genetic fuzzy controllers for uncertain systems yonggon lee and stanislaw h. Żak supported by...
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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
Flowchart of a typical GA
Initial population
Stop ?
START
Fitness evaluation
Generate new population
END
Genetic Operators
YES
NO
Encoding
Fitness evaluation
PlantFLCReference
Signal
Error
Genetic Operations
Genetic Algorithm
+
-
Flowchart of a typical GA
Initial population
Stop ?
START
Fitness evaluation
Generate new population
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
Case 1: GA tunes fuzzy rules only
FLC using heuristically obtained fuzzy rule base
Case 1: GA tunes fuzzy rules only
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
Case 1: GA tunes fuzzy rules only
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
Case 2: GA tunes FMFs only
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
Case 2: GA tunes FMFs only
The best FMFs generated by the GA after 50th generation
Case 2: GA tunes FMFs only
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
Case 3: GA tunes fuzzy rules and FMFs
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
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)
• 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
Case 3: GA tunes fuzzy rules and FMFs
Resulting fuzzy rule base by the GA after 100th generation
Case 3: GA tunes fuzzy rules and FMFs
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.
Thank you