Genetic fuzzy controllers for uncertain systems Yonggon Lee

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

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

Motivation n 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

Genetic Algorithm (GA) n n n 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

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

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

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

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

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

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

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

A model of a ground vehicle* where the lateral forces Fyf (af) and Fyr (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

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

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

Case 1: GA tunes fuzzy rules only Encoding n n n N Z P LP 2 3 4 5 Chromosome 5 n LN 1 n 5 5 4 3 2 4 4 3 2 2 4 3 2 1 1 1 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 where

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

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

Case 2: GA tunes FMFs only n Encoding: real number encoding n Chromosome 0. 1 n 0. 4 0. 1 0. 4 2 4 Genetic operators and other parameters are same as Case 1

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

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

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

Case 3: GA tunes fuzzy rules and FMFs n Chromosome structure* 0. 8 0. 6 Rule 1 IF x 1 IS AND x 2 IS 3. 2 4. 3 5. 5 0 Rule 2 IF x 1 IS AND x 2 IS Rules matrix* 1 1 1 0 No. of inputs No. of rules u IS 10 1. 1 2. 5 0. 4 0. 1 1. 6 2. 0 n then u IS 2. 5 3. 2 4. 2 n 5 Parameter matrix* 4. 3 0. 6 3. 2 5. 5 2. 1 0. 8 1. 1 2. 5 10 1. 6 0 1. 5 2. 0 3. 2 0. 4 2. 5 4. 2 5 x 1 x 2 q * 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

Case 3: GA tunes fuzzy rules and FMFs n n 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.

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

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

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

Motivation n n 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

ABS operation § Minimize stopping distance § Maximize tractive force between tire and road surface n Tractive force = m(Normal force) where = ( ) is road adhesion coefficient Wheel slip :

Wheel slip vs. road adhesion coefficient Road adhesion coefficient ( ) 1. 2 dry asphalt 1 0. 8 0. 6 Wheel lock-up wheel slip = 100 % 0. 4 icy asphalt 0. 2 0 0 10 20 30 40 50 60 Wheel slip ( ) 70 80 90 100 % • Role of ABS : Find and keep the wheel slip value corresponding to maximum road adhesion coefficient

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

Modeling of the braking maneuver* n 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

Vehicle free body model

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

Wheel free body model

Vehicle braking model State variables:

Neural non-derivative optimizer* n n 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.

Block diagram of the 2 D neural optimizer r 3 -d 3 e w y A D z + + r 2 e -d 2 e e yd + -d 1 + + d 1 -A - y y r 2 r 1 A -M + + -2 A B -D r 3 e r 1

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

Encoding a fuzzy rule base as a chromosome

The Genetic Algorithm 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

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

Best chromosome of 146 th generation

Simulation Results Genetic fuzzy ABS controller simulation block diagram

Dry asphalt Reference wheel slips and actual wheel slips Front wheel f and f ref (%) 30 25 20 15 f f ref 10 5 0 0 0. 2 0. 4 0. 6 0. 8 1 1. 2 1. 4 1. 6 1. 8 Time (sec) Rear wheel r and r ref (%) 30 r r ref 25 20 15 10 5 0 0 0. 2 0. 4 0. 6 0. 8 1 Time (sec) 1. 2 1. 4 1. 6 1. 8

Position(m), Speed(m/s) Position (m), Speed (m/s) Position, speed and brake torque 20 15 Position Vehicle speed Front wheel speed Rear wheel speed 10 5 0 0 0. 2 0. 4 0. 6 0. 8 1 Time (sec) 1. 2 1. 4 1. 6 1. 8 Brake torque (Nm) 5000 4000 3000 Front Rear 2000 1000 0 0 0. 2 0. 4 0. 6 0. 8 1 Time (sec) 1. 2 1. 4 1. 6 1. 8

Changing surface The surface is changing from dry asphalt to icy asphalt at 10 m Wheel lock-up 91 m 13. 2 s Fixed slip-ABS 42 m 7. 4 s 45 mph Proposed ABS 31 m 5. 8 s Icy asphalt Panic braking Dry asphalt 20 m Position (m) 100 80 60 40 Proposed ABS Fixed-slip ABS Wheel lock-up 20 0 0 2 4 6 8 Time (sec) 10 12 14

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

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

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

Thank you zak@purdue. edu