1. Ming-Feng Yeh2-65 4. General Fuzzy Systems A fuzzy system is a static nonlinear mapping between its inputs and outputs (i.e., it is not a dynamic system).

2. Ming-Feng Yeh2-66 Universe of Discourse The “universe of discourse” for u i or y i since it provides the range of values (domain) of U i or Y i that can be quantified with linguistic and fuzzy sets. An “effective” universe of discourse [ ,  ]. Width of the universe of discourse:

3. Ming-Feng Yeh2-67 Linguistic Variables Linguistic expressions are needed for the inputs and outputs and the characteristics of the inputs and outputs. Linguistic variables (constant symbolic descriptions of what are in general time-varying quantities) to describe fuzzy system inputs and outputs. Linguistic variables: is to described the input is to described the output for example, “position error”

4. Ming-Feng Yeh2-68 Linguistic Values Linguistic variables and take on “linguistic values” that are used to describe characteristic of the variables. Let denote the jth linguistic value of the linguistic variable defined over the universe of discourse U i.  For example, “speed”

5. Ming-Feng Yeh2-69 Linguistic Rules The mapping of the inputs to the outputs for a fuzzy system is in part characterized by a set of condition  action rules, or in modus ponens (If- Then) form: If premise Then consequent. Multi-input single-output (MISO) rule: the ith MISO rule: Multi-input multi-output (MIMO) rule:

6. Ming-Feng Yeh2-70 Number of Fuzzy Rules If all the premise terms are used in very rule and a rule is formed for each possible combination of premise elements, then there are rules in the rule-base.

7. Ming-Feng Yeh2-71 Fuzzy Quantification of Rules: Fuzzy Implications Multi-input single-output (MISO) rule: Define: These fuzzy sets quantify the terms, in the premise and the consequent of the given If-Then rule, to make a “fuzzy implication” (which is a fuzzy relation).

8. Ming-Feng Yeh2-72 Fuzzy Implications A fuzzy implication: the fuzzy set quantifies the meaning of the linguistic statement “ “, and quantifies the meaning of “ “. Two general properties of fuzzy logic rule-bases 1. Completeness whether there are conclusions for every possible fuzzy controller input. 2. Consistency whether the conclusions that rules make conflict with other rules’ conclusions.

9. Ming-Feng Yeh2-73 Fuzzification Fuzzification: convert its numeric inputs u i  U i into fuzzy sets. Let denote the set of all possible fuzzy sets that can be defined on U i. Given u i  U i, fuzzification transforms u i to a fuzzy set denoted by defined on the universe of discourse U i. Fuzzification operation: where

10. Ming-Feng Yeh2-74 Singleton Fuzzification When a singleton fuzzification is used, which produces a fuzzy set with a membership function defined by Any fuzzy set with this form for its membership function is called a “singleton”. Other fuzzification methods haves not been used very much because they are complexity.

11. Ming-Feng Yeh2-75 Inference Mechanism Two basic tasks – (1) matching: determining the extent to which each rule is relevant to the current situation as characterized by the inputs u i, i = 1, 2, …, n. (2) inference step: drawing conclusions using the current input u i and the information in the rule- base.

12. Ming-Feng Yeh2-76 Matching Assume that the current inputs u i, i = 1, 2, …, n, and fuzzification produces the fuzzy sets representing the inputs. Step 1: combine inputs with rule premises Step 2: determine which rules are on

13. Ming-Feng Yeh2-77 Rule Certainty We use to represent the certainty that the premise of rule i matches the input information when we use singleton fuzzification. An additional “rule certainty” is multiplied by  i. Such a certainty could represent our a priori confidence in each rule’s applicability and would normally be a number between zero and one.

14. Ming-Feng Yeh2-78 Inference Step Alternative 1: determine implied fuzzy sets For the ith rule, the “implied fuzzy set” with membership function Alternative 2: determine the overall implied fuzzy sets. The “overall implied fuzzy set” with membership function

15. Ming-Feng Yeh2-79 Compositional Rule of Inference The overall implied fuzzy set: where Sup-star compositional rule of inference: “sup” corresponds to the  operation, and the “star” corresponds to the  operation.  Sup (supremum): the least upper bound Zadeh’s compositional rule of inference: maximum is used for  and minimum is used for .

16. Ming-Feng Yeh2-80 Defuzzification: Implied Fuzzy Sets Center of gravity (COG): using the center of are and area of each implied fuzzy set Center-average: using the centers of each of the output membership functions and the maximum certainty of each of the conclusions represented with the implied fuzzy sets

17. Ming-Feng Yeh2-81 Defuzzification: Overall Implied Fuzzy Sets Max criterion A crisp output is chosen as the point on the output universe of discourse Y q for which the overall implied fuzzy set achieves a maximum “arg sup x {  (x)}” returns the value of x that results in the supremum of the function being achieve. Sometimes the supremum can occur at more than one point in Y q. In this case you also need to specify a strategy on how to pick one point for (e.g., choosing the smallest value)

18. Ming-Feng Yeh2-82 Defuzzification: Overall Implied Fuzzy Sets Mean of maximum (MOM) A crisp output is chosen to represent the mean value of all elements whose membership in is a maximum. Define as the supremum of the membership function of over Y q. Define a fuzzy set with the following membership function

19. Ming-Feng Yeh2-83 Defuzzification: Overall Implied Fuzzy Sets Center of area (COA) A crisp output is chosen as the center of area for the membership function of the overall implied fuzzy set. For a continuous output universe of discourse Y q, the center of area output is defined by

20. Ming-Feng Yeh2-84 Functional Fuzzy Systems Standard fuzzy system: Functional fuzzy system: The choice of the function g i (·) depends on the application being considered. The function g i (·) can be linear or nonlinear. Defuzzification:

21. Ming-Feng Yeh2-85 Takagi-Sugeno fuzzy system: If a i,0 =0, then the g i (·) mapping is a linear mapping. If a i,0  0, then the g i (·) mapping is called “affine.” Suppose n = 1, R = 2. Takagi-Sugeno Fuzzy System

22. Ming-Feng Yeh2-86 Singleton O/P Fuzzy System If g i = a i,0, then Takagi-Sugeno fuzzy system is equivalent to a standard fuzzy system that uses center-average defuzzification with singleton output membership function at a i,0. The corresponding fuzzy rule is of the form: where b i is a real number.

23. Ming-Feng Yeh2-87 Consequent Forms Type 1: a crisp value (singleton output) Type 2: a fuzzy number (standard fuzzy system) Type 3: a function (functional fuzzy system)

24. Ming-Feng Yeh2-88 Universal Approximation Property Suppose that we use center-average defuzzification, product for the premise and implication, and Gaussian membership functions. Name this fuzzy system f(u). Then, for any real continuous  (u) defined on a closed and bounded set and an arbitrary  > 0, there exists a fuzzy system f(u) such that  : Psi,  : Epsilon