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Fuzzy control theory: A nonlinear case. (English) Zbl 0713.93036
Summary: Sources of nonlinearity in a fuzzy controller include the fuzzification algorithm used for the controller inputs; fuzzy control rules; type of fuzzy logic used for evaluating the fuzzy control rules; and the defuzzification algorithm used for the controller output. We analyze the performance of a simple fuzzy controller with linear and nonlinear defuzzification algorithms. We prove theoretically that such a fuzzy controller, the smallest possible, with two inputs (error and rate change of error) and a nonlinear defuzzification algorithm is equivalent to a nonfuzzy nonlinear proportional-integral (PI) controller with proportional-gain and integral-gain changing with error and rate change of error about a setpoint. Furthermore, this fuzzy controller is precisely equivalent to a conventional linear PI controller if a linear defuzzification algorithm is employed. Computer simulation showed that the performance of the fuzzy controller was almost the same as that of the PI controller when first-order and second-order linear processes were used. Furthermore, the fuzzy controller was significantly better when a first-order with a time delay model was used. More importantly, the simulated result illustrated that the fuzzy controller was stable when a nonlinear process model was controlled, but the PI controller was unstable.

##### MSC:
 93C42 Fuzzy control/observation systems 93C15 Control/observation systems governed by ordinary differential equations
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##### References:
  Buckley, J.J.; Ying, H., Linear fuzzy controller: it is a linear non-fuzzy controller, Inf. sci., (1988), to appear · Zbl 0718.93037  Buckley, J.J.; Ying, H., Fuzzy controller theory: limit theorems for linear fuzzy control rules, Automatica, 25, 469-472, (1988) · Zbl 0685.93003  Mamdani, E.H., Application of fuzzy algorithms for control of simple dynamics plant, (), 1585-1588  ()  Siler, W.; Ying, H., Fuzzy control theory: the linear case, Fuzzy sets systs., 33, 275-290, (1988) · Zbl 0683.93002  ()  Zadeh, L.A., Fuzzy sets, Inf. control, 8, 338, (1965) · Zbl 0139.24606
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