Abstract
This article presents a mathematical framework for studying the existence of optimal feedback control based on IF-THEN fuzzy rules through approximate reasoning, and introduces the notion of an admissible fuzzy controller. The framework consists of two propositions: To guarantee the convergence of optimal solution, a set of fuzzy membership functions (admissible fuzzy controller) which are selected out of L 2 space is convex and compact metrizable for the weak topology. And assuming approximate reasoning to be a functional on the set of membership functions, its continuity is proved. Then, we show the existence of a fuzzy controller which minimize (maximize) the integral performance function of the nonlinear feedback system.
The paper was supported in part by Grant-in-Aid for Young Scientists (B) #19700225 from Japan Society for the Promotion of Science (JSPS).
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References
Shidama, Y., Yang, Y., Eguchi, M., Yamaura, H.: The compactness of a set of membership functions and its application to fuzzy optimal control. The Japan Society for Industrial and Applied Mathematics 1, 1–13 (1996)
Mitsuishi, T., Wasaki, K., Kawabe, J., Kawamoto, N.P., Shidama, Y.: Fuzzy optimal control in L 2 space. In: Proc. 7th IFAC Symposium Artificial Intelligence in Real-Time Control, pp. 173–177 (1998)
Mizumoto, M.M.: Improvement of fuzzy control (IV) - Case by product-sum-gravity method. In: Proc. 6th Fuzzy System Symposium, pp. 9–13 (1990)
Nakamori, Y., Ryoke, M.: Identification of fuzzy prediction models through hyperellipsoidal clustering. IEEE Transactions on Systems, Man and Cybernetics SMC-24(8), 1153–1173 (1994)
Dunford, N., Schwartz, J.T.: Linear Operators Part I: General Theory. John Wiley & Sons, New York (1988)
Mitsuishi, T., Kawabe, J., Wasaki, K., Shidama, Y.: Optimization of Fuzzy Feedback Control Determined by Product-Sum-Gravity Method. Journal of Nonlinear and Convex Analysis 1(2), 201–211 (2000)
Mitsuishi, T., Endou, N., Shidama, Y.: Continuity of Nakamori Fuzzy Model and Its Application to Optimal Feedback Control. In: Proc. IEEE International Conference on Systems, Man and Cybernetics, pp. 577–581 (2005)
Miller, R.K., Michel, A.N.: Ordinary Differential Equations. Academic Press, New York (1982)
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© 2008 Springer-Verlag Berlin Heidelberg
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Mitsuishi, T., Shidama, Y. (2008). Admissible Fuzzy Controller in L 2 Space . In: Nguyen, N.T., Borzemski, L., Grzech, A., Ali, M. (eds) New Frontiers in Applied Artificial Intelligence. IEA/AIE 2008. Lecture Notes in Computer Science(), vol 5027. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69052-8_7
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DOI: https://doi.org/10.1007/978-3-540-69052-8_7
Publisher Name: Springer, Berlin, Heidelberg
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