Construction of a fuzzy relation with reduced dimension for multivariable systems by using genetic algorithm

Fuzzy control is widely used in industry. It is often applied by describing both the input and output in fuzzy variables, and composing the fuzzy relation "R" represented by multi-dimensional matrix, which is based on the "if-then" fuzzy rules. The method of Mamdani and the methods using a GA (genetic algorithm) were proposed to solve the fuzzy relation equation. However, for most multivariable systems there occurs the problem of insufficient memory for calculation. To cope with this problem, Gegov (1994) proposed a method using a two-dimensional fuzzy relation for multivariable systems. Though it is effective in the sense of reducing the calculation, it does not often satisfy all of the fuzzy rules. Especially in the case when outputs are not alike in shape in spite of their similar inputs, it is difficult to find the fuzzy relation representing the system correctly. Therefore, another method suitable for multivariable cases is needed. We propose a technique to construct the reduced dimensional fuzzy relation for multivariable systems. The optimal fuzzy relation can be chosen among the candidates that satisfy all of the rules, by evaluating the number of matrix elements. Here, a GA is adopted for efficient search. In the case that we have several solutions from the search, the set that gives the lowest dimension is selected. Then we decide the value of matrix elements by least square errors that are obtained by applying the same inputs to the resulting (reduced dimensional) fuzzy relation and original one. Finally we show some simulation examples, which show the usefulness of the proposed method.