Delay-dependent stabilization of stochastic interval delay systems with nonlinear disturbances☆
Introduction
Interval systems have been well known for their importance in practical applications. When modeling real-time plants, the parameter uncertainties are unavoidable, which would lead to perturbations of the elements of a system matrix in a state-space model. These uncertainties may arise from variations of the operating point, aging of the devices, identification errors, etc. As a result, the parameters of a system matrix are estimated only within certain closed intervals. In recent years, the stability analysis and stabilization problems of various deterministic interval systems have received considerable research attention, see e.g.[11], [13] and the references therein. Very recently, in [14], the stability analysis problem of a class of stochastic delay interval systems has been considered by using the Razumikhin method.
In view of time delays being commonly residing in practical systems, the past few decades have witnessed significant progress on filtering and control for linear/nonlinear systems with various types of delays, and a large amount of literature has appeared on the general topic of time-delay systems, see e.g. [1], [5], [7], [15], [16], [17], [19], [22], [23]. In particular, the linear matrix inequality (LMI) technique has been extensively used because of its computational efficiency, and a great number of LMI-based results have been published, see e.g. [2], [18], [20]. It is worth mentioning that, since delay-dependent LMI techniques take into account the information on the length of delays, delay-dependent stability criteria tend to be less conservative than the traditionally delay-independent ones especially when the time delays are known and small, see [2], [3], [4], [6], [25], [27], [28]. Moreover, some improved delay-dependent techniques have recently been developed, see e.g. [8], [9], [10], [24] for some up-to-date results.
In real-time systems, the signal transmission is usually a noisy process brought on by random fluctuations from probabilistic causes and, therefore, stochastic modeling has been of vital importance in many branches of science such as biology, economics and engineering applications. Recently, many fundamental results for deterministic systems have been extended to stochastic systems. The robust stability, stabilization, control and filtering problems for stochastic systems have been investigated by many researchers, and a lot of results on these topics have been reported in the literature, see e.g. [2], [18], [21], [26]. It is noticed that the delay-dependent technique has been applied to the analysis and synthesis of stochastic systems in, for example, [3], [27]. Unfortunately, up to now, the stability analysis and stabilization problems for stochastic time-delay interval systems with nonlinear disturbances have not been adequately addressed by delay-dependent technique yet, which remains as an interesting research topic.
In this paper, we deal with the robust stability and stabilization problems for a class of stochastic time-delay interval systems with nonlinear disturbances by developing delay-dependent analysis techniques. The robust stability analysis problem is first dealt with, where the aim is to derive sufficient conditions such that the system is asymptotically stability in the mean square, dependent on the length of the time delay, for all admissible nonlinear disturbances as well as intervally varying uncertain parameters. Then, we tackle the robust stabilization problem where a memoryless state feedback controller is designed to stabilize the closed-loop system. By using Itô's differential formula and the Lyapunov stability theory, sufficient conditions for the solvability of these problems are derived in terms of LMIs, which can be easily checked by resorting to available software packages. A numerical example is exploited to demonstrate the effectiveness of the results obtained.
Notation: In this paper, and denote, respectively, the n-dimensional Euclidean space and the set of all real matrices. is the space of square-integrable vector functions over . refers to the Euclidean norm in , and stands for the usual norm. We let , denote the family of continuous functions from to with the norm , and I denote the identity matrix of compatible dimension. The notation (respectively, ) where X and Y are symmetric matrices, means that is positive semi-definite (respectively, positive definite). For a matrix represents its transpose, (respectively, stands for its maximum (respectively, minimum) eigenvalue and its operator norm is denoted by . is a complete probability space with a filtration satisfying the usual conditions (i.e. the filtration contains all P-null sets and is right continuous). Denote by the family of all -measurable -valued random variables such that , where stands for the expectation of stochastic variable x. The shorthand denotes a block diagonal matrix with diagonal blocks being the matrices . The notation denotes a nth-order block square matrix whose all nonzero blocks are the th block , the th block ,…,the th block , and all other blocks are zero matrices. In symmetric block matrices, the symbol is used as an ellipsis for terms induced by symmetry. Matrices, if not explicitly stated, are assumed to have compatible dimensions.
Section snippets
Problem formulation
For a matrix , define the following matrix interval: where and satisfy for all .
Consider the following stochastic time-delay interval system with nonlinear disturbance:where is the state, is the control input, is an unknown nonlinear exogenous disturbance input, is a
Robust stability analysis
First, let us give the following lemmas which will be used in the proof of our main results. Lemma 1 Schur Complement Given the constant matrices where and . Then if and only if or equivalently, Lemma 2 Let X, Y, F be real matrices of appropriate dimensions with . Then for any scalar , we have Lemma 3 Let and be given constant matrices with appropriate dimensions. Then, for any scalar satisfying , we have Gao and Wang [6]
Delay-dependent robust stabilization
In this section, we aim to propose a design procedure for the state feedback controller that can robustly stochastically stabilize the addressed stochastic delayed interval systems with nonlinear disturbances. Again, a delay-dependent LMI technique will be developed in order to obtain a less conservative condition. The main result of this paper is given in the following theorem. Theorem 2 Consider the system (1)–(2). If there exist positive definite matrices , , , , a matrix Y, and positive
An illustrative example
In this section, to illustrate the usefulness and flexibility of the theory developed in previous section, we present a simple numerical example. Attention is focused on the design of a stabilizing controller for a class of stochastic time-delay interval system with nonlinear disturbance.
The system data of (1)–(2) are as follows:
Using Matlab LMI control
Conclusions
In this paper, we have investigated the robust stability analysis problem as well as the robust stabilization problem for a class of stochastic time-delay interval systems with nonlinear disturbances. A delay-dependent LMI approach has been developed to derive sufficient conditions under which the controlled system is mean-square asymptotically stable, where the conditions are dependent on the length of the time delays. A numerical example has been employed to illustrate the effectiveness of
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This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Nuffield Foundation of the UK under Grant NAL/00630/G, and the Alexander von Humboldt Foundation of Germany.