Delay-dependent filtering for discrete-time fuzzy stochastic systems with mixed delays and sector-bounded nonlinearity
Introduction
Over the past two decades, fuzzy systems in the Takagi–Sugeno (T–S) model have been extensively studied because such systems can be used to deal with analysis and synthesis problems for complex nonlinear systems, such as the stirred tank reactor system, the chaotic Lorenz system, the truck trailer system and the Henon system [1], [2], [3]. It is well known that the time delays are often the main cause for instability and poor performance of systems. Hence, it is important to study their stability analysis and controller synthesis for T–S fuzzy systems with time delay [4], [7], [8], [9], [10].
On the other hand, stochastic modeling has come to play an important role in many branches of science such as biology, economics and engineering applications. Therefore, the system analysis and synthesis for stochastic systems with delays have been studied [11], [12], [13], [14], [15], [16], [17], [18]. Meanwhile, as discussed in [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], mixed delays (or called distributed delays) should be incorporated into the model due to the fact that the signal propagation is often distributed during a certain time period with the presence of an amount of parallel pathways with a variety of axon sizes and lengths. Thus, it is necessary to introduce mixed delays into the modeling of stochastic systems.
Recently, T–S fuzzy model approach has been extended to filer design for systems with delays. Therefore, filtering problem for delay systems have been studied in the literatures, see, for instance, [7], [8], [9], [12], [17], [18], [29], [30], [31], [32], [33] and the references therein. Recently, the robust filtering problem for a class of uncertain nonlinear time-delay stochastic systems is investigated, in which the nonlinearity is described as sector-bounded condition [34], [35], [36], [37], [38], [39]. Then this condition is also applied to the state estimation for neural networks [39], [40], [41]. The filtering problem for fuzzy stochastic systems has been studied in [11], [12], [14], [15], [16], [17], [18], [33], [34], [35]. The filtering problem has been focused on continuous-time stochastic fuzzy systems [12], [33]. It is should be pointed that the results in [17], [18] did not take mixed delays and sector-bounded nonlinearity into account. And the filter design for discrete-time stochastic systems has been studied [11], [13], [14], [15], [16], [34], [35]. For discrete-time systems, in order to pursue less conservative filter design method, some results have been reported on discrete-time systems in the literature. For examples, filter design for discrete-time systems with time-varying delay is considered. A delay-dependent performance analysis result is firstly established for filtering error systems without ignoring any terms in the derivative of Lyapunov functional by considering the relationship between the time-varying delay and its upper bound [44]. More recently, based on a delay-dependent piecewise Lyapunov–Krasovskii functional and establishing a finite sum inequality, delay-dependent robust filter design for discrete-time fuzzy systems with time-varying delay are studied [46]. Then by using a new linearization technique incorporating a bounding technique, the problem of delay-dependent robust filtering design for discrete-time polytopic linear systems with interval-like time-varying delay is considered [47]. Based on a novel fuzzy-basis-dependent Lyapunov–Krasovskii functional combined with Finsler׳s lemma and an improved free-weighting matrix technique for delay-dependent criteria, the delay-dependent robust filtering problem for a class of uncertain discrete-time T–S fuzzy systems with interval-like time-varying state delay is also considered [48]. To the best of the authors’ knowledge, the filtering for discrete-time fuzzy stochastic systems with mixed delay and sector-bounded nonlinearity has not been investigated and remains to be important and challenging.
Motivated by the above discussion, in this paper we are interested in the filtering problems for discrete-time fuzzy stochastic systems with mixed delays and sector-bounded nonlinearity. The mixed delays consist of both discrete and distributed delays, and the nonlinearity under consideration is subject to sector-bounded condition. We deal with both the fuzzy-rule-independent and fuzzy-rule-dependent filter designs for discrete-time fuzzy stochastic systems such that the filtering error systems are mean-square asymptotically stable and the disturbance rejection attenuation is constrained to a prescribed performance index. To reduce the possible conservatism [8], [18], [46], [47], [48], the Lyapunov–Krasovskii functional approach combined with the S-procedure is applied to derive the main results. The existence conditions of the designed filters are proposed in terms of a set of linear matrix inequalities. Finally, numerical examples are given to show the effectiveness of the proposed method.
The main contribution of this paper are summarized as follows:
Both sector-bounded nonlinearity and mixed delays are firstly investigated in the design of filters for discrete-time fuzzy stochastic systems.
Based on the Lyapunov–Krasovskii functional combined with the S-procedure, delay-dependent sufficient conditions are presented to guarantee the existence of the desired filters for discrete-time fuzzy stochastic systems, which are less conservative than those in [8], [18], [46], [47], [48].
Both the fuzzy-rule-dependent and fuzzy-rule-independent filters are firstly considered for discrete-time fuzzy stochastic systems with mixed delays and sector-bounded nonlinearity. Numerical example is given to demonstrate the effectiveness of the proposed filter design, which shows that fuzzy-rule-dependent filter generally leads to less conservative results than fuzzy-rule-independent one.
Notations: Throughout this paper, denotes the n-dimensional Euclidean space, is the set of n×m real matrices. I is the identity matrix. refers to the space of square summable infinite vector sequences. denotes Euclidean norm for vectors and denotes the usual norm. N denotes the set of all natural number, i.e., . is a complete probability space with a filtration satisfying the usual conditions. stands for the transpose of the matrix M. For symmetric matrices X and Y, the notation (respectively ) means that the is positive definite (respectively, positive semi-definite). ⁎ denotes a block that is readily inferred by symmetry. denotes the mathematical expectation operator with respect to the given probability measure .
Section snippets
Problem description
Consider the following discrete-time fuzzy stochastic systems with mixed delays and sector-bounded nonlinearity:
Rule i: IF is η1i and … and is ηpi, THENwhere , r is
Design of fuzzy-rule-dependent filter
The following theorem gives sufficient conditions of the stochastic stability with an performance for the filtering error system (14), (15). Theorem 1 Given a scalar and filter matrices Af(k), Bf(k), Cf(k). The filtering error systems (14), (15) are asymptotically mean-square stable with a prescribed disturbance attenuation level γ, if there exist matrices , , , , , , , scalars and such that the following matrix inequalities hold:
Numerical example
In this section, three examples are given to demonstrate the effectiveness of the proposed method. Example 1 Consider the discrete-time fuzzy stochastic systems (50), (51), (52) with parameters in [18]:
Conclusion
This paper is concerned with the delay-dependent filter design for discrete-time fuzzy stochastic systems with mixed delays and sector-bounded nonlinearity through T–S fuzzy models. By constructing a new Lyapunov functional, the filter design scheme is proposed in terms of linear matrix inequalities (LMIs). And delay-dependent sufficient conditions for the existence of filters are proposed in terms of linear matrix inequalities. Numerical examples are given to illustrate the
Acknowledgments
The authors wish to thank the Editor and the anonymous reviewer very much for their valuable comments and suggestions. This work was supported by the Natural Science Foundation of Jiangsu Province (No. BK20130239) and the Research Fund for the Doctoral Program of Higher Education of China (No. 20130094120015).
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