Genetic fuzzy controllers for uncertain systems Yonggon Lee and Stanislaw H. Żak Supported by...

Genetic fuzzy controllers for uncertain systems

Yonggon Lee and Stanislaw H. Żak

Supported by National Science Foundation under grant ECS-9819310

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

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

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

Flowchart of a typical GA

Initial population

Stop ?

START

Fitness evaluation

Generate new population

END

Genetic Operators

YES

NO

Encoding

Representation of solution in the form of chromosome Depending on the available information, GA is used

to optimizeFuzzy rules onlyFuzzy membership functions onlyFuzzy membership functions and fuzzy rules

Encoding


Initial population

Stop ?

START

Fitness evaluation


END

Genetic Operators

YES

NO

Encoding

Fitness evaluation

PlantFLCReference

Signal

Error

Genetic Operations

Genetic Algorithm

+

-


Initial population

Stop ?

START

Fitness evaluation


END

Genetic Operators

YES

NO

Encoding

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*

124

213

32

31

)(2

xxvx

FlFlI

x

xxm

FFxvx

x

yryf

yryfx

where the lateral forces Fyf f and Fyr r are functions of slip angles

xf v

xxl 131

xr v

xxl 132

* 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

Fuzzy membership functions (FMFs) are known

GA finds fuzzy rules


FLC using heuristically obtained fuzzy rule base


Encoding LN N Z P LP

1 2 3 4 5

Chromosome5 5 5 4 3 5 5 4 3 2 4 4 3 2 2 4 3 2 1 1 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

50

0

22

21 )()( dttetePI

PI

1Fitness


Performance of the best FLC generated by the GA after 50th generation

Case 2: GA tunes FMFs only

Fuzzy rules are known

GA finds fuzzy membership functions

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.5

1

e

LN N Z P LP

m

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.5

1

e.LN N Z P LP

m/s

-6 -4 -2 0 2 4 60

0.5

1

u

LN:1 N:2 Z:3 P:4 LP:5

degree


Encoding: real number encoding

Chromosome

Genetic operators and other parameters are same as Case 1

0.1 0.4 0.1 0.4 2 4


The best FMFs generated by the GA after 50th generation


Performance of the best FLC generated by the GA after 50th generation

Case 3: GA tunes fuzzy rules and FMFs

Fuzzy rule description

Rule i : IF x1 IS AND x2 IS THEN u IS

ixF

1

ixF

2

Input Fuzzy MFs

Each input fuzzy MF is described by four real numbers c, d, l, and r. x

1

l r

dxF

c

d

Fuzzy output: center average defuzzification

: trapezoidal input fuzzy MFs : output fuzzy singletonsix j

F

m

i i

m

i

iqi

qu1

1

11

21

1 nq FFF where m is the number of fuzzy rules , and the firing strength is


Chromosome structure*

Rule 1 IF x1 IS AND x2 IS then u IS 10

0.6

4.33.2 5.5

0.8

2.11.1 2.5

Rule 2 IF x1 IS AND x2 IS then

u IS 53.22.5 4.2

0.4

1.60.1 2.0

0

01

1 1

No. of inputs

No. of rules3.20.6 5.5 2.1 0.8 1.1 2.5 104.3

1.50 2.0 3.2 0.4 2.5 4.2 51.6

x1 x2

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


Population size: 40 Number of generations: 100 Maximum number of rules: 20 Mutation Operator (pm= 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


Resulting fuzzy rule base by the GA after 100th generation


Performance of the GA-generated FLC

Simulation experiment 2

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

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

ABS operation

Tractive force = (Normal force)

where is road adhesion coefficient

Minimize stopping distance Maximize tractive force between tire and road

surface

Wheel slip : % 100speed vehicle

speed ntialcircumfere tire- speed vehicle

• Role of ABS : Find and keep the wheel slip value

corresponding to maximum road adhesion coefficient

Wheel slip vs. road adhesion coefficient

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

Wheel slip ()

Ro

ad a

dh

esio

n c

oef

fici

ent

()

icy asphalt

dry asphalt

%

Wheel lock-upwheel slip = 100

%

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 frontwheel slip

Desired rear wheel slip

Rear wheel

slip

Components of the genetic fuzzy ABS controller

Modeling of the braking maneuver*

Assumption: straight line braking with no steering input

A vehicle free body modelA 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

)()(

)()(

33

21

mmm

mmgx

rftot

rf

)()(

)()(

33

21

mmm

mmgx

rftot

rf

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

Wheel free body model

))()((2

1

))()((2

1

32

31

xRmgRmuRkJ

TxRmgRmuRJ

wrwrbbrr

r

ewfwfbf

f

))()((2

1

))()((2

1

)()(

)()(

2324

2313

33

212

21

xRmgRmuRkJ

x

TxRmgRmuRJ

x

mmm

mmgx

xx

wrwrbbrr

ewfwfbf

rftot

rf

rf xxxxxx 4321 and , , ,

1 1 ] [ 2

4

2

32121

T

wwTrf x

xR

x

xRxxxxy

Vehicle braking model

State variables:

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.

B

-M

z

w

A

-A

y

y

yddy

A

r3

-2A

3

2x

1x

y2x

1x 1x

2x

r2

r1

3

22

11

r3

r2

r1

321

Block diagram of the 2D neural optimizer

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

Encoding a fuzzy rule base as a chromosome

The Genetic Algorithm

dtteT

02 )(

1

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

Fuzzy logic controller (FLC) tuning using GA

+

+ FLC for rear

FLCfor front

uf

ur

_

_fr

VehicleModel

ref

Random signal

GeneticAlgorithm

Best chromosome of 146th generation

Simulation Results

Genetic fuzzy ABS controller simulation block diagram

Reference wheel slips and actual wheel slips

Dry asphalt

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

5

10

15

20

25

30

Time (sec)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

5

10

15

20

25

30

Time (sec)

fa

nd

f re

f (%

) ra

nd

r re

f (%

)

Frontwheel

Rearwheel

r

r ref

f

f ref

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

5

10

15

20

Time (sec)

Po

sitio

n (

m),

Sp

ee

d (

m/s

)

Position Vehicle speed Front wheel speedRear wheel speed

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

1000

2000

3000

4000

5000

Time (sec)

Bra

ke t

orq

ue

(N

m)

FrontRear

Po

sitio

n(m

), S

pe

ed

(m/s

)B

rake

to

rqu

e (

Nm

)

Position, speed and brake torque

The surface is changing from dry asphalt to icy asphalt at 10m

Icy asphalt Dry asphalt20m

Wheel lock-up 91m 13.2s

Fixed slip-ABS 42m 7.4s

Proposed ABS 31m 5.8s

45mph

Panic braking

0 2 4 6 8 10 12 140

20

40

60

80

100

Time (sec)

Po

sitio

n (m

)

Proposed ABS Fixed-slip ABSWheel lock-up

Pos

ition

(m

)

Changing surface

0 2 4 6 8 10 120

20

40

60

80

100

Time (sec)

Whe

el s

lip (

%)

0 2 4 6 8 10 120

20

40

60

80

100

Time (sec)

Whe

el s

lip (

%)

0 2 4 6 8 10 120

20

40

60

80

100

Time (sec)

Whe

el s

lip (

%)

Wheel lock-up

Fixed-slip ABS

Proposed ABS

Wheel slips

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.

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.

[email protected]

Thank you

Genetic fuzzy controllers for uncertain systems Yonggon Lee and Stanislaw H. Żak Supported by...

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