Fuzzy Logic Introduction

A fuzzy logic model is a functional relation between two multidimensional spaces containing the fuzzy sets, which are the central concept of Zadeh theory and are defined as:

A = {(x, ma{x)) : x єX}

where mA (x) is the membership function expressing the degree of elements x in the fuzzy set A.

Different sets are distinguished by different membership functions. Let’s see an intuitive example. Assuming the Julian days set {1,2… 172} corresponds to X from the definition Eq. (7.13). We are familiar with the division of X at j0 = 80 (March, 21) in two sets (seasons): WINTER for j є W = {1,79} and SPRING for j є S = {1,171}. We are used to express that j belongs to W by an application f:X —— {0,1}, showed inFig. 7.5. It is what we call a crisp set. Fuzzy logic relaxes the crossing from W to S by replacing the step-like separation between WINTER and SPRING with a slow passing in a finite interval around j0. Thus, the binary domain {0, 1} is filled with real numbers being turned into a continuous domain and the function f (t) is replaced with the membership functions mWINTER(j) and mSPRiNG (j). From Fig. 7.5 a day up to February 20 (j < 51) certainly is a WINTER day while a day after April 20 (j = 111) certainly is a SPRING day; 1 March, j = 60, is assigned of mWINTER(60) = 0.83 degree to be WINTER and mSPRING(60) = 0.17 degree to be SPRING. Therefore, the membership function reads out the level of confidence for a day to be in the one of the sets WINTER or SPRING.

In fuzzy sets theory a physical variable, as Julian day in previously example is named linguistic variable. The values of a linguistic variable are not numbers, as in the case of deterministic variables, but linguistic values, called attribut, expressed by words or sentences (e. g. WINTER and SPRING). The number of attributes of a linguistic variable and the shape of membership functions depends on the applica­tion, being specified in a heuristic way. Theoretically, the membership function can have any form; in practice triangular and trapezoidal forms are widely used.

Different fuzzy sets are combined through membership functions:

Fuzzy intersection (AND): mArB = min(mA(x),mB(x)), Vx єX (7.14a) Fuzzy reunion (OR): mAUB = max(mA(x),mB(x)) ,Vx єX (7.14b)

Equations (7.14) define the Zadeh fuzzy operators (Zadeh 1965). There are also others definitions of fuzzy logic operators (Zimmermann 1996) but we will use only the definitions Eqs. (7.14) in operations with fuzzy sets.

The map between the input and the output fuzzy spaces is a collection of asso­ciative rules, each reading:

IF (premises) THEN (conclusions)

Every premise or conclusion consists of an expression as:

(variable) IS (attribute) (7.16)

connected through fuzzy operator AND.

The information is carried out from input to output of a fuzzy system in three

steps:

1. Fuzzification is a coding process in which each numerical input of a linguistic variable is converted in membership function values of attributes.

2. Inference is a process itself in two steps:

– The computation of a rule fulfilled by the intersection of individual premises, applying the fuzzy operator AND.

– Often, more rules drive to the same conclusion. To yield the conclusion (i. e., the membership function value of a certain attribute of output linguistic vari­able) the individual confidence levels are joined by applying the fuzzy opera­tor OR.

3. Deffuzification is a decoding operation of the information contained in the output fuzzy sets resulted from the inference process, with the purpose of providing an output crisp value. There are more methods for deffuzification (Zimmermann, 1996); we apply the Center Of Gravity (COG) method, one of the most popular:

X cf myt (x)dx

І

X myt (x)dx

І

In the Eq. (7.18), ci is the center of the membership function (generally, where it reaches its peak), the integral fmyi (x)dx represents the area under the membership function myi(x) corresponding to the attribute i of the output linguistic variable y.

Updated: August 4, 2015 — 12:13 am