Approaches to extended non-quadratic stability and stabilization conditions for discrete-time Takagi–Sugeno fuzzy systems

Abstract

This paper provides simple and effective linear matrix inequality (LMI) characterizations for the stability and stabilization conditions of discrete-time Takagi–Sugeno (T–S) fuzzy systems. To do this, more general classes of non-parallel distributed compensation (non-PDC) control laws and non-quadratic Lyapunov functions are presented. Unlike the conventional non-quadratic approaches using only current-time normalized fuzzy weighting functions, we consider not only the current-time fuzzy weighting functions but also the $l$-step-past ($l⩾0$) and one-step-ahead ones when constructing the control laws and Lyapunov functions. Consequently, by introducing additional decision variables, it can be shown that the proposed conditions include the existing ones found in the literature as particular cases. Examples are given to demonstrate the effectiveness of the approaches.

Introduction

In recent years, growing attention has been paid to the study of stability and stabilization of Takagi–Sugeno (T–S) fuzzy systems (Ding, 2009, Fang et al., 2006, Liu and Zhang, 2003, Sala and Ariño, 2007). A popular approach to solving this problem is the Lyapunov direct method. However, it is well known that the use of a common quadratic Lyapunov function often leads to overly conservative results. For this reason, different techniques based on piecewise Lyapunov functions (Johansson, Rantzer, & Årzén, 1999), non-quadratic Lyapunov functions (Ding and Huang, 2008, Ding et al., 2006, Guerra and Vermeiren, 2004, Mozelli et al., 2009, Tanaka et al., 2003), and the $\kappa$-samples variation approach (Guerra et al., 2009, Kruszewski et al., 2008) appeared in order to provide less conservative results. Not only these, but similar concepts were proposed for robust stability of uncertain linear systems in polytopic domains (Chesi et al., 2003, Chesi et al., 2007, Daafouz and Bernussou, 2001, de Oliveira et al., 1999, de Oliveira et al., 2002, Ebihara et al., 2005, Oliveira and Peres, 2006) as well.

Especially, Guerra and Vermeiren (2004) introduced non-parallel distributed compensation (non-PDC) control laws along with non-quadratic Lyapunov functions and proved that the non-quadratic approaches always include the common quadratic one as a special case. Also, the conservativeness was further reduced in Ding and Huang (2008) and Ding et al. (2006) by introducing an extended non-quadratic Lyapunov function. Although the non-quadratic approaches cover a wider class of systems than the common quadratic one, the conservativeness still remains. This is because the current Lyapunov function $V\left(x\left(k\right)\right)$ and the one-step-ahead one $V\left(x(k+1)\right)$ in the Lyapunov inequality do not share the same fuzzy weighting functions, and this fact is a strong restriction when using the convex-sum property for proving the Lyapunov inequality.

Bearing this in mind, we propose new stability and stabilization conditions for discrete-time T–S fuzzy systems in linear matrix inequalities (LMIs) format. To this end, an extended non-PDC control law and more general non-quadratic Lyapunov functions which incorporate $l$-step-past ($l⩾0$) and one-step-ahead fuzzy weighting functions are proposed. In the Lyapunov inequality, $V\left(x\left(k\right)\right)$ and $V\left(x(k+1)\right)$ partially share the weighting functions with each other and are blended together. The extended structures of the Lyapunov functions and control law provide new decision variables for the LMI problems, and consequently, it is shown that the results of this paper include the existing ones as particular cases. Examples are given to numerically verify improvements over some existing results.

Notations: An asterisk $(\ast )$ inside a matrix represents the transpose of its symmetric term; ${\mathcal{I}}_r≔\{1,2,\dots ,r\}$; ${\mathbb{N}}^{\ast }$ denotes the set of positive integer;

$${\left[X_{i,j}\right]}_{r\times r}≔\left[\begin{array}{ccc}\hfill X_{1,1}\hfill & \hfill \cdots \hfill & \hfill X_{1,r}\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill X_{r,1}\hfill & \hfill \cdots \hfill & \hfill X_{r,r}\hfill \end{array}\right]\text{;}$$$$X_{z(k+l_0)z(k+l_1)\cdots z(k+l_{N-1})}≔\sum _{i_0=1}^r\sum _{i_1=1}^r\cdots \sum _{i_{N-1}=1}^r{h}_{i_0}\left(z(k+l_0)\right)h_{i_1}\left(z(k+l_1)\right)\cdots \times h_{i_{N-1}}\left(z(k+l_{N-1})\right)X_{i_0,i_1,\dots ,i_{N-1}}\text{,}$$

where $l_0,l_1,\dots ,l_{N-1}$ are any integers and $N\in {\mathbb{N}}^{\ast }$.