Introduction to Intelligent Control Part 6

ECE 4951 - Spring 2010

Introduction to Intelligent Control Part 6 Prof. Marian S. Stachowicz Laboratory for Intelligent Systems ECE Department, University of Minnesota Duluth

February 4 - 5, 2010

Fuzzy System Part 2

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

IIntelligent Control I

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References for reading

1. M.S. Stachowicz, Lance Beall, Fuzzy Logic Package, Version-2 for Mathematica 5.1, Wolfram Research, Inc., 2003 - Demonstration Notebook: 2.3, 2.4, 2.5, 2.8.6, 2.8.7 - Manual: 1.6, 1.7, 1.8 2. 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


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Fuzzy system


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Components of Fuzzy Systems

• • • •

Fuzzy knowledge base Fuzzifier Inference engine Defuzzifier


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Fuzzy knowledge base • The knowledge base is made up of fuzzy rules in the form of IF-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.


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


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Fuzzifier

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


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Fuzzifiers

• A fuzzifier maps a crisp point to a fuzzy set. u ∈ U ⊂ Rn to a fuzzy set A ⊂ U


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Fuzzifiers

– singleton fuzzifier, – Gaussian fuzzifier, – triangular fuzzifier.


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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. IIntelligent Control I

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Singleton fuzzifier


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Gaussian fuzzifier • The input is converted into Gaussian FS A(u) = exp[-(u1-u1*)/a1]2*…* exp[-(un-un*)/an]2 - simplifies calculations if MFs are Gaussian - suppress noise in the input.


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Gaussian fuzzifier


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Triangular fuzzifier • The input is converted into triangular fuzzy set A(u) = (1- |u1-u1*|/b1)*…* (1- |un-un*|/bn) - simplifies calculations if MFs are triangular - suppress noise in the input.


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Triangular fuzzifier


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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.


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Defuzzifier

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


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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. IIntelligent Control I

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Center of area CenterOfArea[FS1, ShowGraph -> True] Center of area is 9.21739.


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Mean of max

MeanOfMax[FS1, ShowGraph -> True]; Mean of max is 9.5.


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


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Smallest of max

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


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Largest of max

LargestOfMax[FS1, ShowGraph -> True]; Largest of max is 13.


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FUZZY SYSTEM AS UNIVERSAL APPROXIMATOR

For any given real continuous function g(u) on U g(u) : U ⊂ Rn→R and arbitrary e> 0, there exists a fuzzy system f(u) such that sup | f(u) - g(u) | < e

for all u ∈ U.


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Fuzzy system

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


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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 (xi,g(xi)), where xi can not be chosen arbitrarily.


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UNIVERSAL APPROXIMATOR

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The fuzzy systems f(u): with product inference engine, singleton fuzzifier, center average defuzzifier, and Gaussian membership functions are universal approximators. IIntelligent Control I

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Thank you.

Demo 1 Defuzzification strategies

Demo 2 a Measurement of similarities

Demo 2 b Measurement of fuzziness