Exact fuzzy modeling and optimal control of the inverted pendulum on cart

Presents a method of exact fuzzy (TSK) modeling and optimal control of the inverted pendulum on a cart, which is a benchmark nonlinear dynamic system. Conventionally, the TSK (Takagi-Sugeno-Kang) fuzzy modeling blends local linear models to represent a nonlinear system, which in general does not exactly represent the nonlinear system under consideration. Here, instead of local linear models, a set of 'boundary linear models' and new membership-functions are defined such that the fuzzy blending of these models result in an exact representation of the overall nonlinear system. The SAM theorem of exact fuzzy representation of a scalar function is extended to a class of nonlinear dynamic system. Optimal fuzzy controller design results based on local linear models could be applied for the optimal fuzzy controller design based on 'boundary linear models' where, the fuzzy blending of these boundary linear models gives rise to an exact fuzzy model of the inverted pendulum on the cart. Optimal control laws are designed for each of these fuzzy subsystems, and overall control is again the fuzzy blending of these individual control laws. This procedure results in optimal control solution for the original nonlinear control problem. Optimal fuzzy controller is designed based on this exact fuzzy model and results are compared with a conventional design.