Nonlinear control and rigorous stability analysis based on fuzzy system for inverted pendulum

The basic motion of the inverted pendulum with a DC motor can be described by a set of nonlinear differential equations. In this paper, on the basis of the method proposed by the Kawamoto et al., first the equations including a control input are rewritten into the form of a fuzzy system without any inference and approximation. Next, by applying Tanaka and Sugeno´s (1994) stability theorem to the fuzzy system, the problem of finding a common symmetric positive definite matrix P satisfying two Lyapunov inequalities, that is, the common Lyapunov problem can be derived. Then, the P-region method is introduced for finding the matrix P, and feedback gains are determined under the existence of P. For the inverted pendulum treated here, the stability of both cases of linear and fuzzy control inputs is rigorously analyzed and is guaranteed in the whole region of all the dependent variables. Finally, it is numerically shown that the fuzzy control input gives better control responses than the linear one does