Contribution à l’estimation et à la commande des systèmes non linéaires

Exploring the observation and estimation of non-linear systems, particularly those represented by Takagi-Sugeno fuzzy multi-systems, this research addresses the significant challenges in modeling and creating an observer for such systems. It adopts a multi-model approach and employs non-linear sector decomposition to transform these systems into a polytope form, facilitating the development of a robust observer that can accurately reconstruct both the system's states and its unknown inputs under various conditions. The study contrasts traditional quadratic approaches with an innovative non-quadratic method that uses a line-integral Lyapunov function for the estimation of both state and unknown inputs. This method, in conjunction with a specially designed observer, results in linear matrix inequality conditions, providing a more straightforward solution compared to the complex bilinear matrix inequality commonly associated with TS fuzzy systems. Notably, this approach demonstrates reduced conservatism, thereby enhancing its effectiveness and reliability. Additionally, the research investigates state and unknown input estimation for non-linear systems using TS fuzzy systems in scenarios where premise variables are unmeasurable, and it includes practical examples to illustrate the effectiveness of the proposed methods

Stability and Stabilization of TS Fuzzy Systems via Line Integral Lyapunov Fuzzy Function

This paper is concerned with the stability and stabilization problem of a Takagi-Sugeno fuzzy (TSF) system. Using a non-quadratic function (well-known integral Lyapunov fuzzy candidate (ILF)) and some lemmas, new sufficient conditions are established as linear matrix inequalities (LMIs), which are solved with a stochastic fractal search (SFS). The main advantage of the technique used is its small conservatives. Motivated by the mean value theorem, a state feedback controller based on a non-quadratic Lyapunov function is designed. Unlike other approaches based on poly-quadratic Lyapunov candidates, stability conditions of the closed loop are obtained in LMI regions. It is important to highlight that the time derivatives of membership functions do not appear in the used line integral Lyapunov function, which is the well-known problem of poly-quadratic Lyapunov functions. A numerical example is given to show the advantages and the utility of the integral Lyapunov fuzzy candidate, which provides a wider feasibility region than other Lyapunov functions