1. 1 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 Stuttgart July 2006

2. 2 Abstract Feedback: The simple and best solution Applications to self-optimizing control and stabilization of new operating regimes Sigurd Skogestad, NTNU, Trondheim, Norway Most chemical engineers are (indirectly) trained to be “feedforward thinkers" 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: 1. Implementation of optimal operation by "self-optimizing control". 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. 2. Stabilization of desired operating regimes. 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.

3. 3 Outline About Trondheim I. Why feedback (and not feedforward) ? II. Self-optimizing feedback control: What should we control? III. Stabilizing feedback control: Anti-slug control Conclusion More information: Skogestad on google

4. 4 Trondheim, Norway

5. 5 Trondheim Oslo UK NORWAY DENMARK GERMANY North Sea SWEDEN Arctic circle

6. 6 NTNU, Trondheim

7. 7 Outline About Trondheim I. Why feedback (and not feedforward) ? II. Self-optimizing feedback control: What should we control? III. Stabilizing feedback control: Anti-slug control Conclusion

8. 8 Example G GdGd u d y Plant (uncontrolled system) 1 k=10 time 25

9. 9 G GdGd u d y

10. 10 Model-based control = Feedforward (FF) control G GdGd u d y ”Perfect” feedforward control: u = - G -1 G d d Our case: G=G d → Use u = -d

11. 11 G GdGd u d y Feedforward control: Nominal (perfect model)

12. 12 G GdGd u d y Feedforward: sensitive to gain error

13. 13 G GdGd u d y Feedforward: sensitive to time constant error

14. 14 G GdGd u d y Feedforward: Moderate sensitive to delay (in G or G d )

15. 15 Measurement-based correction = Feedback (FB) control d G GdGd u y C ysys e

16. 16 Feedback PI-control: Nominal case d G GdGd u y C ysys e Input u Output y Feedback generates inverse! Resulting output

17. 17 G GdGd u d y C ysys e Feedback PI control: insensitive to gain error

18. 18 Feedback: insenstive to time constant error G GdGd u d y C ysys e

19. 19 Feedback control: sensitive to time delay G GdGd u d y C ysys e

20. 20 Comment Time delay error in disturbance model (G d ): No effect (!) with feedback (except time shift) Feedforward: Similar effect as time delay error in G

21. 21 Conclusion: Why feedback? (and not feedforward control) Simple: High gain feedback! Counteract unmeasured disturbances Reduce effect of changes / uncertainty (robustness) Change system dynamics (including stabilization) Linearize the behavior No explicit model required MAIN PROBLEM Potential instability (may occur “suddenly”) with time delay/RHP-zero

22. 22 Problem feedback: Effective delay θ Effective delay PI-control = ”original delay” + ”inverse response” + ”half of second time constant” + ”all smaller time constants” Example: Series-cascade of five first-order systems u y y(t) t u

23. 23 PI-control with single measurement

24. 24 Improve control? Some improvement possible with more complex controller –For example, add derivative action (PID-controller) –May reduce θ eff from 3.5 s to 2.5 s –Problem: Sensitive to measurement noise –Does not remove the fundamental limitation (recall waterbed) Add extra measurement and introduce local control –May remove the fundamental waterbed limitation Waterbed limitation does not apply to first-order system –Cascade

25. 25 Add four local P-controllers (inner cascades)

26. 26 Further layers: Process control hierarchy PID RTO MPC

27. 27 Example amazing power of feedback: MIMO Diagonal plant Simulations (and for tuning): Add delay 0.5 in each input Simulations setpoint changes: r 1 =1 at t=0 and r 2 =1 at t=20 Performance: Want |y 1 -r 1 | and |y 2 -r 2 | less than 1 Clear that diagonal pairings are preferred

28. 28 Diagonal pairings Get two independent subsystems:

29. 29 Diagonal pairings.... Simulation with delay included:

30. 30 Off-diagonal pairings (!!?) Pair on two zero elements !! Loops do not work independently! But there is some effect when both loops are closed:

31. 31 Off- diagonal pairings for diagonal plant Example: Want to control temperature in two completely different rooms (which may even be located in different countries). BUT: –Room 1 is controlled using heat input in room 2 (?!) –Room 2 is controlled using heat input in room 1 (?!) TC ?? 1 2

32. 32 Off-diagonal pairings.... –Performance quite poor, but it works because of the “hidden” feedback loop g 12 g 21 k 1 k 2 !! –No failure tolerance Controller design difficult. After some trial and error:

33. 33 Outline About Trondheim Why feedback (and not feedforward) ? II. Self-optimizing feedback control: What should we control? Stabilizing feedback control: Anti-slug control Conclusion

34. 34 Optimal operation (economics) Define scalar cost function J(u 0,d) –u 0 : degrees of freedom –d: disturbances Optimal operation for given d: min u0 J(u 0,x,d) subject to: f(u 0,x,d) = 0 g(u 0,x,d) < 0

35. 35 Estimate d and compute new u opt (d) Probem: Complicated and sensitive to uncertainty ”Obvious” solution: Optimizing control = ”Feedforward”

36. 36 Engineering systems Most (all?) large-scale engineering systems are controlled using hierarchies of quite simple single-loop controllers –Commercial aircraft –Large-scale chemical plant (refinery) 1000’s of loops Simple components: on-off + P-control + PI-control + nonlinear fixes + some feedforward Same in biological systems

37. 37 In Practice: Feedback implementation Issue: What should we control?

38. 38 In Practice: Feedback implementation Issue: What should we control?

39. 39 Further layers: Process control hierarchy y 1 = c ? (economics) PID RTO MPC

40. 40 Implementation of optimal operation Optimal solution is usually at constraints, that is, most of the degrees of freedom are used to satisfy “active constraints”, g(u 0,d) = 0 CONTROL ACTIVE CONSTRAINTS! –Implementation of active constraints is usually simple. WHAT MORE SHOULD WE CONTROL? –We here concentrate on the remaining unconstrained degrees of freedom.

41. 41 Optimal operation Cost J Controlled variable c c opt J opt

42. 42 Optimal operation Cost J Controlled variable c c opt J opt Two problems: 1. Optimum moves because of disturbances d: c opt (d) d LOSS Optimization error

43. 43 Optimal operation Cost J Controlled variable c c opt J opt Two problems: 1. Optimum moves because of disturbances d: c opt (d) 2. Implementation error, c = c opt + n n LOSS

44. 44 Optimal operation Cost J Controlled variable c c opt J opt Two problems: 1. Optimum moves because of disturbances d: c opt (d) 2. Implementation error, c = c opt + n n LOSS WANT FLAT OPTIMUM TO REDUCE EFFECT OF IMPLENTATION ERROR

45. 45 Self-optimizing Control c=c s Self-optimizing Control –Self-optimizing control is when acceptable operation (=acceptable loss) can be achieved using constant set points (c s ) for the controlled variables c (without the need for re- optimizing when disturbances occur). Define loss:

46. 46 Self-optimizing Control – Sprinter Optimal operation of Sprinter (100 m), J=T –Active constraint control: Maximum speed (”no thinking required”)

47. 47 Self-optimizing Control – Marathon Optimal operation of Marathon runner, J=T –Any self-optimizing variable c (to control at constant setpoint)?

48. 48 Self-optimizing Control – Marathon Optimal operation of Marathon runner, J=T –Any self-optimizing variable c (to control at constant setpoint)? c 1 = distance to leader of race c 2 = speed c 3 = heart rate c 4 = level of lactate in muscles

49. 49 Self-optimizing Control – Marathon Optimal operation of Marathon runner, J=T –Any self-optimizing variable c (to control at constant setpoint)? c 1 = distance to leader of race (Problem: Feasibility for d) c 2 = speed (Problem: Feasibility for d) c 3 = heart rate (Problem: Impl. Error n) c 4 = level of lactate in muscles (=Pain! Used in practice)

50. 50 Further examples Central bank. J = welfare. u = interest rate. c=inflation rate (2.5%) Cake baking. J = nice taste, u = heat input. c = Temperature (200C) Business, J = profit. c = ”Key performance indicator (KPI), e.g. –Response time to order –Energy consumption pr. kg or unit –Number of employees –Research spending Optimal values obtained by ”benchmarking” Investment (portofolio management). J = profit. c = Fraction of investment in shares (50%) Biological systems: –”Self-optimizing” controlled variables c have been found by natural selection –Need to do ”reverse engineering” : Find the controlled variables used in nature From this possibly identify what overall objective J the biological system has been attempting to optimize

51. 51 Candidate controlled variables c for self-optimizing control 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

52. 52 Mathematical: Local analysis u cost J u opt c = G u

53. 53 Minimum singular value of scaled gain Maximum gain rule (Skogestad and Postlethwaite, 1996): Look for variables that maximize the scaled gain  (G s ) (minimum singular value of the appropriately scaled steady-state gain matrix G s from u to c)

54. 54 EXAMPLE: Recycle plant (Luyben, Yu, etc.) 1 2 3 4 5 Given feedrate F 0 and column pressure: Dynamic DOFs: N m = 5 Column levels: N 0y = 2 Steady-state DOFs:N 0 = 5 - 2 = 3

55. 55 Recycle plant: Optimal operation mTmT 1 remaining unconstrained degree of freedom

56. 56 Maximum gain rule: Steady-state gain Luyben snow-ball rule: Not promising economically Conventional: Looks good

57. 57 Recycle process: Proposed control structure for case with J = V (minimize energy) Active constraint M r = M rmax Active constraint x B = x Bmin Self-optimizing loop: Adjust L such that L/F is constant

58. 58 Outline About myself Why feedback (and not feedforward) ? Self-optimizing feedback control: What should we control? III. Stabilizing feedback control: Anti-slug control Conclusion

59. 59 Application stabilizing feedback control: Anti-slug control Slug (liquid) buildup Two-phase pipe flow (liquid and vapor)

60. 60 Slug cycle (stable limit cycle) Experiments performed by the Multiphase Laboratory, NTNU

61. 61 Experimental mini-loop

62. 62 p1p1 p2p2 z Experimental mini-loop Valve opening (z) = 100%

63. 63 p1p1 p2p2 z Experimental mini-loop Valve opening (z) = 25%

64. 64 p1p1 p2p2 z Experimental mini-loop Valve opening (z) = 15%

65. 65 p1p1 p2p2 z Experimental mini-loop: Bifurcation diagram Valve opening z % No slug Slugging

66. 66 Avoid slugging? Design changes Feedforward control? Feedback control?

67. 67 p1p1 p2p2 z Avoid slugging: 1. Close valve (but increases pressure) Valve opening z % No slugging when valve is closed Design change

68. 68 Avoid slugging: 2. Other design changes to avoid slugging p1p1 p2p2 z Design change

69. 69 Minimize effect of slugging: 3. Build large slug-catcher Most common strategy in practice p1p1 p2p2 z Design change

70. 70 Avoid slugging: 4. Feedback control? Valve opening z % Predicted smooth flow: Desirable but open-loop unstable Comparison with simple 3-state model:

71. 71 Avoid slugging: 4. ”Active” feedback control PT PC ref Simple PI-controller p1p1 z

72. 72 Anti slug control: Mini-loop experiments Controller ONController OFF p 1 [bar] z [%]

73. 73 Anti slug control: Full-scale offshore experiments at Hod-Vallhall field (Havre,1999)

74. 74 Analysis: Poles and zeros y z P 1 [Bar]DP[Bar]ρ T [kg/m 3 ]F Q [m 3 /s]F W [kg/s] 0.175 -0.00343.2473 0.0142 -0.0004 0.0048 -4.5722 -0.0032 -0.0004 -7.6315 -0.0004 0 0.25 -0.00343.4828 0.0131 -0.0004 0.0048 -4.6276 -0.0032 -0.0004 -7.7528 -0.0004 0 Operation points: Zeros: z P1P1 DPPoles 0.175 70.051.94 -6.11 0.0008±0.0067i 0.25 690.96 -6.21 0.0027±0.0092i P1P1 ρTρT DP FT Topside Topside measurements: Ooops.... RHP-zeros or zeros close to origin

75. 75 Stabilization with topside measurements: Avoid RHP-zeros by using 2 measurements Model based control (LQG) with 2 top measurements: DP and density ρ T

76. 76 Summary anti slug control Stabilization of smooth flow regime = $$$$! Stabilization using downhole pressure simple Stabilization using topside measurements possible Control can make a difference! Thanks to: Håkon Olsen, Espen Storkaas, Heidi Sivertsen and Ingvald Bårdsen

77. 77 Conclusions Feedback is an extremely powerful tool Complex systems can be controlled by hierarchies (cascades) of single- input-single-output (SISO) control loops Control the right variables (primary outputs) to achieve ”self- optimizing control” Feedback can make new things possible (anti-slug)