Defuuzification Techniques for Fuzzy Controllers

Chun-Fu KungSystem Laboratory,

Department of Computer Engineering and Science, Yuan-Ze University, Taiwan, Republic of

China2000/7/26

Jean J. Saade and Hassan B. diab

Outline

Introduction Elements of fuzzy controller Common defuzzification methods New defuzzification technique Conclusion

Introduction

Aiming at improving the performance of fuzzy controller, several useful concepts and approaches have been developed.

Self-organizing controllers, artificial neural network, and fuzzy relational equations.

Defuzzification is a procedure for determining the crisp value that is regarded as the most representative of the output fuzzy sets.

Introduction (cont.)

The mean of maxima (MOM) and the center of gravity (COG) methods have been mostly used to come up with crisp controller outputs.

The min-max weighted average formula (min-max WAF) is another powerful method to compute the crisp values.

Fuzzy Controller

A fuzzy controller is formed by input and output fuzzy sets assigned over the controller input and output variables, a collection of inference rules and a defuzzifier.

We usually using Zadeh’s compositional rule of inference to give an output fuzzy set for each crisp input pair (x0,y0)

Common Defuzzification Method

In order that this output be transformed into a crisp one, three main defuzzification techniques have so far been applied: the MOM, COG and min-max WAF.

COG method:

Min-max method:

dzzCdzzzCzCCOG )()()]([

N

jj

N

jjjcc

11

)]()([ 00 yBxA jjj

Case1 Study

New Technique

We required that the sum of the membership grades of any crisp input value in the different overlapping fuzzy sets defined over an input variable be 1.

Instead of using the minimum operation for AND in order to combine the membership grades of crisp input value in the fuzzy sets, the product of there grade is applied.

COOL -> sco%, WARM -> swa% and HOT -> shp%.

DRY -> sdr%, MOIST -> smo% and WET -> swe%

New Technique (cont.)

)()(

)()(

)()(

31

21

11

tDrtHo

tDrtWa

tDrtCo

)()(

)()(

)()(

32

22

12

tMotHo

tMotWa

tMotCo

)()(

)()(

)()(

33

23

13

tWetHo

tWetWa

tWetCo

),(),(),(

),(),(),(

),(),(),(

333231

232221

131211

wehomohodrho

wewamowadrwa

wecomocodrco

fan

ssfssfssf

ssfssfssf

ssfssfssf

S

2)(),( qpqpf

New Technique (cont.)

Temperature Humidity μ Fan Speed

COOL DRY μ 11COOL MOIST μ 12COOL WET μ 13WARM DRY μ 21WARM MOIST μ 22WARM WET μ 23

HOT DRY μ 31HOT MOIST μ 32HOT WET μ 33

),( drco ssf),( moco ssf),( weco ssf),( drwa ssf),( mowa ssf),( wewa ssf

),( drho ssf),( moho ssf),( weho ssf

),()]()([1 1

00 jiij

n

i

p

jji cBcAfyBxAc

Result

Humidity = 70% , left is Min-Max WAF and right is New method

Result (cont.)

left is MOM, right is COG

Result (cont.)

left is Min-Max WAF, right is New method

Case2 Study (washing machine)

left is MOM, right is COG

Case2 Study (cont.)

left is Min-Max WAF, right is New method

Conclusion

This technique integrates the defuzzification problem into the global setting of the elements of the fuzzy controller.

The new technique doesn’t consider probabilistic averaging and helps achieve performance objectives in an easy and systematic manner.

A nonprobabilistic and parametrized defuzzification method is a research project that has almost been completed.

Conclusion (cont.)

left is Fuzzy Fan, right is Washing Machine (δ=0.5)