Approximation theory of fuzzy systems based upon genuine many-valued implications — SISO cases

doi:10.1016/S0165-0114(01)00166-X

Fuzzy Sets and Systems

Volume 130, Issue 2, 1 September 2002, Pages 147-157

https://doi.org/10.1016/S0165-0114(01)00166-X Get rights and content

Abstract

It is proved that the single input and single output (SISO) fuzzy systems based upon genuine many-valued implications are universal approximators. It is shown theoretically that fuzzy control systems based upon genuine many-valued implications are equivalent to those based upon t-norm implications, the general approach to construct fuzzy systems is given. It is also shown that defuzzifier based upon center of areas is not appropriate to the fuzzy systems based upon genuine many-valued implications.

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Commonly studied desirable properties are (i) Interpolativity, (ii) Continuity, (iii) Robustness, (iv) Approximation Capability, (v) Efficiency and (vi) Monotonicity. For an FRI system with an implicative rule base, these properties have been studied in [3–15]. Among these many desirable properties of an FRI system, monotonicity of the system function is one of the most important properties and lack of which may lead to an unreliable inference system.
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Performance analysis of fuzzy systems based on quintuple implications method
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To improve the quality of approximate reasoning in fuzzy system, quintuple implication principle (QIP) to resolve fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT) problems has been proposed by Zhou et al. [32]. The same approximation as Mamdani-type fuzzy inference can be reached by fuzzy reasoning with QIP method for Gödel implication. It therefore is essential to establish some fundamental properties of fuzzy inference system with QIP method. This paper mainly investigates the robustness and universal approximation capability of fuzzy inference system with QIP method. Firstly, we present the QIP solutions of FMP and FMT for R-, S-, QL-, f- and g-implications. And then the robustness of fuzzy inference system with QIP method is discussed. Finally, we study the universal approximation properties of multiple-input and single-output (MISO) fuzzy systems with QIP method for R-, S- and QL-implications. These results reveal that the QIP method possesses better performance in fuzzy rule-based system.
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Also for the conditionally firing implicative fuzzy rules, one may question whether the universal approximation ability is preserved. As we know, this crucial property is preserved for Mamdani–Assilian systems as well as for the implicative fuzzy systems [29,30,35] and in [49], the authors proved the preservation of the same property for multiplicative systems, which differ from the implicative ones only by the Łukasiewicz t-norm as the operation aggregating rules instead of the minimum. The proof stems from the technique used in [49] with the focus on the difference due to the conditional firing.
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^☆: This work is supported by National Science Foundation of China (Grant No. 19901028) and National 973 Foundational Research Program G1998030417.

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Approximation theory of fuzzy systems based upon genuine many-valued implications — SISO cases☆

Abstract

Automatica

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Inform. Sci.

Fuzzy Sets and Systems

Reasoning with truth values on compacted fuzzy chained rules

IEEE Trans. Systems Man Cybernet. — Part B: Cybernet.

Investigations on the applicability of fuzzy inference

Fuzzy Sets and Systems