Further results on passivity analysis for uncertain neural networks with discrete and distributed delays

Abstract

The problem of passivity analysis of uncertain neural networks (UNNs) with discrete and distributed delay is considered. By constructing a suitable augmented Lyapunov-Krasovskii functional(LKF) and combing a novel integral inequality with convex approach to estimate the derivative of the proposed LKF, improved sufficient conditions to guarantee passivity of the concerned neural networks are established with the framework of linear matrix inequalities(LMIs), which can be solved easily by various efficient convex optimization algorithms. Two numerical examples are provided to demonstrate the enhancement of feasible region of the proposed criteria by the comparison of maximum allowable delay bounds.

Introduction

In the past several years, various neural network models have been investigated due to their extensive applications in areas of pattern recognition, signal processing, associative memories, optimization problems and even mechanics of structures and materials [8], [40]. It should be noted, due to the finite switching speed of amplifiers, time-delay occurs in many neural networks regardless how small it may be. Precisely time-delay is a main factor that can cause performance degradation and/or the instability of neural networks. Thus, delay-dependent stability and stabilization problem for neural network with time-delays receives considerable attention than delay-independent ones because the information on the size of time-delay is utilized in delay-dependent criteria, which lead to reduce the conservatism of stability and stabilization criteria [1], [2], [4], [5], [12], [14], [15], [19], [21], [22], [23], [24], [25], [26], [29], [30], [31], [34], [35], [39], [41], [43].

On the other hand, the passivity theory originated from circuit theory plays an important role in the stability analysis of dynamical systems [11], [20], [37], [46], which relates the input and output to the storage function. It has been extensively applied in many physical systems such as networked control, fuzzy control, sliding mode control and signal processing. Therefore, the study on passivity analysis for UNNs with time-delays has been widely investigated in [3], [6], [7], [9], [13], [16], [17], [18], [44], [48], [49], since parametric uncertainties, which sometimes affect the stability of systems, are also undesirable dynamics in the hardware implementation of neural networks.

The existing passivity criteria can be grouped into delay-independent and delay-dependent ones. In general, delay-dependent criteria are less conservative than the delay-independent ones. The delay-independent passivity analysis is addressed for neural networks with time-delay in [18]. For neural networks with discrete delay, delay-dependent passivity conditions are established for UNNs in [9], [13], [17], [44], [48], [49]. By considering some useful terms which are ignored in previous literatures and utilizing free-weighting matrix techniques, the enhancement of feasible region of passivity criteria is shown in [49]. By constructing an augmented LKF, improved delay-dependent passivity conditions are proposed in [13]. By proposing a complete delay-decomposing approach and utilizing a segmentation technique, improved conditions for passivity of neural networks are presented in [48]. By the consideration of newly constructed LKF, improved sufficient conditions to guarantee the passivity for neural networks with time-varying delays and parameter uncertainties are obtained in [17]. For neural networks with both discrete and distributed delays, the problem of delay-dependent passivity analysis is studied in [3], [6], [7], [16]. In [3], new delay-dependent conditions for this problem are obtained by using a novel LKF together with the linear matrix inequality approach. In [6], based on direct delay decomposition idea and free-weighting matrix approach, several new delay-dependent passive criteria are derived. In [16], by construction of an augmented LKF and utilization of zero equalities, improved passivity criteria for the networks are derived in terms of LMIs via new approaches. In addition, in [7], the robust passivity problem for UNNs with discrete and distributed delays was addressed by delay partitioning and reciprocally convex approaches. Since delay bounds for guaranteeing the passivity are recognized as one of the most important index for checking the conservatism of criteria, all the works [3], [6], [9], [13], [16], [17], [36], [37], [44], [48], [49] show their advantages of the proposed methods via comparison of maximum delay bounds with the previous works. Therefore, how to choose a suitable LKF and obtain an upper bound of time-derivative of it such that the considered neural networks are passive play key roles to reduce the conservatism of passivity criteria. Very recently, one of the most remarkable methods in reducing the conservatism of stability criteria is Wiritinger-based integral inequality [32] which reduced Jensens gap effectively. In [50], by employing a Wirtinger-based integral inequality, less conservative passivity conditions are obtained for neural networks with both discrete and distributed delays. Recently, by proposing a new integral inequality which is more general than some existing integral inequalities, the stability problem of neural networks with time-varying delays is studied in [45], and less conservative delay-dependent stability criteria are presented. By constructing an improved LKF and refined Jensen inequalities, new delay-dependent conditions that ensure the passivity of neural networks with interval time-varying delay are derived in [36]. Therefore, To further improve the issue of delay-dependent passivity for UNNs with both discrete and distributed delays is necessary and interesting.

Motivated by the above discussions, in this paper, the problem of passivity for UNNs with both discrete and distributed delays is further investigated. Firstly, an improved LKF, which fully utilizes information of the neuron activation functions and two delay-product-type terms, is constructed. Secondly, a novel integral inequality, which includes several existing integral inequalities as special cases, is utilized to deal with the estimation of the derivative of the LKF candidate. Thirdly, novel delay-dependent criteria that ensure the passivity of the system are presented in terms of LMIs. Finally, a comparison of maximum delay bounds obtained by our proposed methods with those in many existing literatures [3], [6], [7], [9], [13], [36], [42], [44], [49], [50] is conducted, which supports that passivity criteria provided in this paper can enhance the feasible region effectively and significantly.

Throughout this paper, the following notation is used: C^T represents the transposition of matrix C. ${\mathcal{R}}^n$ denotes n -dimensional Euclidean space and ${\mathcal{R}}^{n\times m}$ is the set of all m × n real matrices. P > 0 means that P is positively definite. Symbol * represents the elements below the main diagonal of a symmetric block matrix, and diag{⋅⋅⋅} denotes a block diagonal matrix. Sym(X) is defined as $Sym\left(X\right)=X+X^T$.