Improved Delay-dependent Robust Stability Analysis for Neutral-type Uncertain Neural Networks with Markovian jumping Parameters and Time-varying Delays
Introduction
In the last few decades, the study of neural networks has shown an increasing research for their wide application in many fields such as pattern recognition, signal processing, optimization problem, knowledge acquisition and so on [1], [2]. It is well known that the stability has been proved to be one of the most important behaviors for neural networks, meanwhile, time delay often appears in neural networks due to the signal transmission lags between neurons, and it is frequently the reason of instability and poor performance in neural networks. Therefore, the stability analysis problem of delayed neural networks have received much attention in recent years, and a number of results related to this problem have been published, see, for example, in [3], [4], [5], [6], [7], [8], [9], [10]. Furthermore, it is common that the time delay occurs not only in system states or outputs but also in the derivatives of system states, the systems containing the information of past state derivatives are called neutral-type systems. Accordingly, the stability analysis of neutral-type neural networks has also been received considerable attention and lots of works were reported in recent years [11], [12], [13], [14], [15].
On the other hand, as an important kind of hybrid systems, Markovian jumping systems have been widely studied in the past decades due to their advantage of modeling many practical dynamic systems, such as manufacturing systems, networked control systems, economics systems, fault-tolerant control systems, etc., and lots of works on stability analysis, controller synthesis and filter design have been focused on the study for Markovian jumping systems [16], [17], [18], [19], [20], [21]. Recently, there were lots of research works on the dynamics analysis for delayed neural networks with Markovian jumping parameters have been reported in the literature [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34]. For example, for neural networks with Markovian jumping parameters and time delays, the problem of stability analysis and passivity analysis have been investigated in [22], [23], [24] and [27], [28], respectively. And the same problems have been proposed in [29], [30], [31], [32] and [33], [34] for the neutral-type delayed neural networks with Markovian jumping parameters. It is worth mentioning that, although there are already many works to deal with the problem of dynamic analysis to those neural networks, they are still conservative to some extent, for example, the technique to deal with the cross products in most of those works was Jensen inequality, it will lead to some conservativeness of the achieved results, which leaves great room for further research.
In this paper, the problem of robust stochastic stability in the mean square for neutral-type uncertain neural networks with Markovian jumping parameters and time-varying delays is investigated. By constructing a novel augmented Lyapunov-Krasovskii functional based on the idea of delay partitioning, and using new effective techniques(reciprocally convex approach and Wirtinger-based integral inequality), some delay-dependent stochastic stability conditions are obtained in terms of LMIs. Numerical examples are given to show the effectiveness of the achieved criteria.
Notation: Throughout this paper, for symmetric matrices X and Y, the notation (respectively, ) means that the matrix is positive semi-definite (respectively, positive definite); I is the identity matrix with appropriate dimension; MT represents the transpose of the matrix M; Rn denotes the n-dimensional Euclidean space; represents a zero matrix with m×n dimensions; denotes the Euclidean norm for vector or the spectral norm of matrices; sym(A) denotes ; (Ω, F, P) is a probability space, where Ω is the sample space, F is the σ-algebra of subsets of the sample space, and P is the probability measure on F; and ; denotes the family of all measurable ; -valued random variables such that , where stands for the expectation operator with respect to some probability measure P. The notations () is used to denote a symmetric positive-definite (positive-semidefinite) matrix. In symmetric block matrices or complex matrix expressions, we use an asterisk ⁎ to represent a term that is induced by symmetry, and stands for a block-diagonal matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.
Section snippets
System description and preliminaries
Consider the neutral-type neural networks with Markovian jumping parameters and mixed delayswhere is the neuron state vector; , is the neuron activation function vectors; is a right-continuous Markov process defined on the probability space which takes values in a finite set with transition probability
Main results
For simplicity of vector and matrix representation, we define
Numerical examples
In this section, numerical examples will be addressed to show the effectiveness of the proposed methods. Example 1 Consider the uncertain neural networks (1) with the following parameters
Conclusion
In this paper, the problem of robust stochastic stability for neutral-type neural networks with Markovian jumping parameters and time varying-delays is considered. By choosing novel augmented Lyapunov-Krasovskii functional and effective techniques to deal with the cross product, some novel delay-dependent stability conditions are obtained in the form of LMIs which can be solved bye Matlab LMI toolbox. Numerical examples have also been given to show that our results are less conservative than
Acknowledgment
The authors would like to thank the Editor and the anonymous reviewers for their valuable comments and suggestions which improve the quality and presentation of this paper. Dr. J.H. Park gives special thanks to his friend, H. Lee, for the continuous support and encouragement of his work.
This work of J. Xia was supported National Natural Science Foundation of China under Grant 61004046, 61104117. Also, the work of J.H. Park was supported Basic Science Research Program through the National
Jianwei Xia is an Associate Professor of the School of Mathematics Science, Liaocheng University. He received the Ph.D. degree in Automatic Control from Nanjing University of Science and Technology in 2007. Since October 2013, he is working as a Postdoctoral Research Associate in the Department of Electrical Engineering, Yeungnam University, Kyongsan, Korea. His research topics are robust control, stochastic systems, and neural networks.
References (39)
- et al.
Stability analysis of delayed cellular neural networks
Neural Networks
(1998) - et al.
Delay-difference-dependent robust exponential stability for uncertain stochastic neural networks with multiple delays
Neurocomputing
(2014) - et al.
On stability analysis for neural networks with interval time-varying delays via some new augmented Lyapunov-Krasovskii functional
Communications in Nonlinear Science and Numerical Simulation
(2014) - et al.
Fault-tolerant control of Markovian jump stochastic systems via the augmented sliding mode observer approach
Automatica
(2014) - et al.
Global exponential stability in Lagrange sense for neutral type recurrent neural networks
Neurocomputing
(2011) - et al.
Global asymptotic stability to a generalized Cohen-Grossberg BAM neural networks of neutral type delays
Neural Networks
(2012) New sufficient conditions for global stability of neutral-type neural networks with time delays
Neurocomputing
(2012)- et al.
A delay partitioning approach to delay-dependent stability analysis for neutral type neural networks with discrete and distributed delays
Neurocomputing
(2013) Improved delay-dependent stability of neutral type neural networks with distributed delays
ISA Transactions
(2013)- et al.
Parameter-dependent robust stability for uncertain Markovian jump systems with time delay
J. Franklin Institute
(2011)
New robust control for uncertain stochastic Markovian jumping systems with mixed delays based on decoupling method
Journal of the Franklin Institute
Filtering for neutral Markovian switching system with mode-dependent time-varying delays and partially unknown transition probabilities
Applied Mathematics and Computation
Stability of stochastic Markovian jumping neural networks with mode-dependent delays
Neurocomputing
Robust stability analysis for Markovian jumping interval neural networks with discrete and distributed time-varying delays
Chaos, Solitons and Fractals
New Passivity Criteria for Fuzzy Bam Neural Networks with Markovian Jumping Parameters and Time-Varying Delays
Reports on Mathematical Physics
Stability analysis for a class of neutral-type neural networks with Markovian jumping parameters and mode-dependent mixed delays
Neurocomputing
Global exponential stability of neutral high-order stochastic Hopfield neural networks with Markovian jump parameters and mixed time delays
ISA Transactions
Delay-dependent stability for neutral-type neural networks with time-varying delays and Markovian jumping parameters
Neurocomputing
Passivity analysis for neural networks of neutral type with Markovian jumping parameters and time delay in the leakage term
Commun. Nonlinear Sci. Numer. Simul.
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Jianwei Xia is an Associate Professor of the School of Mathematics Science, Liaocheng University. He received the Ph.D. degree in Automatic Control from Nanjing University of Science and Technology in 2007. Since October 2013, he is working as a Postdoctoral Research Associate in the Department of Electrical Engineering, Yeungnam University, Kyongsan, Korea. His research topics are robust control, stochastic systems, and neural networks.
Ju H. Park received the Ph.D. degree in Electronics and Electrical Engineering from POSTECH, Pohang, Republic of Korea, in 1997. From May 1997 to February 2000, he was a Research Associate in ERC-ARC, POSTECH. In March 2000, he joined Yeungnam University, Kyongsan, Republic of Korea, where he is currently the Chuma Chair Professor. From December 2006 to December 2007, he was a Visiting Professor in the Department of Mechanical Engineering, Georgia Institute of Technology. Prof Park׳s research interests include robust control and filtering, neural networks, complex networks, multi-agent systems, and chaotic systems. He has published a number of papers in these areas. Prof. Park severs as Editor of International Journal of Control, Automation and Systems. He is also an Associate Editor/Editorial Board member for several international journals, including IET Control Theory and Applications, Applied Mathematics and Computation, Journal of The Franklin Institute, Journal of Applied Mathematics and Computing, etc.
Hongbin Zeng received the BS degree in electrical engineering from Tianjin University of Technology and Education, Tianjin, China in 2003, MS degree in computer science from Central South University of Forestry, Changsha, China in 2006, and PhD degree in control science and engineering from Central South University, Changsha, China in 2012, respectively. Since July 2003, he has been with the Department of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou, China, where he is currently an Associate Professor of automatic control engineering. Now, he is working as a Postdoctoral Research Associate in the Department of Electrical Engineering, Yeungnam University, Kyongsan, Korea. His current research interests are time-delay systems, neural networks and networked control systems.