1. Cascade control of unstable systems with application to stabilization of slug flow
Espen Storkaas and Sigurd Skogestad
Dep. of Chemical Engineering
Norwegian University of Science and Technology
Presented at AdChem’03
Hong Kong

2. Outline Unstable systems Properties of stabilizing control loop
Effect of stabilized poles on higher level control loops
Application to Anti Slug Control

3. Unstable systems G(s)=C(sI-A)-1B+D, unstable if |λi(A)|>0 for any i
Feedback stabilization of G(s) requires active use of inputs
Lower bandwidth limit
Lower limit on H2- and H∞-norms of KS = K(I+GK)-1
Unstable (RHP) zeros and time delay imposes upper limit on bandwidth
Incompatible bandwidth requirements => stabilizing control impossible

4. Stabilizing control G2 unstable with RHP poles pi stabilized by K2
Internal stability
S2=(1+G2K2)-1 contains unstable poles of G2 as zeros
Process as seen from primary layer: G=G1S2K2

5. Effect of stabilizing control on primary control layer
Case 1: Unstable poles of G2 detectable in y1 – Same unstable poles in G1 and G2
No effect of unstable poles in G2 on primary control layer
Zeros in S2 in G=G1S2K2 cancelled by poles in G1
Bandwidth limitations only from G1

6. Effect of stabilizing control on primary control layer
Case 2 : Unstable poles of G2 not detectable in y1
Unstable poles in G2 affects the primary loop as RHP zeros in G=G1S2K2
Bandwidth limitation due to unstable G2: ωB1<min(pi2)
May be benificial to operate at faster instabilities

7. Example – Anti Slug Control
Experiments performed by the Multiphase Laboratory, NTNU

8. System characteristics

9. Measurement selectionK2 r2
Unstable poles at p=0.0008±0.007i
Zeros: P1 P2 ρT Q
0.0142
3.2489
0.0048
Using y2 = Q as secondary control variable

10. Choise of primary control variable y1
y1 = P2 yields case 1 behaviour (instability detectable in y1)
RHP zeros in G1 itself limits performance
y1 = u2 (input reset, G1=1) yields case 2 behaviour
RHP zeros from poles of G2 limits performance K1 r1 K2 r2 z P2

11. Case 1 : Risertop pressure as primary variable

12. Case 2: Input reset (y1=u) in primary loop

13. Conclusions
Stabilizied poles will affect the higher level control loops as unstable zeros when unstable poles are not detectable in primary control variables
May be beneficial to operate at faster instabilities if fast responses are needed for the primary control objective
Problem illustrated with stabilization of severe slugging

14. Acknowledgements Norwegian Research Council for finacial support
ABB and Statoil for supervision and collaboration