Abstract
The general mathematical problem of fuzzy control is an interpolation problem: a list of fuzzy input-output data, usually provided by a list of linguistic control rules, should be realized as argument-value pairs for a suitably chosen fuzzy function. However, contrary to the usual understanding of interpolation, in the actual approaches this interpolation problem is considered as a global one: one uniformly and globally defined function should realize all the fuzzy input-output data.
In standard classes of functions thus this interpolation problem often becomes unsolvable. Hence it becomes intertwined with an approximation problem which allows that the given fuzzy input-output data are realized only approximately by argument-value pairs.
In this context the paper discusses some quite general sufficient conditions for the true solution of the interpolation problem, as well as similar conditions for suitably modified data, i.e. for a quite controlled approximation.
The present work has partly been supported by the German-Czech cooperation project CZE 01/022 of the BMBF.
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.
References
T. Calco, G. Mayor, R. Mesiar (eds.): Aggregation Operators: New Trends and Applications, Physica-Verlag: Heidelberg, 2002.
D. Dubois, H. Prade: On the use of aggregation operations in information fusion processes, Fuzzy Sets Systems 142 (2004), 143–161.
S. Gottwald: Characterizations of the solvability of fuzzy equations. Elektron. Informationsverarb. Kybernet. 22 (1986) 67–91.
S. Gottwald: Fuzzy Sets and Fuzzy Logic. The Foundations of Application — From a Mathematical Point of View, Vieweg: Braunschweig/Wiesbaden and Teknea: Toulouse, 1993.
S. Gottwald, V. Novák, I. Perfilieva: Fuzzy control and t-norm-based fuzzy logic. Some recent results, in: Proc. 9th Internat. Conf. IPMU’2002, ESIA — Université de Savoie, Annecy, 2002, 1087–1094.
L.P. Holmblad, J.J. Ostergaard: Control of a cement kiln by fuzzy logic, in: M.M. Gupta/ E. Sanchez (eds.), Fuzzy Information and Decision Processes. North-Holland Publ. Comp.: Amsterdam, 1982, 389–399.
A. Mamdani, S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, Internat. J. Man-Machine Studies 7 (1975) 1–13.
N.N. Morsi, A.A. Fahmy, On generalized modus ponens with multiple rules and a residuated implication, Fuzzy Sets Systems, 129 (2002), 267–274.
E. Sanchez, Resolution of composite fuzzy relation equations, Information and Control, 30 (1976) 38–48.
L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Systems, Man and Cybernet. SMC-3 (1973) 28–44.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gottwald, S. (2005). Fuzzy Control as a General Interpolation Problem. In: Reusch, B. (eds) Computational Intelligence, Theory and Applications. Advances in Soft Computing, vol 33. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31182-3_17
Download citation
DOI: https://doi.org/10.1007/3-540-31182-3_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22807-3
Online ISBN: 978-3-540-31182-9
eBook Packages: EngineeringEngineering (R0)