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Pad-analysis of fuzzy control stability. (English) Zbl 0714.93029
Summary: By means of an index-represented rule mapping, the response g of a fuzzy controller is decomposed into two parts: f, the classical component and g-f, the nonlinear component, called pad. We intend to analyse the pad in order to know what the relation is between fuzzy and classical PID controllers. Fuzzy controllers mentioned here are defuzzified by the typical center of gravity method.
In the 1-input case, we get that $$f-g=0$$, which means that the fuzzy controller is exactly the same as the classical P-controller, whenever the rule mapping is simple and the linguistic values are a binary family.
In the 2-input case of error and error change for simple rule mapping and for simple linguistic values family, and perfectly for the WXYZ linguistic values family, the fuzzy controller can be viewed as a series connection of a classical PD conroller and a $$1+pad$$-element. The describing function of a $$1+pad$$-element is very simple and fine, and ensures that such fuzzy controller will be stable if the PD controller is.
The similar consequences are also obtained for the case of three inputs.

##### MSC:
 93C42 Fuzzy control/observation systems 93D99 Stability of control systems
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