Multiple Models Approach in Automation: Takagi-Sugeno Fuzzy Systems

Chapter 2

Stability of Continuous Multiple Models

2.1. Introduction

Our objective in this chapter is to study the stability of multiple models. Numerous studies on the subject have initially been inspired from techniques developed in the linear domain. Indeed, Lyapunov’s method, or more specifically the quadratic method, and formulation using linear matrix inequalities have been extensively used [BOY 94, CHA 02b, CHA 08d, CHA 10, MA 98, MAR 99, TAN 98]. In this sense, transformation into a Lur’e problem, interconnected system techniques and the properties of M-matrices have been adapted for nominal and uncertain multiple models [BRE 02, CHA 02b].

This chapter deals with the stability of continuous multiple models, with and without uncertainties. Two types of Lyapunov function are used, quadratic and nonquadratic, and the chapter is set out as follows. In the following section, we propose some sufficient conditions for standard multiple model stability, based on the presence of quadratic Lyapunov functions and the properties of M-matrices. Then, we present some conditions for stability which make use of nonquadratic functions. Finally, we examine conditions for robust stability, which imply two types of uncertainty.

2.2. Stability analysis

The approach proposed in this chapter is principally based on quadratic Lyapunov functions. We need to find a positive definite symmetric matrix, i.e. the associated quadratic Lyapunov function which guarantees asymptotic multiple model stability. ...