Control limitations for unstable plants

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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

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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?
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Outline

• Previous work RHP-pole and RHP-zero
• Introductory example Control of G1/(s-10) with
  P-controller
• Minimum input usage in terms of H2 and H-infinity
  norms
• Inverse response in input
• Examples
• Conclusion

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Feedback control system
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Introductory example
Note inverse response for input (u)
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Introductory example.
RHP- Pole
RHP- zero
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Minimum input energy for Kc20(with closed-loop
pole move to mirror image)
Introductory example.
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Fast response possible with large Kc (and large
u)
Introductory example.
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Inverse response for bicycle caused by underlying
instability
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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!

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Example SISO
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Proof (SISO case)
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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

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2. Performance limitation for stabilized plant
Unstable plant G
Primary output
Secondary measurement (for stabilization)
P
Stabilized plant
y1
r2
G1 G2
u
K2
y2
Question Does original instability (in G2)
impose limitations on the use of
r2 to control y1 (for the
stabilized plant P)
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Special (and common) case Control objective at
the input y1 u
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Challenge for potential World Championship in
bicycle tilting (y1 u)
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Application Anti-slug control
Two-phase flow (liquid and vapor)
Slug (liquid) buildup
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Anti slug-control - control structure
Undesired slug flow (limit cycle) unless feedback
control is used to stabilize a steady flow
regime (desired, but open-loop unstable)
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Anti slug control experimental data
(Statoil/SINTEF)
Pressure (y2)
Controller ON
OFF
INPUT u
Density
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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/