1. Fuzzy Control –Configuration –Design choices –Takagi-Sugeno controller

2. Direct Control Deviations Actions Outputs Ref Controller End-user Inference engine Rule base Plant

3. Building Blocks Fuzzy controller Inference engine Rule base Defuzzi -fication Postpro - cessing Fuzzi- fication Prepro- cessing

4. Nonlinear Input Scaling -505 -100 -50 0 50 100 measured input scaled input

5. If-Then Rule Base 1. If error is Neg and change in error is Neg then output is NB 2. If error is Neg and change in error is Zero then output is NM 3. If error is Neg and change in error is Pos then output is Zero 4. If error is Zero and change in error is Neg then output is NM 5. If error is Zero and change in error is Zero then output is Zero 6. If error is Zero and change in error is Pos then output is PM 7. If error is Pos and change in error is Neg then output is Zero 8. If error is Pos and change in error is Zero then output is PM 9. If error is Pos and change in error is Pos then output is PB

6. Relational Rule Format ErrorChange in errorControl Pos PB PosZeroPM PosNegZero PosPM Zero NegNM NegPosZero NegZeroNM Neg NB

7. Tabular Rule Format Change in error NegZeroPos NegNBNMZero ErrorZeroNMZeroPM PosZeroPMPB

8. Connectives minimum maximum algebraic product probabilistic sum

9. FLS I/O Families -0.500.51 0 1 Input Membership -0.500.51 0 1 Output Membership Neg Zero Pos

10. Examples Of Primary Sets

11. Inference And Terminology AND Aggregation Accumulation Defuzzification Activation  4  5

12. Defuzzification 050100 0 0.5 1 RM BOA COG MOM LM

13. Rule Based Controllers 1.If error is Neg then control is Neg 2.If error is Zero then control is Zero 3.If error is Pos then control is Pos

14. Mamdani Inference

15. FLS Inference

16. Sugeno Inference

17. Singleton Output 1. If error is Pos then control is 10 2. If error is Zero then control is 0 3. If error is Neg then control is -10

18. First Order Output 1. If error is Pos then control is a 2 *error + b 2 2. If error is Neg then control is a 1 *error + b 1

19. Interpolation (Takagi-Sugeno) 050100 0 50 100 150 (a) output 1 2 050100 0 0.5 1 (b) membership

20. Rule Base To Table

21. Look-Up Table Change in error -100-50050100 Error 100040100 200 50-40061121160 0-100-61061100 -50-100-121-61040 -100-200-160-100-400

22. Control Surface -100 0 100 -100 0 100 -200 0 200 E CE u -100-50050100 0 0.2 0.4 0.6 0.8 1 input family membership

23. Linear Controller -100 0 100 -100 0 100 -200 0 200 E CE u -100-50050100 0 0.2 0.4 0.6 0.8 1 input family membership

24. Linear Rule Base

25. Conditions For Linearity Triangular sets, crossing at  = 0.5 Rules: complete  -combination Define  as * Use conclusion singletons, positioned at sum of input peak positions Use sum-accumulation and COGS defuzzification

26. Simplification of 4 rules 1. If error is Neg and change in error is Neg then control is NB 3. If error is Neg and change in error is Pos then control is Zero 7. If error is Pos and change in error is Neg then control is Zero 9. If error is Pos and change in error is Pos then control is PB is

27. Simplification of 9 rules 1. If error is Neg and change in error is Neg then output is NB 2. If error is Neg and change in error is Zero then output is NM 3. If error is Neg and change in error is Pos then output is Zero 4. If error is Zero and change in error is Neg then output is NM 5. If error is Zero and change in error is Zero then output is Zero 6. If error is Zero and change in error is Pos then output is PM 7. If error is Pos and change in error is Neg then output is Zero 8. If error is Pos and change in error is Zero then output is PM 9. If error is Pos and change in error is Pos then output is PB is

28. Summary Of Choices Rule-base related choices: # of inputs and outputs, rules, universes, continuous or discrete, # of membership functions, their overlap and width, singleton conclusions Inference engine choices: Connectives, modifiers, activation operation, aggregation operation, accumulation operation Defuzzification method: COG, COGS, BOA, MOM, LM, RM Pre- and postprocessing: Scaling, quantization, sampling time