A novel Lyapunov–Krasovskii functional approach to stability and stabilization for T–S fuzzy systems with time delay

Abstract

This paper is concerned with the problem of the stability and stabilization for continuous-time Takagi–Sugeno(T–S) fuzzy systems with time delay. A novel Lyapunov–Krasovskii functional which includes fuzzy line-integral Lyapunov functional and membership-function-dependent Lyapunov functional is proposed to investigate stability and stabilization of T–S fuzzy systems with time delay. In addition, switching idea which can avoid time derivative of membership functions is introduced to deal with derivative term. Relaxed Wirtinger inequality is employed to estimate integral cross term. Sufficient stability and stabilization criteria are derived in the form of matrix inequalities which can be solved using the switching idea and LMI method. Several numerical examples are given to demonstrate the advantage and effectiveness of the proposed method by comparing with some recent works.

Introduction

Most of systems are nonlinear, and the direct analysis and control of nonlinear systems are difficult. Takagi–Sugeno(T–S) fuzzy model [1] has been known one as of the powerful tools to approximate systems dynamics of nonlinear systems and great attention has been paid on stability analysis and stabilization of T–S fuzzy systems(see [2], [3], [4], [5], [6], [7]). In addition, it is worth noting that time delay phenomenon often appears in all kinds of engineering systems which can lead to oscillation, performance degradation, or even instability. Therefore, studying on T–S fuzzy systems with time delay has theoretical importance and practical significance.

Usually, Lyapunov analysis method is widely used to develop less conservative stability and stabilization criteria for T–S fuzzy system with time delay, which can be checked by maximum allowable upper bound of the delay and the number of decision variables. For Lyapunov analysis, one way is that integral inequality technique has been developed in order to estimate the lower bounds of integral term such as ${\int }_{t-h}^t\dot{x}\left(s\right)R\dot{x}\left(s\right)ds,$ such as Jensens inequality [8], Wirtinger inequality [9], free-matrix-based inequality [10], double integral inequality [11], Bessel-Legendre inequality [12] and so on. Especially, relaxed Writinger inequality [13], [14] which relaxes Wirtinger inequality [9] can derive lower bounds to estimate integral term.

Another way to obtain less conservative delay-dependent stability and stabilization criteria comes from the construction of suitable Lyapunov–Krasovskii functionals. Generally, the common quadratic Lyapunov functional methods are extensively utilized in investigating T–S fuzzy systems [15], [16], [17], [18], [19]. Although the use of the common quadratic Lyapunov functional provides simpler conditions, which means a positive matrix P needs to meet the linear matrix inequality of each fuzzy rule. Thus, such a method is inherently conservative. In order to reduce conservatism and obtain improved stability and stabilization criteria, the fuzzy weight-dependent Lyapunov function is employed in [20], [21], [22], [23], [24], [25]. However, one fact that cannot be ignored in utilizing the fuzzy weight-dependent Lyapunov function is that the upper bound of the time derivative of membership functions is prior given to obtain stability criteria but it is usually hard to obtain in practice. To avoid the disadvantage, a fuzzy line-integral Lyapunov function is proposed in [26]. In [27], less conservative stability conditions and controller design are obtained by applying discretized Lyapunov–Krasovskii functional. Recently, in [28], less conservative conditions are obtained for T–S fuzzy systems with time delay by utilizing the fuzzy line-integral Lyapunov function and Wirtinger inequality [9]. By employing augmented Lyapunov–Krasovskii functionals, improved stability and stabilization conditions for T–S fuzzy with time-varying delay are proposed in [29]. In [30], non-quadratic Lyapunov functional and triple integral term are introduced to obtain stability and stabilization conditions of fuzzy time-delay systems. Quite recently, membership-function-dependent Lyapunov–Krasovskii functional is utilized in [31], [32] to study T–S fuzzy systems, and a switching idea [32] is proposed to avoid time derivative of membership function and remain its information.

Based on above work and observations, this paper aims to study stability and stabilization conditions of T–S fuzzy systems with time delay by constructing a novel Lyapunov–Krasovskii functional. Meanwhile, relaxed Writinger inequality [13], [14] and free weighting matrix techniques [33] are, respectively, utilized to estimate integral term and to relax conditions. The main contributions and highlights of this paper can be summarized as follows:

• The proposed novel Lyapunov–Krasovskii functional contains the following term:

$$V_{ALF}\left(x_t\right)={\int }_{t-h}^t{\left[\begin{array}{c}x\left(s\right)+{\int }_s^{t}x\left(r\right)dr\\ \dot{x}\left(s\right)+{\int }_s^t\dot{x}\left(r\right)dr\end{array}\right]}^{T}G\left[\begin{array}{c}x\left(s\right)+{\int }_s^{t}x\left(r\right)dr\\ \dot{x}\left(s\right)+{\int }_s^t\dot{x}\left(r\right)dr\end{array}\right]ds.$$

As a result, some new cross terms are introduced to reduce the conservatism of stability and stabilization criteria.

• The novel Lyapunov–Krasovskii functional also includes fuzzy line-integral Lyapunov function and membership-function-dependent Lyapunov function such that it contains much information on T–S fuzzy systems to reduce conservatism of finding maximum delay bounds. Moreover, switching idea in [32] is utilized to deal with derivative term of membership function which not only avoids upper bounds of time derivative of membership functions but also retains information of membership functions.

Furthermore, a delay-dependent stability condition dependent of the information on the time-derivatives of membership functions is presented in Theorem 1. Also, the parallel distributed compensation control design method [34] is used to obtain stabilization condition in Theorem 2. Finally, several examples are given to illustrate the effectiveness and the improvement of our results.

Notation: The superscripts ‘$-1$’and ‘T’ stand for the inverse and the transpose of a matrix, respectively; ${\mathbb{R}}^n$ denotes the n-dimensional real Euclidean space; ${\mathbb{R}}^{n\times m}$ is real matrix space with dimension n × m; P > 0 (P ≥ 0) means that P is positive definite(semi-definite) matrix; I_n is identity matrix with dimension n × n and 0_n is zero matrix with dimension n × n; for any square matrix $X\in {\mathbb{R}}^{n\times m},$ we define $sym\left\{X\right\}=X+X^T$; If the dimensions of a matrix are not explicitly stated, the matrix is assumed to have compatible dimensions.