1. Control limitations for unstable plants
Sigurd Skogestad
Kjetil Havre
Truls Larsson
Department of Chemical Engineering
Norwegian University of Science and Tecnology (NTNU)
N 7491 Trondheim, Norway
IFAC World Congress, Barcelona, July 2002

2. Unstable pole by itself: Any fundamental limitations?
Previous work: Performance limitation for unstable plant when combined with unstable (RHP) zero
“Presence of unstable (RHP) poles impose a lower limit on the system bandwidth which may be incompatible with the upper limit imposed by RHP-zeros and time delays”
Boyd and Desoer (1985)
Doyle (1986), Doyle, Francis and Tannenbaum (1992)
Middleton (1991)
Kwakernaak (1995)
Seron, Braslavsky and Goodwin (1997)
Åstrøm (1997)
Havre and Skogestad (1998), Skogestad and Postlethwaite (1996)
Unstable pole by itself: Any fundamental limitations?

3. Outline Previous work: RHP-pole and RHP-zero
Introductory example: Control of G=1/(s-10) with P-controller
Minimum input usage in terms of H2 and H-infinity norms
Inverse response in input
Examples
Conclusion

4. Feedback control system

5. Introductory example P PI
Note inverse response for input (u)

6. Introductory example….
RHP-
Pole
RHP-
zero

7. Introductory example….
Minimum input energy for Kc=20 (with closed-loop pole move to mirror image)

8. Fast response possible with large Kc (and large u)
Introductory example….
Fast response possible with large Kc (and large u)

9. Inverse response for bicycle caused by underlying instability

10. Performance limitation for unstable plant
Stabilization: Requires the active use of manipulated inputs
Obervations from simulations:
Input usage: Large inputs may be required
Inverse response for input
Quantify effect on control performance!

11. “unavoidable”

12. Example SISO

13. Proof (SISO case)

14. Conclusion input usage:
Instability requires active use of inputs
Quantified by lower bound on norm of KS
u = KS (r – Gd d – n)
Stabilization may be impossible with constraints on input u

15. 2. Performance limitation for stabilized plant
Unstable plant: G
Primary output G1 G2 u y1 y2
Secondary measurement (for stabilization)
Stabilized plant: P r2 y1 u G1 G2 K2 y2
Question: Does original instability (in G2) impose
limitations on the use of r2 to control y1
(for the stabilized plant P)

16. Answer:
Instability detectable in y1 (i.e. G1 contains the instability): No performance limitation for y1
Instability not detectable in y1 (G1 stable): “New” plant P has unstable zero located at unstable pole Performance limitation for for y1
Special (and common) case: Control objective at the input
y1 = u

17. Challenge for potential World Championship in bicycle tilting (y1 = u)

18. Application: Anti-slug control
Two-phase flow
(liquid and vapor)
Slug (liquid) buildup

19. Anti slug-control - control structure
Slug-catcher
PIC SP PT MV
Input u =
Primary output y1
Measurement y2
Undesired slug flow (limit cycle) unless feedback
control is used to stabilize a steady flow regime
(desired, but open-loop unstable)

20. Anti slug control – experimental data (Statoil/SINTEF)
Pressure (y2)
Controller ON
OFF
INPUT u
Density

21. Conclusion RHP-pole: Performance limitations at the plant input (u)
Minimum input usage
RHP-zero: Performance limitations at the plant output (y)
Minimum output variation
See also the home page of Sigurd Skogestad: