This is the most commonly used defuzzification technique. In this method, the overlapping area is counted twice.

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1 Chapter 5 Defuzzificatin Methds Fuzzy rule based systems evaluate linguistic if-then rules using fuzzificatin, inference and cmpsitin prcedures. They prduce fuzzy results which usually have t be cnverted int crisp utput. T transfrm the fuzzy results in t crisp, defuzzificatin is perfrmed. Defuzzificatin is the prcess f cnverting a fuzzified utput int a single crisp value with respect t a fuzzy set. The defuzzified value in FLC (Fuzzy Lgic Cntrller) represents the actin t be taken in cntrlling the prcess. Different Defuzzificatin Methds The fllwing are the knwn methds f defuzzificatin. Center f Sums Methd (COS) Center f gravity (COG) / Centrid f Area (COA) Methd Center f Area / Bisectr f Area Methd (BOA) Weighted Average Methd Maxima Methds First f Maxima Methd (FOM) Last f Maxima Methd (LOM) Mean f Maxima Methd (MOM) Center f Sums (COS) Methd This is the mst cmmnly used defuzzificatin technique. In this methd, the verlapping area is cunted twice. The defuzzified value is defined as : =., Here, n is the number f fuzzy sets, N is the number f fuzzy variables, μ is the membership functin fr the k-th fuzzy set. Example 1

2 The defuzzified value is defined as : =, Here, represents the firing area f rules and k is the ttal number f rules fired and represents the center f area. The aggregated fuzzy set f tw fuzzy sets and is shwn in Figure 1. Let the area f this tw fuzzy sets are and. = ½ * [(8-1) + (7-3)] * 0.5 = ½ * 11 * 0.5 = 55/20=2.75 = ½ * [(9-3) + (8-4)] * 0.3 = ½ * 10 * 0.3 = 3/2 =1.5 Nw the center f area f the fuzzy set is let say = (7+3)/2= 5 and the center f area f the fuzzy set is = (8+4)/2=6. Nw the defuzzified value =.... =.. = 22.75/4.25 = 5.35 µ A A x Figure 1 : Fuzzy sets and Center f gravity (COG) / Centrid f Area (COA) Methd This methd prvides a crisp value based n the center f gravity f the fuzzy set. The ttal area f the membership functin distributin used t represent the cmbined cntrl actin is divided int a number f sub-areas. The area and the center f gravity r centrid f each sub-area is calculated and then the summatin f all these sub-areas is taken t find the defuzzified value fr a discrete fuzzy set. 2

3 Fr discrete membership functin, the defuzzified value dented as using COG is defined as: =. the membership functin, and n represents the number f elements in the sample. Fr cntinuus membership functin, is defined as : = µ µ, Here indicates the sample element, μ is µ x 6 Figure 2 : Fuzzy sets C1 and C2 Example: The defuzzified value using COG is defined as: =, Here N indicates the number f sub-areas, and represents the area and centrid f area, respectively, f sub-area. In the aggregated fuzzy set as shwn in figure 2., the ttal area is divided int six sub-areas. Fr COG methd, we have t calculate the area and centrid f area f each sub-area. These can be calculated as belw. The ttal area f the sub-area 1 is ½ * 2 * 0.5 = 0.5 The ttal area f the sub-area 2 is (7-3) * 0.5 = 4 * 0.5 = 2 The ttal area f the sub-area 3 is ½ * (7.5-7) * 0.2 = 0.5 * 0.5 *0.2 =.05 The ttal area f the sub-area 4 is 0.5* 0.3 =.15 The ttal area f the sub-area 5 is 0.5* 0.3 =.15 The ttal area f the sub-area 6 is ½ *1* 0.3 =.15 Nw the centrid r center f gravity f these sub-areas can be calculated as 3

4 Centrid f sub-area1 will be (1+3+3)/3 = 7/3 =2.333 Centrid f sub-area2 will be (7+3)/2 = 10/2 = 5 Centrid f sub-area3 will be ( )/3 = 21.5/3 =7.166 Centrid f sub-area4 will be (7+7.5)/2 =14.5/2=7.25 Centrid f sub-are5 will be (7.5+8)/2 =15.5/2 = 7.75 Centrid f sub-area6 will be (8+8+9)/3 = 25/3 = Nw we can calculate. and is shwn in table 1. Table 1 Sub area number Area( ) Centrid f area( ) The defuzzified value will be = = ( )/3 =5.008 Center f Area / Bisectr f Area Methd (BOA) This methd calculates the psitin under the curve where the areas n bth sides are equal. The BOA generates the actin that partitins the area int tw regins with the same area. μ dx = μ, where α = min {x x X} and β = max {x x X} Weighted Average Methd This methd is valid fr fuzzy sets with symmetrical utput membership functins and prduces results very clse t the COA methd. This methd is less cmputatinally intensive. Each membership functin is weighted by its maximum membership value. The defuzzified value is defined as : 4

5 = µ. µ Here dentes the algebraic summatin and x is the element with maximum membership functin. µ(x) x Figure 3: Fuzzy set A Example: Let A be a fuzzy set that tells abut a student as shwn in figure 3 and the elements with crrespnding maximum membership values are als given. A = {(P, 0.6), (F, 0.4),(G, 0.2),(VG, 0.2), (E, 0)} Here, the linguistic variable P represents a Pass student, F stands fr a Fair student, G represents a Gd student, VG represents a Very Gd student and E fr an Excellent student. Nw the defuzzified value fr set A will be = = 98/1.4=70 The defuzzified value fr the fuzzy set A with weighted average methd represents a Fair student. Maxima Methds 5

6 This methd cnsiders values with maximum membership. There are different maxima methds with different cnflict reslutin strategies fr multiple maxima. First f Maxima Methd (FOM) Last f Maxima Methd (LOM) Mean f Maxima Methd (MOM) First f Maxima Methd (FOM) This methd determines the smallest value f the dmain with maximum membership value. Example: The defuzzified value f the given fuzzy set will be =4. µ(x) x Last f Maxima Methd (LOM) Determine the largest value f the dmain with maximum membership value. In the example given fr FOM, the defuzzified value fr LOM methd will be = 8 Mean f Maxima Methd (MOM) In this methd, the defuzzified value is taken as the element with the highest membership values. When there are mre than ne element having maximum membership values, the mean value f the maxima is taken. Let A be a fuzzy set with membership functin µ (x) defined ver x X, where X is a universe f discurse. The defuzzified value is let say f a fuzzy set and is defined as, 6

7 =, Example Here, M = { μ ( ) is equal t the height f the fuzzy set A} and M is the cardinality f the set M. In the example as shwn in Fig., x = 4, 6, 8 have maximum membership values and hence M = 3 Accrding t MOM methd, = Nw the defuzzified value will be = = 6. References: 1. N. Mgharreban and L. F. DiLalla Cmparisn f Defuzzificatin Techniques fr Analysis f Nninterval Data, IEEE, Jean J. Saade and Hassan B. Diab. Defuzzificatin Methds and New Techniques fr Fuzzy Cntrllers, Iranian Jurnal f Electrical and Cmputer Engineering, Aarthi Chandramhan, M. V. C. Ra and M. Senthil Arumugam: Tw new and useful defuzzificatin methds based n rt mean square value, Sft Cmputing, Sft Cmputing by D.K. Pratihar, Narsa Publicatin. 7