Genetic fuzzy controllers for uncertain systems

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

2. 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

3.  Genetic algorithms (GAs) can be used to design and fine-tune FLC Motivation • Fuzzy logiccontrol---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

4. 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

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

6. Encoding • 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

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

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

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

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

11. 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

12. A model of a ground vehicle* where the lateral forcesFyf(af)andFyr (ar)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

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

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

15. where Case 1: GA tunes fuzzy rules only • Encoding • Chromosome • Selection: roulette wheel method • Crossover: single point crossover with pc= 0.9 • Mutation: random change from {1, 2, 3, 4, 5} with pm= 0.05 • Population size: 30

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

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

18. 0.1 0.4 0.1 0.4 2 4 Case 2: GA tunes FMFs only • Encoding: real number encoding • Chromosome • Genetic operators and other parameters are same as Case 1

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

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

21. Rule i: IF x1 IS AND x2 IS THEN u IS q1 : trapezoidal input fuzzy MFs q1 : output fuzzy singletons d d 1 x c l r where m is the number of fuzzy rules , and the firing strength is Case 3: GA tunes fuzzy rules and FMFs • Fuzzy rule description • Input Fuzzy MFs Each input fuzzy MF is described by four real numbers c, d, l, and r. • Fuzzy output: center average defuzzification

22. 0.6 0.8 Rule 1 IF x1 IS AND x2 IS then u IS 10 3.2 4.3 5.5 1.1 2.1 2.5 0.4 Rule 2 IF x1 IS AND x2 IS then u IS 0 2.5 4.2 3.2 5 0.1 1.6 2.0 • Rules matrix* • Parameter matrix* 1 1 No. of rules 4.3 0.6 3.2 5.5 2.1 0.8 1.1 2.5 10 1 0 1.6 0 1.5 2.0 3.2 0.4 2.5 4.2 5 No. of inputs x1 x2 q Case 3: GA tunes fuzzy rules and FMFs • Chromosome structure* * 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

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

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

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

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

27. Motivation • 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

28. Wheel slip : ABS operation • Tractive force =m(Normal force) wherem=m(l) is road adhesion coefficient • Minimize stopping distance • Maximize tractive force between tire and road surface

29. 1.2 dry asphalt 1 0.8 Road adhesion coefficient () 0.6 0.4 icy asphalt 0.2 0 0 10 20 30 40 50 60 70 80 90 100 Wheel slip () % Wheel slip vs. road adhesion coefficient Wheel lock-up wheel slip = 100 % • Role of ABS : Find and keep the wheel slip value corresponding to maximum road adhesion coefficient

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

31. Assumption: straight line braking with no steering input Modeling of the braking maneuver* A front wheel free body model A vehicle 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

32. Vehicle free body model

33. Surface of accelerationas a function of l f and l r for dry asphalt

34. Wheel free body model

35. Vehicle braking model State variables:

36. 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.

37. Block diagram of the 2D neural optimizer r3 -d3 d3 r3 e -2A + w B y + r2 A y r2 -DD e e -d2 d2 z e + r1 + -M A e r1 + + -d1 d1 yd + e -A - y

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

39. Encoding a fuzzy rule base as a chromosome

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

41. Fuzzy logic controller (FLC) tuning using GA Genetic Algorithm l ref uf + FLC for front _ Vehicle Model Random signal + FLC for rear ur _ l r l f

42. Best chromosome of 146th generation

43. Simulation Results Genetic fuzzy ABS controller simulation block diagram

44. 30 25 20 Front wheel lfand lf ref (%) 15 10 5 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (sec) 30 lr lr ref lf lf ref 25 Rear wheel 20 lrand lr ref (%) 15 10 5 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (sec) Dry asphalt Reference wheel slips and actual wheel slips

45. 20 15 Position Position (m), Speed (m/s) Position(m), Speed(m/s) Vehicle speed Front wheel speed 10 Rear wheel speed 5 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (sec) 5000 4000 3000 Brake torque (Nm) Brake torque (Nm) Front Rear 2000 1000 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (sec) Position, speed and brake torque

46. 20m 100 80 60 Position (m) Position (m) 40 Proposed ABS Fixed-slip ABS 20 Wheel lock-up 0 0 2 4 6 8 10 12 14 Time (sec) Changing surface The surface is changing from dry asphalt to icy asphalt at 10m Wheel lock-up 91m 13.2s 45mph Fixed slip-ABS 42m 7.4s Proposed ABS 31m 5.8s Panic braking Icy asphalt Dry asphalt

47. 100 80 60 Wheel slip (%) 40 Wheel lock-up 20 0 0 2 4 6 8 10 12 Time (sec) 100 80 Wheel slip (%) 60 40 Fixed-slip ABS 20 0 0 2 4 6 8 10 12 Time (sec) 100 80 60 Wheel slip (%) 40 Proposed ABS 20 0 0 2 4 6 8 10 12 Time (sec) Wheel slips

48. 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.

49.  • 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. Future work • GA-based methods are not suitable for on-line application.

50. Thank you zak@purdue.edu