2. Genetic fuzzy controllers for uncertain systems Yonggon Lee and Stanislaw H. Żak Supported by National Science Foundation under grant ECS-9819310

3. Outline Motivation Genetic algorithm & fuzzy logic controller design Simulation experiment  Step-lane-change maneuver of a ground vehicle  Anti-lock brake system (ABS) control Summary and future research

4. Motivation Fuzzy logic control---a model-free, rule-based, approach that allows to incorporate linguistic description in the controller design of uncertain systems The fine-tuning of a fuzzy logic controller (FLC) is a tedious trial-and-error process A linguistic description, that is, rules, may be unreliable or incomplete Genetic algorithms (GAs) can be used to design and fine-tune FLC 

5. Genetic Algorithm (GA) GAs are derivative-free population based optimization methods GAs operate on strings called chromosomes that represent candidate solutions A GA performs genetic operations on a population of chromosomes to generate new population

6. Flowchart of a typical GA Initial population Stop ? START Fitness evaluation Generate new population END Genetic Operators YES NO Encoding

7. Representation of solution in the form of chromosome Depending on the available information, GA is used to optimize  Fuzzy rules only  Fuzzy membership functions only  Fuzzy membership functions and fuzzy rules Encoding

8. Flowchart of a typical GA Initial population Stop ? START Fitness evaluation Generate new population END Genetic Operators YES NO Encoding

9. Fitness evaluation Plant FLC Reference Signal Error Genetic Operations Genetic Algorithm + -

10. Flowchart of a typical GA Initial population Stop ? START Fitness evaluation Generate new population END Genetic Operators YES NO Encoding

11. Simulation experiment 1 Genetic fuzzy tracking controllers for step-lane-change maneuver of a ground vehicle

12. A model of a ground vehicle* * A. B. Will and S. H. Zak, “Modeling and control of an automated vehicle,” Vehicle System Dynamics, vol. 27, pp. 131-155, March, 1997

13. A model of a ground vehicle* where the lateral forces F yf  f  and F yr  r  are functions of slip angles * A. B. Will and S. H. Zak, “Modeling and control of an automated vehicle,” Vehicle System Dynamics, vol. 27, pp. 131-155, March, 1997

14. Case 1: GA tunes fuzzy rules only Fuzzy membership functions (FMFs) are known GA finds fuzzy rules

15. Case 1: GA tunes fuzzy rules only FLC using heuristically obtained fuzzy rule base

16. Case 1: GA tunes fuzzy rules only Encoding LNNZPLP 12345 Chromosome 5554355432443224321132111 Selection: roulette wheel method Crossover: single point crossover with p c = 0.9 Mutation: random change from {1, 2, 3, 4, 5} with p m = 0.05 Population size: 30 where

17. Case 1: GA tunes fuzzy rules only Performance of the best FLC generated by the GA after 50 th generation

18. Case 2: GA tunes FMFs only Fuzzy rules are known GA finds fuzzy membership functions

19. Case 2: GA tunes FMFs only Encoding: real number encoding Chromosome Genetic operators and other parameters are same as Case 1 0.10.40.10.424

20. Case 2: GA tunes FMFs only The best FMFs generated by the GA after 50 th generation

21. Case 2: GA tunes FMFs only Performance of the best FLC generated by the GA after 50 th generation

22. Case 3: GA tunes fuzzy rules and FMFs Fuzzy rule description Rule i : IF x 1 IS AND x 2 IS THEN u IS   Input Fuzzy MFs Each input fuzzy MF is described by four real numbers c, d, l, and r. x 1 lr d c d Fuzzy output: center average defuzzification : trapezoidal input fuzzy MFs   : output fuzzy singletons where m is the number of fuzzy rules, and the firing strength is

23. Case 3: GA tunes fuzzy rules and FMFs Chromosome structure* Rule 1 IF x 1 IS AND x 2 IS then u IS 10 0.6 4.33.25.5 0.8 2.11.12.5 Rule 2 IF x 1 IS AND x 2 IS then u IS 5 3.2 2.54.2 0.4 1.60.12.0 0 01 11 No. of inputs No. of rules 3.20.65.52.10.81.12.5104.3 1.502.03.20.42.54.251.6 x1x1 x2x2  Rules matrix* Parameter matrix* * S. J. Kang, C. H. Woo, and K. B. Woo, “Evolutionary design of fuzzy rule base for nonlinear system modeling and control,” IEEE Transactions on Fuzzy Systems, vol. 8, pp. 47-45, Feb, 2000

24. Case 3: GA tunes fuzzy rules and FMFs Population size: 40 Number of generations: 100 Maximum number of rules: 20 Mutation Operator (p m = 0.1) changes the number of fuzzy rules changes the index element of the rules matrix Parameter mutation changes the parameters of MFs Adjust any chromosome so that it is feasible. Post-processing Rule mutation

25. Case 3: GA tunes fuzzy rules and FMFs Resulting fuzzy rule base by the GA after 100 th generation

26. Case 3: GA tunes fuzzy rules and FMFs Performance of the GA-generated FLC

27. Simulation experiment 2 Genetic neural fuzzy control of an anti-lock brake system (ABS)

28. Anti-lock brake system (ABS) minimizes stopping distance by preventing wheel lock-up during braking The performance of ABS is strongly related to the road surface condition Design a controller that identifies the road surface condition to be used for better braking performance Motivation

29. ABS operation Tractive force =  (Normal force) where  is road adhesion coefficient  Minimize stopping distance  Maximize tractive force between tire and road surface Wheel slip :

30. Role of ABS : Find and keep the wheel slip value corresponding to maximum road adhesion coefficient Wheel slip vs. road adhesion coefficient 0102030405060708090100 0 0.2 0.4 0.6 0.8 1 1.2 Wheel slip ( ) Road adhesion coefficient (  ) icy asphalt dry asphalt % Wheel lock-up wheel slip = 100 %

31. 1. Vehicle brake system 2. Non-derivative optimizer for optimal wheel slips 3. Fuzzy logic controller (FLC) tuned using genetic algorithm (GA) FLC Non-derivative optimizer x.. Brake torques Acceleration Front wheel slip Desired front wheel slip Desired rear wheel slip Rear wheel slip Components of the genetic fuzzy ABS controller

32. Modeling of the braking maneuver* Assumption: straight line braking with no steering input A vehicle free body model A front wheel free body model * A.B. Will and S. H. Żak,“Antilock braking system modeling and fuzzy control,” Int. J. of Vehicle Design, Vol. 24, No.1, pp. 1-18, 2000

33. Vehicle free body model

34. Surface of acceleration as a function of  f and  r for dry asphalt

35. Wheel free body model

36. Vehicle braking model State variables:

37. Neural non-derivative optimizer * works for convex function derivative free optimizer: objective function may be non-differentiable robust to disturbances with bounded time derivative modular structure: easily modifiable to new problem with different dimension * M. C. M Teixeira and S. H. Żak, “Analog Neural Nonderivative Optimizers,” IEEE Trans. Neural Networks, vol. 9, no. 4, pp. 629-638, 1998.

38.   B  -M z w A -A   y      y ydyd A   r3r3 -2A  33     y r2r2 r1r1  3 22  2 11  1 r3r3 r2r2 r1r1 Block diagram of the 2D neural optimizer

39. Fuzzy logic controller tuning using GA Input fuzzy sets: triangle membership functions Output fuzzy sets: singletons Product inference and center average defuzzification Fuzzy logic controller

40. Encoding a fuzzy rule base as a chromosome

41. The Genetic Algorithm Selection : roulette wheel method Fitness : where T is the simulation time Crossover : crossover rate 0.9 for input – weighted average for output - one point crossover Mutation : mutation rate 0.02 replace with random value

42. Fuzzy logic controller (FLC) tuning using GA + + FLC for rear FLC for front ufuf urur _ _ ff rr Vehicle Model  ref Random signal Genetic Algorithm

43. Best chromosome of 146th generation

44. Simulation Results Genetic fuzzy ABS controller simulation block diagram

45. Reference wheel slips and actual wheel slips Dry asphalt

46. Position, speed and brake torque

47. The surface is changing from dry asphalt to icy asphalt at 10m Icy asphaltDry asphalt 20m Wheel lock-up 91m 13.2s Fixed slip-ABS 42m 7.4s Proposed ABS 31m 5.8s 45mph Panic braking 02468101214 0 20 40 60 80 100 Time (sec) Position (m) Proposed ABS Fixed-slip ABS Wheel lock-up Position (m) Changing surface

48. Wheel slips

49. Summary Designs of FLCs using GAs are illustrated for the step-lane-change maneuver of a ground vehicle system and for an ABS system The proposed genetic neural fuzzy ABS controller showed excellent performance in the simulations. The proposed controller design method can be utilized in other practical applications.

50. Future work GA-based methods are not suitable for on-line application. Intelligent control design methods  vary neural or fuzzy component on-line to learn the system behavior and to accommodate for the changes in environment  preserve the closed-loop system stability  Development of efficient self-organizing radial basis function network.

51. zak@purdue.edu Thank you