Fuzzy control theory: The linear case.

*(English)*Zbl 0683.93002Summary: A linear fuzzy proportional-integral (PI) controller with one input and one output is defined in terms of piecewise linear membership functions for fuzzification; control rules; and defuzzification algorithm. It is shown that a linear fuzzy controller is not equivalent to a linear non- fuzzy PI controller if the rules are evaluated using the Zadeh, probability or Lukasiewicz fuzzy logic alone.

However, the linear fuzzy controller is precisely equivalent to a linear non-fuzzy PI controller if mixed fuzzy logic is used to evaluate the control rules, when the fuzzy logics used are selected with due regard to prior associations implied by the control rule operands themselves. This finding has relevance to the selection of fuzzy logics in fuzzy expert systems and for fuzzy logics in general.

However, the linear fuzzy controller is precisely equivalent to a linear non-fuzzy PI controller if mixed fuzzy logic is used to evaluate the control rules, when the fuzzy logics used are selected with due regard to prior associations implied by the control rule operands themselves. This finding has relevance to the selection of fuzzy logics in fuzzy expert systems and for fuzzy logics in general.

##### MSC:

93A10 | General systems |

94D05 | Fuzzy sets and logic (in connection with information, communication, or circuits theory) |

03B52 | Fuzzy logic; logic of vagueness |

##### Keywords:

linear fuzzy proportional-integral (PI) controller; fuzzification; defuzzification algorithm; fuzzy logics; fuzzy expert
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\textit{W. Siler} and \textit{H. Ying}, Fuzzy Sets Syst. 33, No. 3, 275--290 (1989; Zbl 0683.93002)

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##### References:

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