Stable adaptive fuzzy controllers with application to inverted pendulum tracking

An adaptive fuzzy controller is constructed from a set of fuzzy IF-THEN rules whose parameters are adjusted on-line according to some adaptation law for the purpose of controlling the plant to track a given-trajectory. In this paper, two adaptive fuzzy controllers are designed based on the Lyapunov synthesis approach. We require that the final closed-loop system must be globally stable in the sense that all signals involved (states, controls, parameters, etc.) must be uniformly bounded. Roughly speaking, the adaptive fuzzy controllers are designed through the following steps: first, construct an initial controller based on linguistic descriptions (in the form of fuzzy IF-THEN rules) about the unknown plant from human experts; then, develop an adaptation law to adjust the parameters of the fuzzy controller on-line. We prove, for both adaptive fuzzy controllers, that: (1) all signals in the closed-loop systems are uniformly bounded; and (2) the tracking errors converge to zero under mild conditions. We provide the specific formulas of the bounds so that controller designers can determine the bounds based on their requirements. Finally, the adaptive fuzzy controllers are used to control the inverted pendulum to track a given trajectory, and the simulation results show that: (1) the adaptive fuzzy controllers can perform successful tracking without using any linguistic information; and (2) after incorporating some linguistic fuzzy rules into the controllers, the adaptation speed becomes faster and the tracking error becomes smaller.