Probably© the smoothest PID tuning rules in the world: Lower limit on controller gain for acceptable disturbance rejection Sigurd Skogestad Department.

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Presentation transcript:

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 © Carlsberg

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

Motivating example 1 d y u PI controller n

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

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

Ziegler-Nichols PI settings (with noise)

”Smooth” PI-settings (with noise)

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 ?

Closed-loop disturbance rejection ymax -ymax

Requirement:

Minimum controller gain at low frequencies: where Alternatively,

Kc u

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 !

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

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, 291-309

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

Discussion Smooth control: Averaging level control h q LC

Discussion Smmoth control: Averaging level control h q Controllability

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

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