By Prof. Andrzej Piegat (auth.)
In the final ten years, a real explosion of investigations into fuzzy modeling and its functions on top of things, diagnostics, choice making, optimization, development acceptance, robotics, and so on. has been saw. The charm of fuzzy modeling effects from its intelligibility and the excessive effectiveness of the versions bought. as a result of this the modeling might be utilized for the answer of difficulties that could no longer be solved until now with any identified traditional equipment. The ebook presents the reader with a sophisticated advent to the issues of fuzzy modeling and to at least one of its most vital purposes: fuzzy regulate. it truly is in response to the newest and most important wisdom of the topic and will be used not just by means of regulate experts but in addition by means of experts operating in any box requiring plant modeling, method modeling, and structures modeling, e.g. economics, company, drugs, agriculture,and meteorology.
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Extra info for Fuzzy Modeling and Control
Disadvantages of the harmonie membership function 1. The harmonie funetion is symmetrieal, and at non-uniform distribution of membership funetions in the space of diseourse this makes the reaIization of the postulate of partition of unity diffieult and negatively influenees the quality of a model in regions indifferently eovered by membership funetions. 38). 4 Types of membership functions of fuzzy sets iL(x) Jl(x) 47 ji(x) x b-a b b-a b+a b b+a Fig. 45. 5 o ii(x) I I I I -1----- I : : X b-a b b+a Fig.
Such ranges can come into existence during adjusting (adapting) fuzzy models based on input/output data (self-Iearning algorithms). The consequence is that zones arise which are insensitive to a change of the given input variable. 34 2. --,. abc d e abc x de x Fig. 27. Conversion of fuzzy sets not satisfying the condition of a partition of unity (a) into sets satisfying this condition (b) The condition of a partition of unity is sometimes also called the condition of overlapping or covering membership functions to unity.
A(X) = a~Yfl1 (1. 25 0) - 1'--2-'--3-'Ll'--5-'13'--7-'8 . Comparing the primary membership function ~A(X) with the function reproduced on the basis of the a-cuts, Fig. 16, we can see that the reproduction is not exact. Its accuracy can be improved by the increase of the number of a-cuts or their better placement. ~A (x) 000 26 2. 2 --1- o : : :: I I I I I I --1---r--j---r-- 234 I I 5 6 7 :: -r--j I I 8 9 x Fig. 16. Primary membership function J-tA(X) (continuous line) and the function J-t:A(x) reproduced on the basis of a-cuts (dotted line) • The power (cardinality) of a fuzzy set The power IIAII or cardinal number, denoted by card(A), of a discrete fuzzy set, is the sum of the membership grades of all elements in the fuzzy set.