Improved delay-dependent stability criteria for T–S fuzzy systems with time-varying delay

Abstract

This paper is concerned with the robust stability of uncertain T–S fuzzy systems with time-varying delay. A novel Lyapunov–Krasovskii functional is established by employing the idea of combining delay-decomposition with state vector augmentation. Then, by employing some integral inequalities and the reciprocally convex approach, some less conservative delay-dependent stability criteria are obtained. The proposed stability conditions are formulated in the form of linear matrix inequalities (LMIs), which can be solved efficiently with Semi-Definite Programming (SDP) solvers. Finally, four numerical examples are provided to show that the proposed conditions are less conservative than existing ones.

Introduction

During the past two decades, Takagi–Sugeno (T–S) fuzzy systems[1] have been an active topic due to the fact that it can combine the flexibility of fuzzy logic theory and fruitful linear system theory into a unified framework to approximate complex nonlinear systems [2], [3], [4]. On the other hand, time-delays often occur in many dynamic systems such as biological systems, chemical processes,communication networks and so on, which is usually a source of instability and deteriorated performance. Therefore, stability analysis for T–S fuzzy systems with time-delay has received more interest and achieved fruitful results [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30].

The stability criteria can be classified into two types: one is delay-dependent and the other is delay-independent. The delay-dependent criteria are less conservative than delay-independent ones since they consider the length information of the delay. In [21], the delay-dependent stability problem of T–S fuzzy systems with time-varying delay was addressed. Recently, a free-weighting matrix approach, which was proposed in [31], has been employed to derive some delay-dependent results in stability analysis or stabilization for T–S fuzzy systems in [22], [23], [24]. By retaining ignored terms in estimating the upper bound of derivative of Lyapunov–Krasovskii functional and employing an improved free-weighting matrix approach to consider the relationship between the time-varying delay and its upper bound, some less conservative conditions are established in [26]. To further reduce the conservativeness of the derived results, the other efforts are paid to the construction of Lyapunov–Krasovskii functionals. In [28], an augmented Lyapunov–Krasovskii functional approach that introduces a triple integral and some augmented vectors was employed to investigate the stability problem of T–S fuzzy systems with time-varying delay. Nevertheless, there are too many decision variables included in the derived condition, which leads to a large amount of computing time. In addition, some delay-partitioning-based approaches were adopted to deal with the stability of T–S fuzzy systems with time-varying delay in [10], [29], [30]. However, as pointed out in [32], the integral terms including the upper bound of the time-varying delay are decomposed while the integral term including the time-varying delay is not taken fully into account, which inevitably leads to conservativeness. Therefore, there is still plenty of room for improvement.

In this paper, we discuss the stability of T–S fuzzy systems with time-varying delay. A novel Lyapunov–Krasovskii functional is established by employing the idea of combining delay-decomposition with state vector augmentation. Some LMI-based stability criteria are obtained via introducing some fuzzy-weighting matrixes to express the relationship of the T–S fuzzy models and some new bounding techniques to estimate the derivative of Lyapunov–Krasovskii functional. The effectiveness and the improvements of the proposed method are demonstrated by four numerical examples.

Notations. Through this paper, $N^T$ and $N^{-1}$ stands for the transpose and the inverse of the matrix $N$, respectively; I is the identity matrix of appropriate dimensions; ${\mathbb{R}}^n$ denotes the $n$-dimensional Euclidean space with vector norm $\Vert ·\Vert \text{;}P>0(P⩾0)$ means that $P$ is symmetric and positive definite (semi-positive definite) matrix; diag{$\cdots$} denotes a block-diagonal matrix; the symmetric terms in a symmetric matrix are denoted by $\ast$, e.g., $\left[\begin{array}{cc}\hfill X\hfill & \hfill Y\hfill \\ \hfill \ast \hfill & \hfill Z\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill X\hfill & \hfill Y\hfill \\ \hfill Y^T\hfill & \hfill Z\hfill \end{array}\right]$.