Introduction to Intelligent Control Part 6

Transcription

1 ECE Spring 2010 ntroduction to ntelligent Control Part 6 Prof. Marian S. Stachowicz Laboratory for ntelligent Systems ECE Department, University of Minnesota Duluth February 4-5, 2010

2 Fuzzy System Part 2

3 Outline Fuzzy System - Fuzzifier - Defuzzifier Fuzzy Decision Making 3

4 References for reading 1. M.S. Stachowicz, Lance Beall, Fuzzy Logic Package, Version-2 for Mathematica 5.1, Wolfram Research, nc., Demonstration Notebook: 2.3, 2.4, 2.5, 2.8.6, Manual: 1.6, 1.7, J. S. R. Jang, C.T. Sun, E. Mizutani, Neuro-Fuzzy and Soft Computing, Prentice Fall, 1997 Chapter 3, 4 3. G.J. Klir, Bo Yuan, Fuzzy Set and Fuzzy Logic, Prentice Hall, 1995 Chapters 5 4

5 Fuzzy system 5

6 Components of Fuzzy Systems Fuzzy knowledge base Fuzzifier nference engine Defuzzifier 6

7 Fuzzy knowledge base The knowledge base is made up of fuzzy rules in the form of F-THEN statements. This is the main part of the fuzzy system since the other components of the system are there to implement the rules in an appropriate manner. 7

8 nference engine The inference engine decides how to process the rules in the knowledge base using the fuzzy inputs from the fuzzifier. 8

9 Fuzzifier The fuzzifier decides how the crisp input will be converted into a fuzzy input to be used by the inference engine. 9

10 Fuzzifiers A fuzzifier maps a crisp point to a fuzzy set. u U R n to a fuzzy set A U 10

11 Fuzzifiers singleton fuzzifier, Gaussian fuzzifier, triangular fuzzifier. 11

12 Singleton fuzzifier The input is converted into fuzzy singletons A(u) = 1 at u, A(u) = 0 elsewhere - simplifies calculations - cannot suppress noise in the input. 12

13 Singleton fuzzifier 13

14 Gaussian fuzzifier The input is converted into Gaussian FS A(u) = exp[-(u * * 1 -u 1* )/a 1 ] 2 * * exp[-(u n -u n* )/a n ] 2 - simplifies calculations if MFs are Gaussian -suppress noise in the input. 14

15 Gaussian fuzzifier 15

16 Triangular fuzzifier The input is converted into triangular fuzzy set A(u) = (1- u 1 -u 1* /b 1 ) * * (1- u n -u n* /b n ) - simplifies calculations if MFs are triangular -suppress noise in the input. 16

17 Triangular fuzzifier 17

18 Summary The singleton fuzzifier simplifies the computation involved in any fuzzy inference engine. The Gaussian and triangular fuzzifiers also simplify the computation in the fuzzy inference engine with Gaussian and triangular MFs. The Gaussian and triangular fuzzifiers can suppress noise in the input, but singleton fuzzifier cannot. 18

19 Defuzzifier The defuzzifier decides how to convert the fuzzy result from the inference engine back into a crisp value. 19

20 Defuzzifiers A defuzzifier maps a fuzzy set to a crisp point. center of area, mean of max, bisector of area, smallest of max, largest of max. 20

21 Center of area CenterOfArea[FS1, ShowGraph-> True] Center of area is

22 Mean of max MeanOfMax[FS1, ShowGraph-> True]; Mean of max is

23 Bisector of area BisectorOfArea[FS1, ShowGraph-> True]; Bisector of area is 9. 23

24 Smallest of max SmallestOfMax[FS1, ShowGraph -> True]; Smallest of max is 6. 24

25 Largest of max LargestOfMax[FS1, ShowGraph-> True]; Largest of max is

26 FUZZY SYSTEM AS UNVERSAL APPROXMATOR For any given real continuous function g(u) on U g(u) : U R n R and arbitrary e> 0, there exists a fuzzy system f(u) such that sup f(u) -g(u) < e for all u U. 26

27 Fuzzy system The goal of fuzzy systems is to approximate the nonlinear function g(x). 27

28 Knowledge about g(x) 1. The analytic formula for g(x) is know at the start. 2. We do not know the formula for g(x), but we can determine g(x) for any arbitrary x U. 3. We don't know the formula for g(x), and we only have a finite set of input-output pairs (x i,g(x i )), where x i can not be chosen arbitrarily. 28

29 UNVERSAL APPROXMATOR The fuzzy systems f(u): - with product inference engine, - singleton fuzzifier, - center average defuzzifier, - and Gaussian membership functions are universal approximators. 29

30 Thank you.

31 Demo 1 Defuzzification strategies

32 Demo 2 a Measurement of similarities

33 Demo 2 b Measurement of fuzziness