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

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?

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

Feedback control system

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

Introductory example…. RHP- Pole RHP- zero

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

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

Inverse response for bicycle caused by underlying instability

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!

“unavoidable”

Example SISO

Proof (SISO case)

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

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)

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

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

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

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)

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

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: http://www.chembio.ntnu.no/users/skoge/