Probably© the smoothest PID
tuning rules in the world:
Lower limit on controller gain for acceptable
disturbance rejection
Sigurd Skogestad
Department of Chemical Engineering
Norwegian University of Science and Technology (NTNU)
Trondheim, Norway
Adchem`03, Hong Kong, 12-14 Jan. 2004
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©
Carlsberg
Outline
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Motivating example (Ziegler Nichols PI-settings)
Minimum requirements for closed-loop disturbance rejection
Derivation of ”smooth” PI-settings
Issues
– Standard factory settings
– Averaging level control
– Controllability
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Motivating example
d
PI
1
y
u
controller
n
3
Ziegler-Nichols PI settings (no noise, n=0)
4
Measurement noise
0.4
0.2
n=
0
- 0.2
- 0.4
0
5
10
20
time
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Ziegler-Nichols PI settings (with noise)
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”Smooth” PI-settings (with noise)
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Conclusion so far
• Most tuning methods (including Ziegler Nichols):
Fastest possible response subject to achieving acceptable stability margins
(maximum controller gain Kc)
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•
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Motivating example: Ziegler-Nichols settings unnecessary aggressive
Main problem: The controller gain Kc is too large
BUT: We need control for disturbance rejection
QUESTION: What is the minimum required controller gain Kc ?
Closed-loop disturbance rejection
d0
-d0
ymax
-ymax
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Requirement:
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Minimum controller gain at low frequencies:
where
Alternatively,
where
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Kc
13
u
Minimum controller gain:
Industrial practice: Variables (instrument ranges) often scaled such that
Minimum controller gain is then
Minimum gain for smooth control )
Common default factory setting Kc=1 is reasonable !
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Special case: Input (“load”) disturbance (gd=g)
• In this case: |u0| = |d0| (exact)
• Minimum gain for PI- and PID-controller:
• Recall motivating example. Has
• So minimum controller gain for acceptable disturbance rejection is
d
c
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u
g
y
Integral time
• No systematic method for detuning Ziegler-Nichols controller
• BETTER: Start with IMC-based settings where closed-loop time
constant is a tuning parameter.
• For example, Skogestad’s IMC tuning rules (SIMC)* :
CONCLUSION: Obtain τC from Kc and from this obtain τI
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*S. Skogestad (2003), Simple analytic rules for model reduction and PID controller tuning, J. Process Control, 13, 291-309
Back to example
Does not quite reach 1 because d is
step disturbance (not not sinusoid)
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Discussion
• Smooth control: Averaging level control
q
h
LC
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Discussion
• Smmoth control: Averaging level control
q
h
LC
• Controllability
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Discussion
• Smooth control: Averaging level control
q
h
LC
• Controllability
• Generalization to multivariable systems
– Closed-loop disturbance gain for decentralized control
– Correct for interactions
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Conclusion
• Conventional tuning rules (e.g. ZN):
– Give fastest possible control
– subject to achieving good stability margins
• Many practical cases:
– Want smoothest possible control
– subject to achieving acceptable disturbance rejection
– Agrees with default factory settings
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