Elsevier

Neurocomputing

Volume 286, 19 April 2018, Pages 67-74
Neurocomputing

Brief papers
On designing state estimators for discrete-time recurrent neural networks with interval-like time-varying delays

https://doi.org/10.1016/j.neucom.2018.01.054Get rights and content

Abstract

This paper deals with designing state estimators for a class of discrete-time recurrent neural networks with interval-like time-varying delays. Based on a delay bi-decomposition idea, a proper Lyapunov functional is introduced, which takes into account more information on the interval-like time-varying delay and the neuronal activation function. This Lyapunov functional, together with an improved reciprocally convex inequality, is employed to derive a sufficient condition to design suitable Luenberger-type estimators by solutions to linear matrix inequalities. An example is taken to show the effectiveness of the proposed method.

Introduction

The last two decades have witnessed successful applications of recurrent neural networks (RNNs) to a number of information processing systems, including parallel computation, learning ability and model processing, etc. [1]. In the implementation of a real RNN, time-delays commonly occur due to the finite switching speed of amplifiers and the inherent communication time of neurons [2], [3], [4]. Time-delays are often a source of oscillation, divergence, and instability of practical engineering systems [5], [6], [7]. Therefore, delayed neural networks have become a hot research topic and some important issues, such as stability, passivity and dissipativity are addressed for various dynamic neural networks with time-delays, see, e.g. [8], [9], [10], [11], [12].

State estimation on delayed neural networks is an important issue from both theoretical and practical points of view, due to the fact that the neuron states in relatively large scale neural networks are not completely available for practical applications. The state estimation problem of a delayed neural network aims to design an estimator based on its output measurements to estimate the neuron states. This problem is first considered in [3]. However, the obtained condition is delay-independent, which is somewhat conservative, especially for a neural network with a small time-delay. In [13], a free-weighted matrix approach is introduced to derive some delay-dependent condition, leading to the notion of delay-dependent state estimation. Since then, delay-dependent state estimation has attracted more attention than the delay-independent one, aiming to seek less conservative sufficient conditions on the existence of proper state estimators. By employing a descriptor model transformation approach, a free-weighted matrix approach and an integral inequality approach, delay-dependent state estimation for neural networks with time-varying delays is extensively investigated, and a number of less conservative criteria for the estimator design are reported in the literature, see, e.g. [14], [15], [16], [17]. Nevertheless, all the above-mentioned results are based on neural networks in the continuous-time domain, while relatively few results on neural networks in the discrete-time domain. Although the delay-dependent state estimation issue on discrete-time neural networks with time-varying delays is addressed in [18], [19], [20], there is still much room for improvement.

It should be noted that most results on delay-dependent state estimation are based on the Lyapunov functional approach, in which the chosen Lyapunov functional usually includes a double and/or a triple integral/summation term [21]. Once the increment of the Lyapunov functional is calculated, the second step comes from the inequalities to be used to derive some sufficient condition which can be verified through solving a tractable numerical optimization problem [7], [22]. In this situation, and in the discrete-time domain, those cross terms appearing in the forward difference of the Lyapunov functional are bounded using some summation inequalities. In the recent years, some novel summation inequalities as a discrete counterpart of the Wirtinger-based integral inequality are proposed to analysis the stability of discrete time-delay systems [23], [24]. For the discrete-time neural networks with interval-like time-varying delays, a reciprocally convex combination should be dealt with during estimating the forward difference of the Lyapunov functional. The lower bound for such a combination is essential to derive less conservative conditions. Recently, an improved reciprocally convex inequality is reported [25], which enables us to develop a less conservative result to design suitable state estimators for the discrete-time neural networks with interval-like time-varying delays.

This paper focuses on the state estimation on the discrete-time recurrent neural networks with interval-like time-varying delays. Based on a delay bi-decomposition idea, a proper Lyapunov functional is introduced, which fully uses the information of delay and activation function. This Lyapunov functional and the Abel-based finite-sum inequality [24] are used to yield a sufficient stability criterion for the resulting error system under study, where the reciprocally convex combination occurring in the forward difference of the Lyapunov functional is bounded by an improved reciprocally convex inequality recently reported in the literature [25]. A less conservative approach is given to design desired state estimators in terms of solutions to linear matrix inequalities. An example is finally given to show the validity of the proposed result.

The notations of the paper are standard. The symmetric elements in a symmetric matrix are denoted by ‘*’, and He{A}=A+AT.

Section snippets

Problem formulation

Consider the following discrete-time delayed recurrent neural networks: {x(k+1)=Ax(k)+W0f(x(k))+W1f(x(kh(k)))+J,x(k)=ϕ(k),k[h2,0],where x(k)=col{x1(k),x2(k),,xn(k)} with xi(k) being the state of the ith neuron; f(x(k))=col{f1(x1(k)), f2(x2(k)), , fn(xn(k))} with fi(xi(k)) being the activation function of the ith neuron, which is continuous and satisfies fi(0)=0 and lifi(a)fi(b)abli+,a,bR,ab,where li and li+ are two real constants. J=col{J1,J2,,Jn} with Ji the external input on

Main results

In this section, we present an approach to design proper state estimators for the neural network (1). For simplicity of presentation, we denote Z¯1=diag{Z1,3h1+1h11Z1},Z¯2=diag{Z2,3Z2},Z¯3=diag{Z3,3Z3},Υ1=[In0n],L3=diag{l1l1+,l2l2+,,lnln+},L4=diag{l1+l1+2,l2+l2+2,,ln+ln+2},I1=[Υ10n×2n0n×2n0n×2n0n×2n0n0n0n0n0n0n×p],I2=[0n×2nΥ10n×2n0n×2n0n×2n0n0n0n0n0n0n×p],I5=[0n×2n0n×2n0n×2n0n×2nΥ10n0n0n0n0n0n×p],I6=[0n×2n0n×2n0n×2n0n×2n0n×2nIn0n0n0n0n0n×p],I10=[0n×2n0n×2n0n×2n0n×2n0n×2n0n0n0n0nIn0n×p

Numerical simulation

In this section, we take an example to show the effectiveness of Proposition 1.

Example 1

Consider the neural network (1) with A=(0.40000.30000.3),W0=(0.20.20.100.30.20.20.10.2),C=(100010),W1=(0.20.100.20.30.10.10.20.3),H=(0.40000.40).Suppose that the time-varying delay h(k) satisfies (3) and the nonlinear function f(x)=col{f1(x1),f2(x2),f3(x3)} is given by fi(s)=tanh(0.7s)0.2s,i=1,2,3.

It is not difficult to verify that the activation function f(x) satisfies (2) with li=0.2,li+=0.5,i=1,2,3.

Conclusions

The problem of state estimation for a class of discrete-time recurrent neural networks with interval-like time-varying delays has been studied. Based on the delay bi-decomposition, a proper Lyapunov functional has been introduced to exploit more information on the time-varying delay and the nonlinear activation function. An improved reciprocally convex inequality has been employed to formulate an LMI-based sufficient condition on the existence of suitable state estimators for the neural network

Lili Liu received her M.S. degree in applied mathematics from Shaanxi Normal University, Xi’an, China, in 2005 and the Ph.D. degree in applied mathematics from Xi’an Jiaotong University, China, in 2011. She is currently an associate professor in School of Mathematics and Information Science from Shaanxi Normal University, Xi’an, China. Her research interests include singular systems, switched systems, neural networks and time delay systems.

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Lili Liu received her M.S. degree in applied mathematics from Shaanxi Normal University, Xi’an, China, in 2005 and the Ph.D. degree in applied mathematics from Xi’an Jiaotong University, China, in 2011. She is currently an associate professor in School of Mathematics and Information Science from Shaanxi Normal University, Xi’an, China. Her research interests include singular systems, switched systems, neural networks and time delay systems.

Shihua Zhu received the M.Sc. degree in applied mathematics from Xi’an Jiaotong University, Xi’an, China, in 1994.In 1994, he joined Central South University, Changsha, China, where he is currently an associate professor with the School of Mathematics and Statistics, and is a research fellow with the Yuelu Research Center for Science and Technology Achievements Transformation of Colleges, Changsha, China. He chaired three national projects and co-chaired five projects from National Natural Science Foundation of China. His current research interests include software development, signal processing and machine learning.

Baowei Wu received his B.Sc. and M.Sc. in mathematics from Shaanxi Normal University, China, in 1982 and 1985 respectively, his Ph.D. in mathematics from Xi’an Jiaotong University, China, in 1998. He is currently a full professor in School of Mathematics and Information Science from Shaanxi Normal University, Xi’an, China. His current research interests include switched systems, positive systems, fractional order systems and time delay systems.

Yue-E Wang received the M.S. degree in School of Mathematics and Information Science from Shaanxi Normal University, Xi’an, China, in 2010, and the Ph.D. degree in Control Theory and Control Engineering from Northeastern University, China, in 2014. She is currently an associate researcher in School of Mathematics and Information Science from Shaanxi Normal University, Xi’an, China. Her research interests include switched systems and time delay systems.

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