Polytopic invariant and contractive sets for closed-loop discrete fuzzy systems

Abstract

In this work a procedure for obtaining polytopic λ-contractive sets for Takagi–Sugeno fuzzy systems is presented, adapting well-known algorithms from literature on discrete-time linear difference inclusions (LDI) to multi-dimensional summations. As a complexity parameter increases, these sets tend to the maximal invariant set of the system when no information on the shape of the membership functions is available. λ-contractive sets are naturally associated to level sets of polyhedral Lyapunov functions proving a decay-rate of λ. The paper proves that the proposed algorithm obtains better results than a class of Lyapunov methods for the same complexity degree: if such a Lyapunov function exists, the proposed algorithm converges in a finite number of steps and proves a larger λ-contractive set.