Delay-dependent stabilization of stochastic interval delay systems with nonlinear disturbances

Abstract

In this paper, a delay-dependent approach is developed to deal with the robust stabilization problem for a class of stochastic time-delay interval systems with nonlinear disturbances. The system matrices are assumed to be uncertain within given intervals, the time delays appear in both the system states and the nonlinear disturbances, and the stochastic perturbation is in the form of a Brownian motion. The purpose of the addressed stochastic stabilization problem is to design a memoryless state feedback controller such that, for all admissible interval uncertainties and nonlinear disturbances, the closed-loop system is asymptotically stable in the mean square, where the stability criteria are dependent on the length of the time delay and therefore less conservative. By using Itô's differential formula and the Lyapunov stability theory, sufficient conditions are first derived for ensuring the stability of the stochastic interval delay systems. Then, the controller gain is characterized in terms of the solution to a delay-dependent linear matrix inequality (LMI), which can be easily solved by using available software packages. A numerical example is exploited to demonstrate the effectiveness of the proposed design procedure.

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, ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times m}$ denote, respectively, the n-dimensional Euclidean space and the set of all $n\times m$ real matrices. $L_2[0,\infty )$ is the space of square-integrable vector functions over $[0,\infty )$. $|·|$ refers to the Euclidean norm in ${\mathbb{R}}^n$, and $\parallel ·{\parallel }_2$ stands for the usual $L_2[0,\infty )$ norm. We let $\tau >0$, $C([-\tau ,0];{\mathbb{R}}^n)$ denote the family of continuous functions $\phi$ from $[-\tau ,0]$ to ${\mathbb{R}}^n$ with the norm $\parallel \phi \parallel ={\sup }_{-\tau ⩽\theta ⩽0}|\phi (\theta )|$, and I denote the identity matrix of compatible dimension. The notation $X⩾Y$ (respectively, $X>Y$) where X and Y are symmetric matrices, means that $X-Y$ is positive semi-definite (respectively, positive definite). For a matrix $M,$ $M^{\mathrm{T}}$ represents its transpose, ${\lambda }_{\max }(M)$ (respectively, ${\lambda }_{\min }(M))$ stands for its maximum (respectively, minimum) eigenvalue and its operator norm is denoted by $\parallel M\parallel =\sup \{|\mathit{Mx}|:|x|=1\}=\sqrt{{\lambda }_{\max }(M^{\mathrm{T}}M)}$. $(\Omega ,\mathcal{F},\{{\mathcal{F}}_t{\}}_{t⩾0},\mathcal{P})$ is a complete probability space with a filtration $\{{\mathcal{F}}_t{\}}_{t⩾0}$ satisfying the usual conditions (i.e. the filtration contains all P-null sets and is right continuous). Denote by $L_{{\mathcal{F}}_0}^p([-h,0];{\mathbb{R}}^n)$ the family of all ${\mathcal{F}}_0$-measurable $C([-\tau ,0];{\mathbb{R}}^n)$-valued random variables $\xi =\{\xi (\theta ):-\tau ⩽\theta ⩽0\}$ such that ${\sup }_{-\tau ⩽\theta ⩽0}\mathbb{E}|\xi (\theta ){|}^p<\infty$, where $\mathbb{E}\{x\}$ stands for the expectation of stochastic variable x. The shorthand $\mathrm{diag}\{M_1,\dots ,M_n\}$ denotes a block diagonal matrix with diagonal blocks being the matrices $M_1,\dots ,M_n$. The notation ${\mathfrak{M}}_n[(M_1{)}_{i_1,j_1},(M_2{)}_{i_2,j_2},\dots ,(M_r{)}_{i_r,j_r}]$ denotes a nth-order block square matrix whose all nonzero blocks are the $i_1{j}_1$th block $M_1$, the $i_2{j}_2$th block $M_2$,…,the $i_r{j}_r$th block $M_r$, 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.