What is Lyapunov’s (Second) Approach

This chapter presents two fuzzy adaptive variable structure controllers for a class of uncertain multi-input multi-output nonlinear systems with actuator nonlinearities (i.e. with sector nonlinearities and dead-zones). The design of the first controller concerns systems with symmetric and positive definite control-gain matrix, while the design of the second one is extended to the case of non-symmetric control-gain matrix thanks to an appropriate matrix decomposition, namely the product of a symmetric positive-definite matrix, a diagonal matrix with diagonal entries +1 or -1, and a unity upper triangular matrix. An appropriate adaptive fuzzy-logic system is used to reasonably approximate the uncertain functions. A Lyapunov approach is adopted to derive the parameter adaptation laws and prove the stability of the closed-loop control system. Finally, some simulation results are carried out to show the effectiveness of the proposed controllers.