Fuzzy Observer, Fuzzy Controller Design, and Common Hurwitz Matrices for a Class of Uncertain Nonlinear Systems

In the chapter, fuzzy state observers and fuzzy controllers are developed for a type of uncertain nonlinear systems. The systems are represented by more general fuzzy modeling. Many interesting results are obtained as follows: First, by constructing Lyapunov function approaches and inequalities tools, the adaptive observer laws including Riccati equations, two differentiators, and many solvability conditions about the above Riccati equations are presented. Second, based on Lyapunov function approaches and the same inequalities tools, the proposed controllers are designed to guarantee the stability of the overall closed-loop systems, and many solvability conditions on the proposed controllers are analyzed too. More importantly, we give the structure of the common stable matrixes for this kind of control problem, including their disturbance structures and Lie algebra conditions. Finally, numerical simulations on the magnetic levitation systems show the effectiveness of our approaches.