An LMI approach to delay-dependent state estimation for delayed neural networks

Abstract

This paper is concerned with the state estimation problem for a class of neural networks with time-varying delay. Comparing with some existing results in the literature, the restriction such as the time-varying delay was required to be differentiable or even its time-derivative was assumed to be smaller than one, are removed. Instead, the time-varying delay is only assumed to be bounded. A delay-dependent condition is developed to estimate the neuron states through observed output measurements such that the error-state system is globally asymptotically stable. The criterion is formulated in terms of linear matrix inequality (LMI), which can be checked readily by using some standard numerical packages. An example with simulation results is given to illustrate the effectiveness of the proposed result and the improvement over the existing ones.

Introduction

Various classes of neural networks have been increasingly studied in the past few years, due to their practical importance and successful applications in many areas such as combinatorial optimization, signal processing and communication [8], [13], [16]. These applications greatly depend on the dynamic behaviors of the underlying neural networks. As is well known, time delay may occur in the process of information storage and transmission in neural networks. In electronic implementation of neural networks, the time delay is often time-variant, and even varies dramatically with time because of the finite switch speed of amplifiers and faults in the electrical circuits. Up to now, the stability analysis for delayed neural networks has attracted considerable attention, and a large amount of results have been available in the literature, see, for example, [1], [3], [4], [5], [7], [9], [15], [17], [21], [22], [23], [24], [25], [26], [27], [28].

On the other hand, the neuron states are not often completely available in the network outputs in many applications. Therefore, the state estimation problem of neural networks becomes significant for many applications [14], [20]. The main objective of the problem is to estimate the neuron states through available output measurements such that the dynamics of the error-state system is globally stable. Recently, the state estimation problem for neural networks has attracted certain attention and some progresses have been made [6], [12], [19]. Wang et al. firstly investigated the state estimation problem for neural networks with time-varying delay in [20]. Under the precondition that the time-derivative of the time-varying delay was smaller than 1, a linear matrix inequality (LMI) condition was derived to guarantee the existence of the expected state estimator. In [14], the problem of state estimation was also addressed for delayed neural networks under a weak assumption that the time-varying delay was required to be differentiable. However, the proposed condition was expressed in terms of a matrix inequality, not an LMI, which corresponds to a nonlinear programming problem. The authors in [18] dealt with the state estimation problem for a class of neural networks with discrete and distributed delays. The existing results related to this issue can be generally classified into two categories: delay-independent criteria [20] and delay-dependent criteria [14], [18]. The delay-independent case is irrespective of the size of the time delay. While the delay-dependent case is relevant to the size of the time delay. Generally speaking, the delay-dependent case is considered to be less conservative than the delay-independent case, especially when the size of time delay is small.

In this paper, the state estimation problem is studied for a class of neural networks with time-varying delays. As mentioned above, the time-varying delay in [14], [20] must satisfy the assumptions that it was differentiable and its time-derivative was less than 1 or a constant. This would greatly limit the applications of the results proposed in [14], [20]. Here, such restrictions on the time-varying delay are removed. By defining a new Lyapunov–Krasovskii functional, the boundedness of the time-varying delay is only required. A delay-dependent condition is developed to estimate the neuron states through available output measurements such that the error-state system is globally asymptotically stable. The criterion is formulated in terms of an LMI, which can be checked efficiently by using some standard numerical packages [2], [10].

Notations: For a real square matrix X, the notation $X>0$ ($X⩾0,X<0,X⩽0$) means that X is symmetric and positive definite (positive semi-definite, negative definite, negative semi-definite, respectively). The shorthand notation $\mathrm{diag}\{M_1,M_2,\dots ,M_N\}$ denotes a block diagonal matrix with diagonal blocks being the matrices $M_1,M_2,\dots ,M_N$. I is the identity matrix with appropriate dimension. The superscript “T” represents the transpose. $|·|$ is the Euclidean norm in ${\mathbb{R}}^n$. Let $L_{{\mathcal{F}}_0}^b([-d,0];{\mathbb{R}}^n)$ denote the family of all ${\mathcal{F}}_0$-measurable $C([-d,0];{\mathbb{R}}^n)$-valued variables $\xi =\{\xi (\theta ):-d⩽\theta ⩽0\}$ such that ${\sup }_{-d⩽\theta ⩽0}|\xi (\theta ){|}^p<\infty$. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible dimensions for algebra operations.