1. 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, Jan. 2004
© Carlsberg

2. Outline 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

3. Motivating example 1 d y u PI
controller n

4. Ziegler-Nichols PI settings (no noise, n=0)

5. Measurement noise
0.4
0.2
n =
- 0.2
- 0.4 10 20 30
time

6. Ziegler-Nichols PI settings (with noise)

8. ”Smooth” PI-settings (with noise)

9. Conclusion so far Most tuning methods (including Ziegler Nichols):
Fastest possible response subject to achieving acceptable stability margins
(maximum controller gain Kc)
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 ?

10. Closed-loop disturbance rejection
ymax
-ymax

11. Requirement:

12. Minimum controller gain at low frequencies:
where
Alternatively,

13. Kc u

14. Common default factory setting Kc=1 is reasonable !
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 !

15. 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 g c d y u

16. 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
*S. Skogestad (2003), Simple analytic rules for model reduction and PID controller tuning, J. Process Control, 13,

17. Back to example Does not quite reach 1 because d is
step disturbance (not not sinusoid)

18. Discussion
Smooth control: Averaging level control h q LC

19. Discussion Smmoth control: Averaging level control h q Controllability

20. Discussion Smooth control: Averaging level control h q Controllability
Generalization to multivariable systems
Closed-loop disturbance gain for decentralized control
Correct for interactions h q LC

21. Conclusion Conventional tuning rules (e.g. ZN): Many practical cases:
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