Resilient H∞ filtering for discrete-time uncertain Markov jump neural networks over a finite-time interval☆
Introduction
During the past few years, much attention has been devoted to the study of neural networks (NNs) due to the fact that NNs can be extensively applied in a variety of areas, such as fixed-point computation, pattern recognition and associative memory [21], [22]. On the other hand, as a special class of switched systems, Markov jump systems (MJSs) can model some practical systems which encounter random abrupt in their structures or parameters. When a system modeled by neural networks experiences random changes, it is natural to consider the issue of Markov jump neural networks (MJNNs) and recently many important results on MJNNs have been reported in the literature, see, for instance [1], [10], [13], [19], [20], [27].
In practice, some state variables of certain system cannot be measured directly. In order to cope with this gap, the filtering or state estimation problem has been gaining a remarkable momentum [11], [26]. When a system encounters the external disturbances, one of the most popular ways is the filtering scheme which is concerned with the design of the state estimators such that the filtering error is limited in a prescribed range according to the index. Currently, the exponential filtering problem for a class of discrete-time switched neural networks with random time-varying delays is studied in [12]. In [29], the event-based filtering issue for sampled-data systems was solved. In [18], the filtering issue of Markov jump nonlinear systems with general uncertain transition probabilities was investigated. It should be noted that the above mentioned papers focus on stochastic stability over an infinite-time interval. Nevertheless, in many practical applications, it is required that the system state can achieve stabilization in a short time. Therefore, there is no doubt that many researchers devote themselves to studying the finite-time filtering problem and a great number of important results have been reported in the last few years [5], [6], [8].
In addition, the packet dropouts are usually inevitable due to the limited capacity in transmission mechanism, sudden data loss of some measurements, or data inaccessibility at certain times. To overcome this shortage, a Bernoulli distributed stochastic process was proposed which can describe the data dropout phenomenon reasonably [2], [7], [9]. Under this circumstance, the filtering over networks can achieve better performance by taking the packet dropouts into account. Furthermore, apart from the packet dropouts, inaccuracies or uncertainties usually occur in the implementation of the filters. The uncertainties could give rise to instability to the filtering system. To circumvent this obstacle, many researchers commit themselves to designing a resilient (or namely, non-fragile) filter which can be insensitive with respect to filter gain uncertainties [25]. Very recently, in [3], the problem of resilient filter design for discrete-time Takagi–Sugeno (T–S) fuzzy systems was solved. In [14], the resilient filtering problem for a class of discrete-time stochastic systems subject to randomly occurring gain variations, channel fadings, as well as randomly occurring nonlinearities was investigated. However, to the best of the authors׳ knowledge, the resilient finite-time filtering with packet dropouts in MJNNs has not yet received adequate research attention which motivated the recent work.
In this paper, we tackle the resilient finite-time filtering problem for discrete-time uncertain MJNNs with packet dropouts. The main contributions lie in some sufficient conditions provided to guarantee the filtering error system is finite-time bounded with a prescribed performance level. Furthermore, the packet dropouts phenomenon described by a stochastic Bernoulli distributed process is also considered which makes the desired filter structure more general. With the help of linear matrix inequalities methodology, a resilient finite-time filter is designed which ensures the finite-time boundedness of the closed-loop system. A numerical example is presented to demonstrate the effectiveness of the main results.
The rest of this paper is organized as follows. The addressed problem is formulated in Section 2. Our main results are presented in Section 3, where some performance conditions are given. The method to calculate the parameters of the filter is also given. Section 4 provides an example to demonstrate the effectiveness of the proposed method. Finally, we conclude the paper in Section 5.
Notation. The notations used throughout this paper are standard. denotes the n-dimensional Euclidean space and is the set of all real matrices. For symmetric matrix W, The notation (respectively, ) means that the matrix W is positive semi-definite (respectively, positive definite). The notation XT represents the transpose of the matrix X; I and 0 represent the identity matrix and zero matrix with appropriate dimension, respectively. and are used to denote, respectively, the smallest and largest eigenvalue of the matrix X; denotes the mathematical expectation; refers to the Euclidean vector norm; is the space of square-summable infinite vector sequences over . We employ an asterisk to represent a term that is induced by symmetry. Matrices, if not explicitly stated, are assumed to have compatible dimensions. is used to denote the set of non-negative integers.
Section snippets
Problem formulation
Let Co be a closure of the convex hull, which is generated by real numbers or real matrices . In this paper, we will consider a class of uncertain Markov jump neural networks (MJNNs). Suppose that the parameter uncertainties belong to the fixed convex polytope, and the neural network can be described by the following state equations :where is the activation of the ith
Main results
In this section, we will discuss the problem of the resilient finite-time filtering for discrete-time uncertain Markov jump neural networks with packet dropouts. Theorem 1 Given scalars , an integer , and a positive matrix . System is stochastically finite-time bounded with a prescribed performance level , if there exist matrices , diagonal matrices such that the following matrix inequalities and (14) hold for each
A numerical example
In this section, we present a simulation example to illustrate the effectiveness of the proposed resilient finite-time filter design method for discrete-time uncertain Markov jump neural networks with packet dropouts. Example 1 Consider the discrete-time Markov jump neural networks (6), (7), (8) with the following parameters:
Conclusions
In this paper, the resilient finite time filtering problem for discrete-time uncertain Markov jump neural networks with packet dropouts has been studied. Based on the linear matrix inequalities method, some sufficient conditions which ensure the filtering error system to be finite time bounded with a prescribed performance level are given. In addition, the resilient property and the data packet dropouts phenomenon are also taken into account which make our paper more common. A numerical
Mengshen Chen is now a M.S. candidate at the School of Electrical and Information Engineering, Anhui University of Technology, China. His current research interests include complex networks, Markov jump systems, switched systems, robust control and filtering.
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Mengshen Chen is now a M.S. candidate at the School of Electrical and Information Engineering, Anhui University of Technology, China. His current research interests include complex networks, Markov jump systems, switched systems, robust control and filtering.
Long Zhang received the B.Ec. degree in materials processing engineering from Anhui University of Technology, Ma׳anshan, China, in 2007, and the Ph.D. degree in materials processing engineering from Huazhong University of Science and Technology, Wuhan, China, in 2011.
He is currently a lecturer with the School of Metallurgical Engineering, Anhui University of Technology, China. His research interests are mainly focused on control engineering and 3D printing technology of modeling material.
Hao Shen received the B.Sc. degree in communication engineering from Anhui University of Technology, Ma׳anshan, China, in 2006, and the Ph.D. degree in control theory and control engineering from Nanjing University of Science and Technology, Nanjing, China, in 2011.
He is currently an associate professor with the School of Electrical and Information Engineering, Anhui University of Technology, China. He was a postdoctoral researcher with Department of Electrical Engineering, Yeungnam University, Republic of Korea, From March 2013 to February 2014. His current research interests include stochastic hybrid systems, network control systems, robust control and filtering, nonlinear systems.
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This work was supported by the National Natural Science Foundation of China under Grant 61304066, 61473171, 61503002, 51405002, the Natural Science Foundation of Anhui Province under Grant 1308085QF119, the Research Project of State Key Laboratory of Mechanical System and Vibration under Grant MSV201509, the Major Science and Technology Project of Anhui Province under Grant 1301041023.