A new type of a fuzzy controller HEKRDLA Josef

We discuss the properties of fuzzy controllers (approximators) based on compositional fuzzy inference rules. Special attention is paid to Mamdani and residuumbased fuzzy controllers 1, 2, 3]. Let and denote the input and the output space, respectively, and let F ( ) denote the collection of all fuzzy subsets. Suppose that we are given a rule base = ( i i )n=1 of if-then rules i
I nt
;:::;n
( ), is g with
X
Int3] The conclusion Int ( ) belongs to the convex hull of ( i )i2F , where = f j Supp i \ Supp 6= ;g. We show that the above axioms (which we consider quite natural) cannot be satis ed by the classical types of fuzzy controllers 5, 6]. As an alternative, we propose a new type of fuzzy controller based on modi cations of the compositional rule, a fuzzy controller with conditionally ring rules 4], which allows to satisfy the above axioms under quite general conditions. We expect that it could have better performance in applications than previous controllers.
References
1] B. de Baets, A note on Mamdani controllers. Technical report FUM.BDB.96.12 , University Gent, 1996. 2] D. Driankov, H. Hellendoorn, M. Reinfrank, An Introduction to Fuzzy Control. Springer, Berlin, Heidelberg, 1993. 3] E.H. Mamdani, S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller. J. Man-Machine Stud. 7, (1975) 1{13. 4] B. Moser, M. Navara, Fuzzy controllers with conditionally ring rules. To appear. 5] B. Moser, M. Navara, Conditionally ring rules extend the possibilities of fuzzy controllers. In: Proc. Int. Conf. Computational Intelligence for Modelling, Control and Automation, M. Mohammadian (ed.), IOS Press, Amsterdam, Netherlands, 1999, 242{245. 6] B. Moser, M. Navara, Which triangular norms are convenient for fuzzy controllers? In: Proc. EUSFLAT-ESTYLF Joint Conf. 99 , Universitat de les Illes Balears, Palma (Mallorca), Spain, 1999, 75{78. 2
X Y : X ;Y
\If 2
x X X Y Y i X Y
X
iHale Waihona Puke then 2 i ,"
y Y
where i 2 F ( ), i 2 F ( ), = 1 . A fuzzy controller, considered as a mapping Int : F ( ) ! F ( ), should satisfy the following axioms:
;:::;n
The authors gratefully acknowledge the support of the project VS 96049 of the Czech Ministry of Education, the grant 201/97/0437 of the Grant Agency of the Czech Republic and the project Aktion Osterreich { Tschechien 23p16.
NAVARA Mirko
Center for Machine Perception, Faculty of Electrical Engineering Czech Technical University Technicka 2, 166 27 Praha Czech Republic E-mail: navara@cmp.felk.cvut.cz
i ;:::;n X Y
Int2] For each normal observation 2 F ( ), the conclusion, not contained in all consequents, i.e., there is an index 2 f1 Int ( ) 6 i .
X X i X Y X Y F i X X
MOSER Bernhard
Fuzzy Logic Laboratorium Linz-Hagenberg, Johannes Kepler University A-4040 Linz Austria E-mail: moser@ ll.uni-linz.ac.at, bernhard.moser@pct.at
1
Int1] If the observation coincides with one of the premises then the resulting conclusion coincides with the corresponding consequent, i.e., 8 2 f1 g: Int ( i ) = i .
A new type of a fuzzy controller
HEKRDLA Josef
Department of Mathematics, Faculty of Electrical Engineering Czech Technical University Technicka 2, 166 27 Praha Czech Republic E-mail: hekrdla@math.feld.cvut.cz