Elsevier

Neurocomputing

Volume 151, Part 2, 5 March 2015, Pages 770-775
Neurocomputing

Short Communication
Robust delay-depent stability criteria for uncertain neural networks with two additive time-varying delay components

https://doi.org/10.1016/j.neucom.2014.10.023Get rights and content

Abstract

This paper considers the problem of robust stability of uncertain neural networks with two additive time varying delay components. The activation functions are monotone nondecreasing with known lower and upper bounds. By constructing of a modified augmented Lyapunov function, some new stability criteria are established in terms of linear matrix inequalities, which is easily solved by various convex optimization techniques. Compared with the existing works, the obtained criteria are less conservative due to reciprocal convex technique and an improved inequality, which provides more accurate upper bound than Jensen inequality for dealing with the cross-term. Finally, two numerical examples are given to illustrate the effectiveness of the proposed method.

Introduction

In the past few decades, neural networks have been one of the hottest issues for their successful applications in various fields such as signal processing, pattern recognition, model identification and optimization problem [1], [2], [3], [4]. In hardware implementation of neural networks, it is well known that time delay is frequently occurred, and the existence of time delay may cause instability and poor performance [5], [6], [7], [8], [9], [10]. Therefore, much effort has been devoted to the delay depent stability analysis of delayed neural networks [11], [12], [13], [14], [15], [16], [17], [18], [19], since delay-depent stability criteria are generally less conservative than delay-indepent ones especially when the size of the time delay is small. In recent years, much attentions have been received in robustness analysis for uncertain neural networks due to the existence of modeling errors, external disturbance, and parameter fluctuations [20], [21], [22], [23], [24].

In the systems considered above, the time delay in a state was assumed to appear in singular form. However, in practical situations especially networked controlled systems, signals sometimes transmitted from one point to another two segments of networks. Therefore, a system with two additive time varying delay components has been considered as a new model of time-delay systems due to variable transmission conditions [25], [26], [27], [28], [29], [30], [31], [32], [33], [34]. Zhao et al. [30] proposed a stability criterion for neural networks with two additive time-varying delay components by using free-weighting matrix method. Han et al. [31] improved the result in [30] by employing a convex polyhedron method. Tian et al. [32] derived a less conservative stability criterion by constructing augmented Lyapunov functional and using reciprocally convex method. In [33], some less conservative results were derived by using convex polyhedron method or reciprocally convex method. Recently, the delay-depent stability criteria for generalized neural networks with two delay components were investigated in [34]. Although these results and analytic tools are elegant for the stability of neural networks with two additive time-varying delay components, there still have room for further improvement. On one hand, Jenson inequality [30], [31], [32], [33], [34], which neglects some terms, was employed to estimate the upper bound of some derivative of Lyapunov–Krasovskii functional. On the other hand, activation functions were not fully utilized for constructing the Lyapunov–Krasovskii functional. In addition, to the best of our knowledge, the robust stability for neural networks with two additive delay components has not been investigated by any researchers.

Inspired by the discussion above, in this paper, we aim at giving a robust stability criterion of uncertain neural networks with additive time-varying delay components, and providing a less conservative stability criterion of delayed neural networks without uncertainties by some new techniques. Two time additive time-varying delays are defined as h(t)=h1(t)+h2(t), where h1(t) is the time delay induced from sensor to controller and h2(t) is the delay induced from controller to actuator [25], [26], [27], [28], [29], [30], [31], [32], [33], [34]. Based on the modified augmented Lyapunov–Krasovskii, which is fully utilized the information of activation functions, some stability criteria are derived by reciprocally convex method [35] and an improved inequality [36], which provides more accurate upper bound than Jensen inequality for dealing with the cross-term. Finally, two numerical examples are given to demonstrate that the proposed conditions which are less conservative than some existing ones.

Notations: Throughout this paper, denotes the elements below the main diagonal of a symmetric block matrix. I denotes the identity matrix with appropriate dimensions, Rn denotes the n dimensional Euclidean space, Rm×n is the set of all m×n real matrices, and . refers to the Euclidean vector norm and the induced matrix norm. For symmetric matrices A and B, the notation A>B (respectively, AB) means that the matrix AB is positive definite (respectively, nonnegative). diag{} denotes the block diagonal matrix.

Section snippets

Problem statement

The uncertain delayed neural networks is described byẏ(t)=(C+ΔC)y(t)+(A+ΔA)g(y(t))+(B+ΔB)g(y(th1(t)h2(t)))+J,where y(t)=[y1(t),y2(t),,yn(t)]TRn is the neuron state vector associated with n neurons, C=diag{c1,c2,,cn}>0, g(y(·))=[g1(y1(·)),g2(y2(·)),,gn(yn(·))]TRn denotes the continuous activation function, J=[J1,J2,,Jn]T is an exogenous input vector, ARn×n and BRn×n are the connection weight matrix and the delayed connection weigh matrix, respectively. h1(t) and h2(t) are two

Main results

In this section, we first propose a stability criterion for uncertain neural networks with two additive time-varying delay components. For the sake of simplicity of matrix and vector representation, eiR15n×n(i=1,2,,15) are defined as block entry matrices (for example e4=[000I00000000000]T). The other notations are defined asξT(t)=[xT(t)xT(th1)xT(th1(t))xT(th2)xT(th2(t))xT(th)xT(th(t))fT(x(t))fT(x(th1(t)))fT(x(th2(t)))fT(x(th(t)))ẋT(t)1hh(t)(thth(t)x(s)ds)T1h(t)(th(t)tx(s)ds)TpT

Numerical examples

In this section, two numerical examples are given to show the effectiveness of the proposed method.

Example 4.1

Consider the system of (28) with the following parameters:C=[2002],A=[1111],B=[0.88101],K1=[0000],K2=[0.4000.8].The maximum value of upper bound h2 compared with the results in [31], [32], [33], [34] with different h1 under μ1=0.7,μ2=0.1 and μ1=0.7,μ2=0.2 is listed in Table 1, Table 2, respectively. From Tables 1 and 2, one can know clearly that the results obtained by Corollary 3.1 can provide

Conclusion

In this paper, the delay-depent stability problem for uncertain delayed neural networks with two additive time-varying delay components has been studied. Based on the modified augmented Lyapunov functional, some new delay-depent asymptotic stability criteria are derived by using reciprocally convex method and more accurate upper bound of the integral form. The effectiveness of the theoretical results has been demonstrated by two numerical examples.

Acknowledgments

This research was supported by the Daegu University Research Scholarship Grants and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2063350 and 2014R1A1A4A01003860).

Yajuan Liu received the B.S. degree in Mathematics and Applied Mathematics from Shanxi Normal University, Linfen, China, in 2010, and M.S. degree in Applied Mathematics, University of Science and Technology Beijing, Beijing, China, in 2012. She is currently working toward the Ph.D. degree in Electronic Engineering in Daegu University, Deagu, Korea. Her current research interests include nonlinear systems and control of time-delay systems.

References (38)

Cited by (41)

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Yajuan Liu received the B.S. degree in Mathematics and Applied Mathematics from Shanxi Normal University, Linfen, China, in 2010, and M.S. degree in Applied Mathematics, University of Science and Technology Beijing, Beijing, China, in 2012. She is currently working toward the Ph.D. degree in Electronic Engineering in Daegu University, Deagu, Korea. Her current research interests include nonlinear systems and control of time-delay systems.

S.M. Lee received the B.S. degree in Electronic Engineering from Gyungbuk National University in 1999, and Ph.D. degree at Department of Electronic Engineering from POSTECH, Korea, in 2006. Currently, he is an Assistant Professor at Division of Electronic Engineering in Daegu University. His main research interests include robust control theory, nonlinear systems, model predictive control and its industrial applications.

H.G. Lee received the M.S. and Ph.D. degrees from the School of Computer Science and Engineering, Seoul National University, Seoul, Korea, in 2001 and 2007, respectively. He was a Senior Engineer with Samsung Electronics from 2007 to 2010 and a Post-Doctoral Research Fellow with the Georgia Institute of Technology, Atlanta from 2010 to 2012. Currently he is an Assistant Professor with the School of Computer and Communication Engineering, Daegu University, Korea. His current research interests include embedded system design, low power system, chip multi-processor, and flash-based storage design.

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