Some Refinements on Stability Analysis and Stabilization of Second Order T-S Models Using Line-Integral Lyapunov Functions

Abstract

This paper deals with new conditions for stability analysis and controller design for Takagi-Sugeno (T-S) models in the non-quadratic framework. The aim of this study is to provide some improvements on results that do not require the knowledge of the bounds of the membership function derivatives, i.e. the ones that employ line-integral Lyapunov candidate functions. First, sight improvements on recent stability conditions are proposed by the use of the Finsler's lemma and with unconstrained slack decision variables. Then, non-PDC controller design is proposed through two new approaches. The first proposed approach is based on the use of the Finsler's lemma but requires a scalar parameter to be fixed in advance. The second proposed approach employs another gimmick to avoid any unknown parameters and introducing new slack decision variables. It is finally noticed that these results can be solved as linear matrix inequalities (LMI) for first and second order systems, bilinear matrix inequalities (BMI) for the third order and then remain more complex as the system's order increases.