Fuzzy PID Control Jan Jantzen [email protected] 2013 1.
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Transcript of Fuzzy PID Control Jan Jantzen [email protected] 2013 1.
2
Summary
• Reduce design choices• Tuning
3
Design Procedure
• Build and tune a conventional PID controller first.• Replace it with an equivalent linear fuzzy controller.• Make the fuzzy controller nonlinear.• Fine-tune the fuzzy controller.
Relevant whenever PID control is possible, or already implemented
4
Single Loop Control
Load Noise
The controller should preferably be able to follow the reference r, reject load changes l and noise disturbances n, but these requirements are in conflict with each other. We would like to transfer PID tuning methods to the fuzzy controller in order to have a tuning method.
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Rule Base With 4 Rules
1. If error is Neg and change in error is Neg then control is NB3. If error is Neg and change in error is Pos then control is Zero7. If error is Pos and change in error is Neg then control is Zero9. If error is Pos and change in error is Pos then control is PB
The four rules can handle many cases, and they are sufficient for a linear controller.
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Textbook PID Controllers
t
di
p t
eTe
TeKu
0 d
dd
1
n
j s
nndsj
inpn T
eeTTe
TeKu
1
11
sn
innpn Te
TeeKu
11
Continuous version
Discrete version
Incremental, discrete version
7
Fuzzy P controller
f
Rule base
uGU
UGE
Ee
GUneGE
GUneGEfnU
)(
)()(
GUGEK p
Gain on error
Gain on control
Provided that the rule base acts like the identity function
By comparison with the P controller equation
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FP Rule Base
1. If E(n) is Pos then u(n) is 100
2. If E(n) is Neg then u(n) is -100
With a proper choice of membership functions the controller will act like a linear P controller
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Fuzzy PD Controller
GE
GCETGUGEK
neGE
GCEneGUGE
GUneGCEneGE
GUneGCEneGEfnU
dp
,
))()((
))()((
)(),()(
eGE
GCE
f
Rule base
E
CE
uGU
U
de/dt
Provided that the rule base acts like a summation
Now we know what the gains do
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FPD Rule Base
1. If E(n) is Neg and CE(n) is Neg then u(n) is -2003. If E(n) is Neg and CE(n) is Pos then u(n) is 07. If E(n) is Pos and CE(n) is Neg then u(n) is 09. If E(n) is Pos and CE(n) is Pos then u(n) is 200
With a proper choice of membership functions the controller will act like a linear PD controller. Four rules are sufficient.
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Fuzzy PD+I Controller
CE
eGE
f
PD rules
GCE
++ GU
E
GIEIE
u Ude/dt
edt
GUTjeGIEneGCEneGEfnUn
js
1
)()(),()(
It is better that the integral action bypasses the rule base. It saves rules.
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Fuzzy Incremental Controller
eGE
GCE
f
Rule base
E
CEGCU 1/s U
CUcu
de/dt
n
jsTGCUneGCEneGEfnU
1
)(),()(
The output is a change to the previous state
This is an integrator. It could be a valve position, for instance.
The increment. It is a change to the sum of all previous signals.
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Fuzzy - PID Gain Relation
Controller Kp 1/Ti Td
FP GE×GU
FInc GCE×GCU GE/GCE
FPD GE×GU GCE/GE
FPD+I GE×GU GIE/GE GCE/GE
It tells what each fuzzy gain does to the proportional gain, the derivative gain, and the integral gain. Conversely, given values for Kp, Ti and Td we can find one or more sets of values for the fuzzy gains. Very important table.
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Tuning
lKK
Knr
KK
KKx
pp
p
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Process gain
If we increase Kp, we suppress load changes.
If we increase Kp, the response will be more sensitive to noise.
If we increase Kp too much, the system might oscillate or even become unstable
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Ziegler-Nichols Tuning
• Increase Kp until oscillation, Kp = Ku
• Read period Tu at this setting
• Use Z-N table for approximate controller gains
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Ziegler-Nichols (freq. method)
Controller Kp Ti Td
P 0.5Ku
PI 0.45Ku Tu/1.2
PID 0.6Ku Tu/2 Tu/8
Given values for Ku and Tu, the table provides the gains in the three controller cases. Easy, but often the result is a poorly damped system.
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Z-N oscillation of 1/(1+s)3
The ultimate gain Ku = 8, and the ultimate period is Tu = 15/4 s
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PID control of 1/(1+s)3 Response to a reference step
Response to a load step
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Fuzzy FPD+I control of 1/(1+s)3
The response is the same as for PID control Trajectory on
the control surface, which is a plane
The membership functions are linear
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Hand-Tuning
1. Set Td = 1/Ti = 0
2. Tune Kp to satisfactory response, ignore any final value offset
3. Increase Kp, adjust Td to dampen overshoot
4. Adjust 1/Ti to remove final value offset
5. Repeat from step 3 until Kp large as possible
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Quick reference to controllers
Controller Advantage Disadvantage
FP Simple Maybe too simple
FPD Less overshoot Noise sensitive, derivative kick
FInc Removes steady state error, smooths control signal Slow
FPD+I All in oneWindup, derivative kick
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Scaling
eGE
GCE
f
Rule base
E
CE
uGU
Uα
α
1/αde/dt
1 GUneGCEneGEGUneGCEneGE
The linear controller is invariant towards scaling. In the nonlinear controller we can use it to avoid saturation in the input universes.
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Summary
1. Design crisp PID
2. Replace it with linear fuzzy
3. Make it nonlinear
4. Fine-tune it
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ADVANCED SECTION
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Nyquist 1/(s+1)3 with PID
-2 0 2-2
-1
0
1
2Kp = 4.8, Ti = 15/8, Td = 15/32
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Tuning Map 1/(s+1)3
-2 0 2-2
0
2000
a)-2 0 2
-2
0
2001
b)-2 0 2
-2
0
2010
c)-2 0 2
-2
0
2011
d)
-2 0 2-2
0
2100
e)-2 0 2
-2
0
2101
f)-2 0 2
-2
0
2110
g)-2 0 2
-2
0
2111
h)