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11/16/2015
1
FUZZY CONTROL
Main bibliography
� J.M.C. Sousa and U. Kaymak. Fuzzy Decision Making
in Modeling and Control. World Scientific Series in
Robotics and Intelligent Systems, vol. 27, Dec. 2002.
� Fakhreddine O. Karray and Clarence De Silva. Soft
Computing and Intelligent Systems Design. Addison
Wesley, 2004.
� Michael Negnevitsky. Artificial Intelligence: A Guide to
Intelligent Systems. Addison-Wesley, Pearson
Education, 2002.
391
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Fuzzy control
� Controller designed by using If-Then rules instead of
mathematical formulas (knowledge-based control).
� Early motivation: mimic experienced operators.
� Fuzzy reasoning: interpolation between discrete
outputs.
� Currently: also controllers designed on the basis of a
fuzzy model (model-based fuzzy control).
� A fuzzy controller represents a nonlinear mapping
(but completely deterministic!).
392
Fuzzy control: history
1965 First publication on fuzzy sets (Zadeh)
1974 Fuzzy control applied to a laboratory system
(Mamdani)
1982 First industrial application of fuzzy control (to a
cement kiln)
1985 Sendai subway train control, consumer products
(Japan)
200? Large number of (micro)controllers: fuel injection,
cameras, washing machines, etc.
393
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Fuzzy control schemes
� PID fuzzy control (nonlinear)
� Fuzzy supervisory control
� Fuzzy model-based control
394
Low-level fuzzy control
395
Fuzzy LogicController Process
yr
d
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Fuzzy control: basic elements
396
1 1 2 2: is is is
is
k k k k
n nk
R x A x A x A
u B
If and and and
then
…
Fuzzy PD controller rule table
397
∆e
e(k) NB NS ZE PS PB
NB NB NB NS NS ZE
NS NB NS NS ZE PS
ZE NS NS ZE PS PS
PS NS ZE PS PS PB
PB ZE PS PS PB PB
8: is NS is ZE is NSR e e u∆If and then
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Mapping of FLC
� Fuzzy rules associate fuzzy regions in the antecedent
space with fuzzy regions in the consequent space.
398
Fuzzy inference mechanism
1. Establish fuzzy relation
2. Inference: sup-t composition
3. Defuzzification
399
1( , ) ( ) ( ), 1, ,k
nk k
j j uj
u x u k Kµ µ µ=
= ∧ =∧x …
R
1( , ) ( , )
RR
== ∨
Kk
ku uµ x x
[ ]( ) sup ( ) ( , )R′ ′
∈
= ⊗B AX
y uµ µ µx
x x
( )
( )
Bcog u U
Bu U
u u duu
u du
µ
µ
′∈
′∈
=∫∫
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Membership functions
400
Max-min inference
401
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7
Max-product inference
402
Comparison of inferences
403
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Design of a fuzzy controller
1. Determine input(s) and output(s).
2. Determine membership functions.
3. Define the rule base, based on e.g. expert knowledge.
4. Test the controller for typical test signals.
5. Fine-tune the controller (the designer can go back to
step 1 if necessary).
404
Types of PID fuzzy controllers
� PD fuzzy controller
� PI fuzzy controller
� PID fuzzy controller
405
: is is is ∆∆i i i
e uR e e A u AIf and then
: is is is ∆ ∆∆ ∆i i i
e uR e e A u AIf and then
2
2: is is is is ∆ ∆∆∆ ∆ ∆i i i i
e ueR e e A e A u AIf and and then
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Example: demo sltank
1. If level is OK then no_change valve
2. If level is low then open fast valve
3. If level is high then close fast valve
4. If level is OK and rate is positive then close slow valve
5. If level is OK and rate is negative then open slow valve
406
Rh
PID control
407
0 50 100 150 200 250 3000.4
0.6
0.8
1
1.2
1.4
1.6
time
level
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PD fuzzy control
408
0 50 100 150 200 250 3000.4
0.6
0.8
1
1.2
1.4
1.6
1.8
time
level
Example: proportional control
� Controller's input-output mapping
409
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Proportional control: rules
1. If error is Negative Big then control input is Negative Big.
2. If error is Zero then control input is Zero.
3. If error is Positive Big then control input is Positive Big.
410
Example: friction compensation
� DC motor with static friction.
� Fuzzy rules to represent “normal” proportional
control.
� Additional rules to prevent undesirable states.
Model of the DC motor
411
1
out_1J.s+b
1
Load
1
s
Dead Zone
L.s+R
K(s)
Armature
K
1
in_1
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Proportional controller
412
MuxMotor
Control u
Angle
P
Proportional controller output
413
0 5 10 15 20 25 30
-0.1
-0.05
0
0.05
0.1
0.15
time [s]
shaft
an
gle
[ra
d]
0 5 10 15 20 25 30-1.5
-1
-0.5
0
0.5
1
1.5
time [s]
co
ntr
ol
inp
ut
[V]
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Fuzzy control rule base
� Proportional Rules:
1. If error is Negative Big then control input is Negative Big.
2. If error is Zero then control input is Zero.
3. If error is Positive Big then control input is Positive Big.
� Additional rules:
4. If error is Negative Small then control input is not
Negative Small.
5. If error is Positive Small then control input is not Positive
Small.
414
Input-Output mapping
415
-0.5
0.5
1
-0.15 -0.1 -0.05 0 0.05 0.1 0.15 e
u
local nonlinearity
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Fuzzy control results
416
0 5 10 15 20 25 30
-0.1
-0.05
0
0.05
0.1
0.15
time [s]
shaft
an
gle
[ra
d]
0 5 10 15 20 25 30-1.5
-1
-0.5
0
0.5
1
1.5
time [s]
co
ntr
ol
inp
ut
[V]
Supervisory fuzzy control
� Example: If y is low then reduce Kp and increase Kd.
417
PIDController Process
yr
FuzzySupervisor
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