Continuity of Fuzzy Approximate Reasoning and Its Application to Optimization
This paper describes a mathematical framework for studying a nonlinear feedback control. The fuzzy control discussed here is the nonlinear feedback control in which the feedback laws are determined by IF-THEN type fuzzy production rules through approximat
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University of Marketing and Distribution Sciences, Kobe 651-2188, Japan takashi [email protected] 2 Shinshu University, Nagano 380-8553, Japan
Abstract. This paper describes a mathematical framework for studying a nonlinear feedback control. The fuzzy control discussed here is the nonlinear feedback control in which the feedback laws are determined by IF-THEN type fuzzy production rules through approximate reasoning introduced by Nakamori. To prove existence of optimal control, we applied compactness of a set of membership functions in L2 space and continuity of the approximate reasoning, and prepared some propositions concerning approximate reasoning of Nakamori model. By considering fuzzy optimal control problems as problems of finding the minimum (maximum) value of the integral cost (benefit) function on an appropriate set of membership functions, the existence of fuzzy optimal control is shown. Keywords: Fuzzy control, Nakamori model, Optimization, Functional analysis.
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Introduction
In 1965 Zadeh introduced the notion of fuzziness [1], and then Mamdani has applied it to the field of control theory using what is called Mamdani method [2]. This method is one of the ways to represent numerically the control given by human language and sensitivity, and it has been applied in various practical control plants. However, unlike the theory of classical control and modern control, systematized considerations have not yet been discussed sufficiently. Moreover, in practical use, fuzzy membership functions, which represent input and output states in optimal control system, are decided on the basis of the experience of experts in each peculiar plant. The authors have been studying to establish the automatic and computational determination of fuzzy membership functions, which give optimal controls in fuzzy control system. We also have been studying to find algorithms that compute optimal solutions. The optimization of fuzzy control discussed in this paper is different from conventional method such as classical control and modern control e.g. a linear matrix inequality approach [3]. We consider fuzzy
The paper was supported in part by Grant-in-Aid for Young Scientists (B) #19700225 from Japan Society for the Promotion of Science (JSPS).
M.A. Orgun and J. Thornton (Eds.): AI 2007, LNAI 4830, pp. 529–538, 2007. c Springer-Verlag Berlin Heidelberg 2007
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T. Mitsuishi and Y. Shidama
optimal control problems as problems of finding the minimum (maximum) value of the performance function with feedback law constructed by approximate reasoning with the way of functional analysis [4] [5] [6] [7] [8]. In our previous studies, some approximate reasoning: Mamdani method and T-S fuzzy model, etc., were analyzed about its two kind of continuity for the optimization. Using these approaches and results, one can obtain optimal fuzzy control using almost all approximate reasoning. However Nakamori method contains another calculation different from some major methods above, therefore we need to study its continuity and existence of
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