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Cascade control of unstable systems with application to stabilization of slug flow

Espen Storkaas and Sigurd SkogestadDep. of Chemical Engineering

Norwegian University of Science and Technology

Presented at AdChem’03 Hong Kong

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Outline

• Unstable systems

• Properties of stabilizing control loop

• Effect of stabilized poles on higher level control loops

• Application to Anti Slug Control

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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

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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

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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

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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

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Example – Anti Slug Control

Experiments performed by the Multiphase Laboratory, NTNU

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System characteristics

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Measurement selection

• Unstable poles at p=0.0008±0.007i• Zeros:

P1 P2 ρT Q

-0.0034 0.0142 -0.0004 -4.1173

3.2489 0.0048 -0.0042

-0.0004

• Using y2 = Q as secondary control variable

K2r2

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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

K2r2K1

r1

P2z

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Case 1 : Risertop pressure as primary variable

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

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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

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Acknowledgements

• Norwegian Research Council for finacial support

• ABB and Statoil for supervision and collaboration