Passivity analysis of stochastic delayed neural networks with Markovian switching

Abstract

In this paper, the problem of passivity analysis is investigated for a class of stochastic delayed neural networks with Markovian switching. By applying Lyapunov functional and free-weighting matrix, delay-dependent/independent passivity criteria are presented in terms of linear matrix inequalities. The results herein include existing ones for neural networks without Markovian switching as special cases. An example is given to demonstrate the effectiveness of the proposed criteria.

Introduction

In the past decades, neural networks have been extensively used in signal processing, pattern recognition, associate memories, control, and other areas. Some of these applications require that the equilibrium points of the designed network be stable. So, it is important to study the stability of neural networks. In reality, time-delay are frequently encountered in neural networks, where a time delay is often a source of instability and oscillations. Recently, lots of sufficient conditions have been proposed to verify the asymptotical or exponential stability of delayed neural networks, see [6], [8], [14], [16], [21], [22], [26], [27], [28], [34], [35], [36], [37], [39], [40] and the references cited therein for some recent publications.

In implementation or applications of neural networks, it is not uncommon for the parameters of neural networks (e.g. connection weights and biases) change abruptly due to unexpected failure or designed switching [17]. In such a case, neural networks can be represented by a switching model which can be regarded as a set of parametric configurations switching from one to another according to a given Markovian chain. On the other hand, when one models real nervous systems, stochastic disturbances and parameter uncertainties are unavoidable to consider. Because in real nervous system, synaptic transmission is a noisy process brought on by random fluctuation from the release of neurotransmitters, and in hardware implementation of neural networks, the network parameters of the neural system may be subjected to some changes due to the tolerances of electronic components employed in the design. The robust stability of several stochastic switching neural networks with time delays was analyzed [13], [24], [32], [35].

The passivity theory intimately related to the circuit analysis method [3] has played an important role in the analysis of the stability of dynamical system, complexity, signal processing, chaos control and synchronization and fuzzy control [1], [5], [7], [11], [25], [29], [31]. In the first place, many systems need to be passive in order to attenuate noises effectively. In the second place, the robustness measure (such as robust stability or robust performance) of a system often reduces to a subsystem or a modified system that is passive. Passivity analysis is a major tool for studying stability of uncertain or nonlinear systems, especially for high-order systems, and thus the passivity analysis approach has been used in control problems for a long time to deal with robust stability problems for complex uncertain systems. The essence of the passivity theory is that the passive properties of a system can keep the system's internal stability. Recently, many authors have studied the passivity of delayed systems [9], [11], [12], [18], [19], [31], [37] and the passivity of delayed neural networks [4], [10], [15], [20], [23], [30], [33], [38], because it can lead to general conclusions on the stability using only input–output characteristics. In the stochastic neural networks, most authors have not considered the input vector in diffusion part. To the best of our knowledge, however, few authors have considered the passivity conditions for stochastic delayed neural networks with Markovian switching.

Motivated by the above discussions, in this paper, we investigate the robust passivity of stochastic delayed neural networks with Markovian switching. The parameter uncertainties are norm-bounded. By utilizing Lyapunov–Krasovskii functional and free-weighting matrix, a unified linear matrix inequality (LMI) approach is developed to establish sufficient conditions for neural networks to be robust passivity. Note that LMIs can be easily solved by using the Matlab LMI toolbox, and no tuning of parameters is required [2]. It is worth mentioning that the passivity criteria of the neural networks with Markovian switching include the passivity criteria of neural networks without Markovian switching as special cases [10], [23]. An example is provided to show the validity of the proposed passivity conditions.

Notations: The notations are quite standard. Throughout this paper, Rⁿ and $R^{n\times m}$ denote, respectively, the n-dimensional Euclidean space and the set of all $n\times m$ real matrices. W^T and W⁻¹ denote, respectively, the transpose of, and inverse of a square matrix W. The notation $X\ge 0$ (respectively, $X>0$), where X is symmetric matrices, means that X is positive semidefinite (respectively, positive definite). I_n is the n×n identity matrix. $|·|$ is the Euclidean norm in Rⁿ, $\parallel ·\parallel$ is the operator norm of a square matrix. The shorthand diag $\{M_1,M_2,\dots ,M_n\}$ denotes a block diagonal matrix with diagonal blocks being the matrices $M_1,M_2,\dots ,M_n$. Moreover, let $(\Omega ,\mathcal{F},\{{\mathcal{F}}_t{\}}_{t\ge 0},P)$ be a complete probability space with a filtration $\{{\mathcal{F}}_t{\}}_{t\ge 0}$ satisfying the usual conditions (i.e. the filtration contains all p-null sets and is right continuous). w(t) be a scalar Brownian motion defined on the probability space. $\mathrm{E}[·]$ is the mathematical expectation operator with respect to the given probability measure P. Sometimes, the arguments of a function or a matrix will be omitted in the analysis when no confusion can arise.