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# 指導 教授 : 曾 慶 耀 學 號 : 10167016 學 生 : 高 嘉 佳 - PowerPoint PPT Presentation

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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### Deriving Analytical Input–Output Relationship ForFuzzy Controllers Using Arbitrary Input Fuzzy SetsAnd Zadeh Fuzzy AND Operator

Outline
• Introduction
• A Class of Mamdani Fuzzy Controllers
• The General Derivation Technique
• General Relationship Between Shape of Input Fuzzy Sets and Shape of ICS’ Boundaries
• Conclusion

input combination

Introduction 1/2
• Compared to conventional control theory, analysis and design of fuzzy controllers are substantially more challenging
• Many fuzzy controllers are constructed via the “knowledge engineering approach”
• Knowing the explicit structure information will enable one to insightfully understand how and why fuzzy control works.
Introduction 2/2
• Deriving Analytical Input–Output Relationship for Fuzzy Controllers Using Arbitrary Input Fuzzy Sets and Zadeh Fuzzy AND Operator
DEVELOPMENT OF A GENERAL TECHNIQUE FOR DERIVING ANALYTICAL INPUT–OUTPUT STRUCTURE OF THE FUZZY CONTROLLERS
• A Class of Mamdani Fuzzy Controllers
• The General Derivation Technique
• General Relationship Between Shape of Input Fuzzy Sets and Shape of ICS’ Boundaries
(a)A Class of Mamdani Fuzzy Controllers

SP(n)=輸出命令訊號

E(n) ∈[L1 , R1]

divided into M-1

R(n) ∈[L2 , R2]

divided into N-1

e(n). r(n):輸入變數

Fig. 1. M example (arbitrary) fuzzy sets of arbitrary types defined for E(n)

B. The General Derivation Technique

Fig. 2. Two example fuzzy sets

TABLE I

RESULT OF ZADEH AND OPERATION IN EACH FUZZY RULE FOR THE OVERALL INPUT SPACE DIVISION SHOWN IN FIG. 4

1B2

2C1

1A2

2D1

TABLE II

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?

• Q2.When fuzzy rules are individually evaluated, can a linear boundary be formed if input fuzzy sets are in shapes other than the trapezoidal ones? If so, what are the conditions for this to happen?
• Q3.What are the conditions under which all ICs’ boundaries are formed by lines when all the fuzzy rules are evaluated simultaneously?

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.

total Φ fuzzy rules and the mth rule is

• Proof:

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

f(a1*X1+b1)=g(a2*X2+b2)

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.

• This completes the sufficiency proof.
• necessary condition

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.

Conclusion
• Some general necessary and sufficient conditions relating the shape of input fuzzy sets to the shape of input space divisions have be established
• Examined the design issue of selecting input fuzzy set type by examining the new input–output structures that we derived
• The trapezoidal or triangular fuzzy sets be used as the first choice