Observer Design for Polytopic Systems: Application to Chaotic System Reconstruction

Abstract

Many studies concerning design of controllers and observers for a class of nonlinear systems described in polytopic representation are carried out. Such representation includes Takagi-Sugeno models, LPV models, switching models, PLDI... Particularly, T-S models are obtained by interpolation of M local LTI (linear time invariant) models throughout convex functions. The choice of the number of local models may be intuitively chosen by considering some operating regimes. Each LTI model can be obtained by using a direct linearization of an a priori nonlinear model around operating points, or alternatively by using an identification procedure. Based on the Lyapunov method and Linear Matrix Inequalities (LMI) formulation, sufficient conditions have been derived for controllers and observers design. Recently, systems subject to unknown inputs are considered for measurable and immeasurable decision variables. Unknown inputs can result either from model uncertainty, faults or due to the presence of unknown external excitation. These different results have been widely applied in the field of fault diagnosis (FDI), fault tolerance (FTC) and also for secure communications. Indeed, the increasing need of secure communications leads to the development of many techniques which make difficult the detecting of transmitted message. Based on unknown inputs observer design, many works have been carried out on secure communication and chaotic system reconstruction problem. In this framework, unknown inputs Takagi-Sugeno fuzzy observer has been exstensively used. The design of such observers is considered based on LMI and Lyapunov methods. The pole placement in an LMI region is also considered to improve the observer performances. Examples are given to illustrate a chaotic cryptosystem procedure where the plaintext (message) is encrypted using chaotic signals at the drive system side and the plaintext is retrieved via the designed unknown input observer.