Tuning fuzzy control rules by the ? constrained method which solves constrained nonlinear optimization problems

Learning of fuzzy control rules can be considered as solving a constrained nonlinear optimization problem, in which the objective function is not differentiable. In this case, usually the problem is solved by the combination of a direct search method and penalty method. However, it is difficult to know what value of the penalty coefficient leads to a feasible solution and how much a search point for a solution satisfies constraints. In this research, we represent the satisfaction level of constraints by fuzzy constraints of fuzzy programming. We propose α level comparison, which compares the search points based on the satisfaction level. We propose the α constrained method, which converts constrained problems to unconstrained problems using α level comparisons. We also propose the α constrained Powell method by applying α constrained method to Powell's direct search method. Through some examples and the learning of fuzzy control rules, we show that a feasible solution can be obtained easily by our method with confirming the satisfaction level.