation operator with respect to some probability measure $P$.