Fuzzy numbers & Fuzzy Equation.pptx

Fuzzy numbers & Fuzzy Equation Vikas Kaduskar Asst Professor B.V.D.U. College of Engineering

Discussion Points Fuzzy Number. Arithmetic Operations on Fuzzy Interval. Arithmetic Operations on Fuzzy number. Fuzzy Equation. Defuzzification Methods. Fuzzy Logic Controller. Applications-FLC: Washing Machine; Vacuum Cleaner • Extension Principle. • • • • • • •

FUZZY NUMBERS • Among the various types of fuzzy sets, of special significance are fuzzy sets that are defined on the set R of real numbers. Membership functions of these sets, which have the form A:R → [0,1]

• clearly have a quantitative meaning and may, under certain conditions, be viewed as fuzzy numbers or fuzzy intervals. To view them in this way, they should capture our intuitive conceptions of approximate numbers or intervals, such as "numbers that are close- to a given real number" or "numbers that are around a given interval of real numbers."

To qualify as a fuzzy number, a fuzzy set A on R must possess at least the following

three properties: (1) A must be a normal fuzzy set; (ii) αA must be a closed interval for every a α belongs to (0, 1); (iii) the support of A, 0+A, must be bounded.

• Special cases of fuzzy numbers include ordinary real numbers and intervals of real numbers, as illustrated in Fig. 1.1: (a) an ordinary real number 1.3; (b) an ordinary (crisp) closed interval [1.25, L35]; (c) a fuzzy number expressing the proposition "close to 1.3;" and (d) a fuzzy number with a flat region (a fuzzy interval).

ARITHMETIC OPERATIONS ON INTERVALS • Fuzzy arithmetic is based on two properties of fuzzy numbers: 1) each fuzzy set, and thus also each fuzzy number, can fully and uniquely be represented by its α-cuts (2) α -cuts of each fuzzy number are closed intervals of real numbers for all α belongs to [0, 1].

• These properties enable us to define arithmetic operations on fuzzy numbers in terms of arithmetic operations on their α-cuts arithmetic operations on closed intervals). • Let * denote any of the four arithmetic Operations on closed intervals: addition +, subtraction —, multiplication . , and division /. Then,

• is a general property of all arithmetic operations on closed intervals, 'except that [a, b]/[d, e] is not defined when 0 E [d e]. That is, the result of an arithmetic operation on closed intervals is again a closed interval. • The four arithmetic operations on closed intervals are defined as follows:

• The following are a few examples illustrating the interval-valued arithmetic operations

• Arithmetic operations on closed intervals satisfy some useful properties. To overview them, Let A =[a1 a2] B = [b1 ,b2] C = [c1 c2], 0 =[0,0] Using these symbols, the properties are formulated as follows:

ARITHMETIC OPERATIONS ON FUZZY NUMBERS • we present two methods for developing fuzzy arithmetic. One method is based on interval arithmetic, which is overviewed in prev. Sec. The other method employs the extension principle, by which operations on real numbers are extended to operations on fuzzy number. • fuzzy numbers. We assume in this section that fuzzy numbers are represented by continuous membership functions.

• Let A and B denote fuzzy numbers and let * denote any of the four basic arithmetic operations. Then, we define a fuzzy set on R, A * B, by defining its α-cut, α(A * B), as • α(A * B)= αA * αB (1) • A * B can be expressed as •

(2)

• As an example of employing, consider two triangular-shape fuzzy numbers A and B defined as follows:

Using eq. 1 & 2 we can obtain

FUZZY EQUATIONS • One area of fuzzy set theory in which fuzzy numbers and arithmetic operations on fuzzy numbers play a fundamental role are fuzzy equations . These are equations in which coefficients and unknowns are fuzzy numbers, and formulas are constructed by operations of fuzzy arithm etic. Such equations have a great potential applicability.

Equation A + X = B • The difficulty of solving this fuzzy equation is caused by the fact that X = B — A is not the solution. To see this, let us consider closed intervals, A=[a1,a2]; B= [b1;b2] • which may be viewed as special fuzzy numbers. Then, B A = [b1 -a2, b2 — a1]

• Therefore, X = B - A is not a solution of the equation

• two ordinary equations of real numbers, a1 +x1= b1 a2 +x2 =b2 • whose solution is x1 = b1 — a1 and x2 = b 2 — a2. Since X must be an interval, it is required that x1 ≤ x2. That is, the equation has a solution 1ff b 1 — a1 ≤ b2 - a2. the solution is X = [b1-a1, b2 -a2].

LATTICE OF FUZZY NUMBERS • As is well known, the set R of real numbers is linearly ordered. For every pair of real numbers, x and y, either x ≤y or y ≤ x.The pair (R,