Deriving Analytical Input–Output Relationship For Fuzzy Controllers Using Arbitrary Input Fuzzy Sets And Zadeh Fuzzy AND Operator. 指導 教授 : 曾 慶 耀 學 號 : 10167016 學 生 : 高 嘉 佳. O utline. Introduction A Class of Mamdani Fuzzy Controllers The General Derivation Technique
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指導教授:曾 慶 耀
學 號 :10167016
學 生 :高 嘉 佳
E(n) ∈[L1 , R1]
divided into M-1
R(n) ∈[L2 , R2]
divided into N-1
Fig. 1. M example (arbitrary) fuzzy sets of arbitrary types defined for E(n)
Zadeh fuzzy logic AND operator is used to realize the AND operations in the rules.
Fig. 2. Two example fuzzy sets
Fig. 3 for each of the four fuzzy rules when they are evaluated individually.
Fig4. superimposing the four individual input space divisions shown in Fig. 3.
RESULT OF ZADEH AND OPERATION IN EACH FUZZY RULE FOR THE OVERALL INPUT SPACE DIVISION SHOWN IN FIG. 4
MATHEMATICAL INPUT–OUTPUT RELATION OF THE FUZZY CONTROLLER FOR THE OVERALL INPUT SPACE DIVISION (FIG. 4)
Q1.Does the above observation from some example controllers actually hold true for any two-input fuzzy controllers with trapezoidal input fuzzy sets?
Theorem 1: For fuzzy controllers using two input variables and Zadeh AND operator, the boundaries for all ICs are lines if and only if all input fuzzy sets are mathematically of the same type.
X1 ,X2 are two input fuzzy sets of the same mathematical type
μPi=f(a1 X1+b1) andμQj=f(a2 X2+b2)
a1* X1+b1= a2 *X2+b2
μPi=f(a1*X1+b1) and μQj=g(a1*X2+b2)
a1* X1- a2 *X2+b1-b2=0
Theorem 2: For fuzzy controllers using three input variables and Zadeh AND operator, the boundaries for all ICs are planes in the three-dimensional rectangular coordinate system if and only if all input fuzzy sets are of the same type of mathematical function.
boundaries formed by all the fuzzy rules are the planes of this nature, all the ICs must be either cuboids or cubes; they cannot be any other shape.
Theorem 3: For fuzzy controllers using n(n>3) input variables and Zadeh AND operator, the boundaries for all ICs are hyperplanesin the -dimensional rectangular system if and only if all input fuzzy sets are of the same type of mathematical function.