Mumbai University > Computer Engineering > Sem 7 > Soft Computing

Marks: 10 Marks

Year: May 2016

Defuzzification

Defuzzification refers to the way a crisp value is extracted from a fuzzy set as a representative value. In general, there are five methods for defuzzifying a fuzzy set A of a universe of discourse Z, as shown in Figure 3.

A brief explanation of each defuzzification strategy are shown as follows:

Centroid of area zCOA:

$^z COA=\dfrac{\int_z \mu _A(z)z dz}{\int_z \mu_A(z) dz}$'

where μA(z) is the aggregationed output MF. This is the most widely adopted defuzzification strategy, which is reminiscent of the calculation of expected values of probability distributions.

Bisector of area zBOA: zBOA satisfies

$$\int^{zBOA}_\alpha\mu_A(z)dz=\int^{\beta}_{zBOA}\mu_A(z)dz,$$

where α=min{z|z ∈ Z} and β=max{z|z ∈ Z}. That is, the vertical line z=zBOA partitions the region between z=α, z=β, y=0 and y=μA(z) into two regions with the same area.

Mean of maximum zMOM: zMOM is the average of the maximizing z at which the MF reach a maximum μ*. In symbols

$$^zMOM=\dfrac{\int_{Z'}^Z dz}{\int_{Z'dz}}$$

where Z' = {z | μA(z)=μ * }. In particular, if μA(z) has a single maximum at z=z *, then zMOM=z *. Moreover, if μA(z) reaches its maximum whenever z ∈ [zleft, zright] (This is the case in Figure 3), then zMOM = (zleft + zright)/2. The mean of maximum is the defuzzification strategy employed in Mamdani's fuzzy logic controllers.

Smallest of maximum zSOM: zSOM is the minimum (in terms of magnitude) of the maximizing z.

Largest of maximum zLOM: zLOM is the maximum (in terms of magnitude) of the maximizing z. Because of their obvious bias, zSOM and zLOM are not used as often as the other three defuzzification methods.

The calculation needed to carry out any of these five defuzzification operations is time-consuming unless special hardware support is available. Furthermore, these defuzzification operations are not easily subject to rigorous mathematical analysis, so most of the studies are based on experimental results. This leads to the propositions of other types of fuzzy inference systems that do not need defuzzification at all; two of them are introduced:

1.Sugeno Fuzzy Models (Also known as Takagi-Sugeno or TSK Fuzzy Model);

2.Tsukamoto Fuzzy Models ;