1. 1 Lecture 4 The Fuzzy Controller design

2. 2 By a fuzzy logic controller (FLC) we mean a control law that is described by a knowledge-based system consisting of IF...THEN rules with vague predicates and a fuzzy logic inference mechanism. The rule base is the main part of the FLC. It is formed by a family of logical rules that describes the relationship between the input e and the output u of the controller. The main difference between conventional control system and fuzzy logic controlled system is not only in the type of logic (Boolean or fuzzy) but in the inspiration. The former attempted to increase the efficiency of conventional control algorithms; the latter were based on the implementation of human understanding and human thinking in control algorithms.

3. 3 Fuzzy controller architecture. 4.1 fuzzy logic controller structure The fuzzy controller has four main components:

4. 4 1. A rule-base (a set of If-Then rules), which contains a fuzzy logic quantification of the expert's linguistic description of how to achieve good control. 2. An inference mechanism (also called an "inference engine" or "fuzzy inference" module), which emulates the expert's decision making in interpreting and applying knowledge about how best to control the plant. 3. A fuzzification interface, which converts controller inputs into information that the inference mechanism can easily use to activate and apply rules. 4. A defuzzification interface, which converts the conclusions of the inference mechanism into actual inputs for the process.

5. 5 4.2 Fuzzy control algorithm consider a fuzzy controller with three inputs and a single output only Given a MISO controller with inputs x1, x2, x3 and output y and assuming that the linguistic control rules are of the form then the membership function of the output of the controller is given by: operator -- implies max-min or max-product

6. 6 4.2 Fuzzy control algorithm the degree of fulfillment of the j-th rule is defined by the degree of fulfillment a measure of how closely the inputs to the controller match the control rules. They can be viewed conveniently as weights that are assigned to every rule. max-product implication Mamdani max-min implication max-product implication Mamdani max-min implication

7. 7 4.2 Fuzzy control algorithm any instant k the membership function of the output of the controller is: Mamdani max-min implication max-product implication

8. 8 4.2 Fuzzy control algorithm Steps in the fuzzification algorithm 1 、 determination of the minimum intercepts for each input, i.e., their membership value. 2 、 determination of the degrees of fulfillment of every rule. 3 、 determination of the composite membership function of the output of the controller.

9. 9 4.2 Fuzzy control algorithm Example Graphical interpretation of fuzzification assume that the controller has two inputs and a single output. Assume that the first input to t he controller Input_1 ( x1) is specified by 5 fuzzy sets, while Input_2 ( x2) is specified by 3 fuzzy sets. The linguistic variables are assumed to be VL=Very_Low, LO=LOw, ZO=ZerO, LH=Little_H ig h, MH=Mediu m_ High, and VH=Very _ High. Assume that the following 15 control rules constitute the rule base:

10. 10 4.2 Fuzzy control algorithm rule matrix

11. 11 4.2 Fuzzy control algorithm For simplicity, assume, furthermore that the fuzzy sets of the inputs and outputs are triangular universes of discourse percentages of their maximum permissible values the opening of a servo-valve pressure deviation temperature deviation

12. 12 4.2 Fuzzy control algorithm The first five rules the instantaneous inputs to the controller are - 20% and -50% respectively.

13. 13 4.2 Fuzzy control algorithm membership value

14. 14 4.2 Fuzzy control algorithm

15. 15 4.2 Fuzzy control algorithm Larsen implication,

16. 16 4.2 Fuzzy control algorithm

17. 17 4.2.2 Defuzzification yield a single crisp value that uniquely specifies the desired control action. there is no theoretical basis for deciding Simplicity and speed of computation are invariably the primary requirements in industrial controllers

18. 18 4.2.2 Defuzzification 1. Center of gravity (COG) defuzzification

19. 19 4.2.2 Defuzzification 2. Center-average

20. 20 3. Max criterion (maximum method) one of the variable values at which the fuzzy subset has its maximum truth value is chosen as the crisp value for the output variable. 4.2.2 Defuzzification

21. 21 4.2.2 Defuzzification 4. Mean of maximum

22. 22 4.2.2 Defuzzification 5. Center of area (COA) defuzzification Weighted average

23. 23 4.2.2 Defuzzification ={ 0.0/0+0.0/10+0.33/20+0.67/30+1.0/40 +1.0/50+0.75/60+0.5/70+0.25/80+0.0/90+0.0/100} Examples ( water temperature middle) COA =(0·0.0 ＋ 10·0.0 ＋ 20·0.33 ＋ 30·0.67 ＋ 40·1.0 ＋ 50·1.0 ＋ 60·0.75 ＋ 70·0.5 ＋ 80·0.25 ＋ 90·0.0 ＋ 100·0.0) /(0.0 ＋ 0.0 ＋ 0.33 ＋ 0.67 ＋ 1.0 ＋ 1.0 ＋ 0.75 ＋ 0.5 ＋ 0.25 ＋ 0.0 ＋ 0.0) =48.2 maximum(40 +50)/2=45 Mean of maximum

24. 24 4.4 Design Considerations Shape of the fuzzy sets basic criteria in selecting the shapes of membership functions  Experience in similar controllers  computational ease In practice triangular and trapezoidal functions are generally used there is no theory which can guide the designer on the best shape to use for a specific application.

25. 25 4.4 Design Considerations Coarseness of the fuzzy sets The number of fuzzy sets that are required to specify a variable is termed the coarseness of the controller and determines the accuracy of the controller. fuzzy sets for coarse-fine control

26. 26 4.4 Design Considerations Completeness of the fuzzy sets The fuzzy control algorithm must lead to a unique control action for any set of inputs. This property is termed completeness and depends on the contents of the knowledge-base as well as the number and shape of the fuzzy sets used to describe the inputs and outputs of the controller. overlap