A new method to guarantee performance and robustness in optimal fuzzy controller design for perturbed nonlinear systems based on Piecewise Discontinues Lyapunov Functions

Abstract

In this paper we present a systematic procedure to design robust fuzzy controller for exponentially stabilizing affine nonlinear systems, based on their TS fuzzy model. For robust design we consider modeling error in TS model and as well as perturbation in the original nonlinear system. Minimization of cost function along with mapping closed loop poles to desired poles are considered simultaneously in controller design. As a result, the desired specified performance in transient response can be achieved. Piecewise Discontinues Lyapunov Functions (PDLF) are utilized in our proposed method. To avoid difficulties in boundary conditions in PDLF we opt to design an online controller and check the regions and boundaries continuously. The constraints required to guarantee the exponential stability of the original nonlinear systems and optimal controller design with guaranteeing desired performance are presented in the LMI form. The y well developed. The power of these methods is that searching Lyapunov function and feedback gain can be stated as a convex optimization problem and the task of finding the common Lyapunov function can be readily be formulated into an LMI problem. However this approach is too conservative and there are lots of stable systems that we can not find a common positive definite Lyapunov function for all subsystems. Piecewise quadratic Lyapunov function approach have been considered to avoid conservativeness of quadratic Lyapunov function approaches. Piecewise quadratic Lyapunov function (PLF) are divided in two categories, one is continuous (PCLF) in boundaries and one of them is discontinuous (PDLF) on boundaries. It was shown that PDLF in contrast with PCLF results in fewer LMIs. To apply all mentioned methods, the system must be presented by a Takagi-Sugeno model and as it was demonstrated TS modeling enables us to deal with high order complicated nonlinear systems. Most of works so far have used PCLF for controller design and stability analysis, but PDLF have been used mainly for stability analysis and there are no reports about using PDLF for controller design. The main reason is difficulties in boundary conditions. effectiveness and applicability of the proposed method is examined on an inverted pendulum system.