Abstract
The purpose of this study is to consider the fuzzy optimal control based on the functional analysis. We used a mathematical approach to compute optimal solutions. The feedback of fuzzy control is evaluated through approximate reasoning using the center of sums defuzzification method or the height method on IF-THEN fuzzy rules. 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 continuous function space is compact metrizable. 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 minimizes (maximizes) the integral performance function of the nonlinear feedback fuzzy system.
This work was supported by JSPS KAKENHI Grant Numbers 24700235, 23730395.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Mamdani, E.H.: Application of fuzzy algorithms for control of simple dynamic plant. Proc. IEE 121(12), 1585–1588 (1974)
Mizumoto, 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(7), 1153–1173 (1994)
Tanaka, K., Sugeno, M.: Stability Analysis of Fuzzy Systems and Construction Procedure for Lyapunov Functions. Transactions of the Japan Society of Mechanical Engineers (C) 58(550), 1766–1772 (1992)
Mitsuishi, T., Wasaki, K., Kawabe, J., Kawamoto, N.P., Shidama, Y.: Fuzzy optimal control in L2 space. In: Proc. 7th IFAC Symposium Artificial Intelligence in Real-Time Control, pp. 173–177 (1998)
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., Shidama, Y.: Continuity of Fuzzy Approximate Reasoning and Its Application to Optimization. In: Orgun, M.A., Thornton, J. (eds.) AI 2007. LNCS (LNAI), vol. 4830, pp. 529–538. Springer, Heidelberg (2007)
Mitsuishi, T., Shidama, Y.: gCompactness of Family of Fuzzy Sets in L2 Space with Application to Optimal Control. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E92-A(4), 952–957 (2009)
Mitsuishi, T.: Continuity of Approximate Reasoning Using Center of Sums Defuzzification Method. In: Proc. of IEEE 35th International Convention of Information Communication Technology, Electronics and Microelectronics MIPRO 2012, pp. 1172–1175 (2012)
Miller, R.K., Michel, A.N.: Ordinary Differential Equations. Academic Press, New York (1982)
Riesz, F., Sz.-Nagy, B.: Functional Analysis. Dover Publications, New York (1990)
Dunford, N., Schwartz, J.T.: Linear Operators Part I: General Theory. John Wiley & Sons, New York (1988)
Ross, T.J.: Fuzzy Logic With Engineering Application. John Wiley and Sons Ltd., UK (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mitsuishi, T., Terashima, T., Shimada, N., Homma, T., Sawada, K., Shidama, Y. (2012). Continuity of Defuzzification on L2 Space for Optimization of Fuzzy Control. In: Huang, R., Ghorbani, A.A., Pasi, G., Yamaguchi, T., Yen, N.Y., Jin, B. (eds) Active Media Technology. AMT 2012. Lecture Notes in Computer Science, vol 7669. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35236-2_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-35236-2_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35235-5
Online ISBN: 978-3-642-35236-2
eBook Packages: Computer ScienceComputer Science (R0)