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Feedback: The simple and best solution. Applications to self-optimizing control and stabilization of new operating regimes Sigurd Skogestad Department of Chemical Engineering Norwegian University of Science and Technology (NTNU) Trondheim
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Applications to self-optimizing control and stabilization of new operating regimes
Sigurd Skogestad Department of Chemical Engineering Norwegian University of Science and Technology (NTNU) Trondheim
WebCAST Feb. 2006
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and they immediately think of “model inversion'' when it comes doing control. Thus, they prefer to rely on models instead of data, although simple feedback solutions in many cases are much simpler and certainly more robust. The seminar starts with a simple comparison of feedback and feedforward control and their sensitivity to uncertainty. Then two nice applications of feedback are considered:
The idea is to turn optimization into a setpoint control problem, and the trick is to find the right variable to control. Applications include process control, pizza baking, marathon running, biology and the central bank of a country.
Here feedback control can lead to completely new and simple solutions. One example would be stabilization of laminar flow at conditions where we normally have turbulent flow. I the seminar a nice application to anti-slug control in multiphase pipeline flow is discussed.
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Trondheim
Oslo UK NORWAY DENMARK GERMANY
North Sea
SWEDEN Arctic circle
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u d y Plant (uncontrolled system)
1 k=10
time
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Gd u d y
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u d y ”Perfect” feedforward control: u = - G-1 Gd d Our case: G=Gd → Use u = -d
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Gd u d y
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Gd u d y
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Gd u d y
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Gd u d y
(in G or Gd)
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d
Gd u y
ys e
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d
Gd u y
ys e
Feedback generates inverse!
Resulting output
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Gd u d y
ys e
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Gd u d y
ys e
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Gd u d y
ys e
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feedback (except time shift)
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Conclusion: Why feedback? (and not feedforward control)
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– u0: degrees of freedom – d: disturbances
minu0 J(u0,x,d)
subject to: f(u0,x,d) = 0 g(u0,x,d) < 0
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Estimate d and compute new uopt(d) Probem: Complicated and sensitive to uncertainty
”Obvious” solution: Optimizing control = ”Feedforward”
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hierarchies of quite simple single-loop controllers
– Commercial aircraft – Large-scale chemical plant (refinery)
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Issue: What should we control?
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Further layers: Process control hierarchy y1 = c ? (economics)
PID RTO MPC
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– Implementation of active constraints is usually simple.
– We here concentrate on the remaining unconstrained degrees of freedom.
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Cost J Controlled variable c c copt
J Jopt
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Cost J Controlled variable c c copt
J Jopt
Two problems:
d
n
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Effect of implementation error
BAD Good Good
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c=cs
– Self-optimizing control is when acceptable
using constant set points (cs) for the controlled variables c (without the need for re-optimizing when disturbances occur).
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– Any self-optimizing variable c (to control at constant setpoint)?
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– Any self-optimizing variable c (to control at constant setpoint)?
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– Any self-optimizing variable c (to control at constant setpoint)?
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– Active constraint control:
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– Response time to order – Energy consumption pr. kg or unit – Number of employees – Research spending Optimal values obtained by ”benchmarking”
investment in shares (50%)
– ”Self-optimizing” controlled variables c have been found by natural selection – Need to do ”reverse engineering” :
been attempting to optimize
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Intuitive
1. The optimal value of c should be insensitive to disturbances (avoid problem 1) 2. Optimum should be flat (avoid problem 2 – implementation error). Equivalently: Value of c should be sensitive to degrees of freedom u. “Want large gain”
Charlie Moore (1980’s): Maximize minimum singular value when selecting temperature locations for distillation
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u cost J uopt c = G u
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Maximum gain rule (Skogestad and Postlethwaite, 1996): Look for variables that maximize the scaled gain (Gs) (minimum singular value of the appropriately scaled steady-state gain matrix Gs from u to c)
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Self-optimizing control: Recycle process
J = V (minimize energy)
Nm = 5
3 economic (steady- state) DOFs 1 2 3 4 5 Given feedrate F0 and column pressure:
Constraints: Mr < Mrmax, xB > xBmin = 0.98
DOF = degree of freedom
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Active constraint Mr = Mrmax Active constraint xB = xBmin
One unconstrained DOF left for optimization: What more should we control?
Remaining DOF:L
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Luyben snow-ball rule: Not promising economically Conventional: Looks good
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Recycle process: Loss with constant setpoint, cs
Large loss with c = F (Luyben rule) Negligible loss with c =L/F
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Recycle process: Proposed control structure
for case with J = V (minimize energy)
Active constraint Mr = Mrmax Active constraint xB = xBmin
Self-optimizing loop: Adjust L such that L/F is constant
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Application stabilizing feedback control:
Slug (liquid) buildup Two-phase pipe flow (liquid and vapor)
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Experiments performed by the Multiphase Laboratory, NTNU
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0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Riser slugging Steady flow Pulsing flow Usg [m/s] Uso [m/s] Flow map with open valve
Steady flow Steady/Pulsing Pulsing flow Pulsing/Slugging Riser slugging
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p1 p2 z
Experimental mini-loop Valve opening (z) = 100%
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p1 p2 z
Experimental mini-loop Valve opening (z) = 25%
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p1 p2 z
Experimental mini-loop Valve opening (z) = 15%
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p1 p2 z
Experimental mini-loop: Bifurcation diagram
Valve opening z %
No slug Slugging
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p1 p2 z
Avoid slugging:
Valve opening z %
No slugging when valve is closed
Design change
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Avoid slugging:
p1 p2 z
Design change
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Minimize effect of slugging:
p1 p2 z
Design change
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Valve opening z %
Predicted smooth flow: Desirable but open-loop unstable Comparison with simple 3-state model:
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Avoid slugging:
PT PC ref
Simple PI-controller p1 z
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Controller ON Controller OFF
p1 [bar] z [%]
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Anti slug control: Full-scale offshore experiments at Hod-Vallhall field (Havre,1999)
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Analysis: Poles and zeros
0.0048 3.4828 0.0131
0.25
0.0048 3.2473 0.0142
0.175 FW [kg/s] FQ [m3/s] ρT [kg/m3] DP[Bar] P1 [Bar] y z
Operation points: Zeros:
0.96 1.94
DP Poles P1 z
0.0027±0.0092i 69
0.25
0.0008±0.0067i 70.05
0.175
P1 ρT DP FT Topside
Topside measurements: Ooops.... RHP-zeros or zeros close to origin
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Stabilization with topside measurements: Avoid “RHP-zeros by using 2 measurements
density ρT
20 40 60 80 100 120 0.1 0.2 0.3 0.4 z [−] time [min] 20 40 60 80 100 120 65 70 75 P1 [Bar] 20 40 60 80 100 120 50 51 52 53 54 P2 [Bar]
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Thanks to: Espen Storkaas + Heidi Sivertsen and Ingvald Bårdsen
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input-single-output (SISO) control loops