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 University of Alberta, Edmonton, 29 April 2004

2. 2 Outline About myself I. Why feedback (and not feedforward) ? II. Self-optimizing feedback control: What should we control? III. Stabilizing feedback control: Anti-slug control Conclusion

3. 3 Trondheim, Norway

4. 4 Trondheim Oslo UK NORWAY DENMARK GERMANY North Sea SWEDEN Arctic circle

5. 5 NTNU, Trondheim

6. 6 Sigurd Skogestad Born in 1955 1978: Siv.ing. Degree (MS) in Chemical Engineering from NTNU (NTH) 1980-83: Process modeling group at the Norsk Hydro Research Center in Porsgrunn 1983-87: Ph.D. student in Chemical Engineering at Caltech, Pasadena, USA. Thesis on “Robust distillation control”. Supervisor: Manfred Morari 1987 - : Professor in Chemical Engineering at NTNU Since 1994: Head of process systems engineering center in Trondheim (PROST) Since 1999: Head of Department of Chemical Engineering 1996: Book “Multivariable feedback control” (Wiley) 2000,2003: Book “Prosessteknikk” (Norwegian) Group of about 10 Ph.D. students in the process control area

7. 7 Research: Develop simple yet rigorous methods to solve problems of engineering significance. Use of feedback as a tool to 1. reduce uncertainty (including robust control), 2.change the system dynamics (including stabilization; anti-slug control), 3.generally make the system more well-behaved (including self-optimizing control). limitations on performance in linear systems (“controllability”), control structure design and plantwide control, interactions between process design and control, distillation column design, control and dynamics. Natural gas processes

8. 8 Outline About myself I. Why feedback (and not feedforward) ? II. Self-optimizing feedback control: What should we control? III. Stabilizing feedback control: Anti-slug control Conclusion

9. 9 Example G GdGd u d y Plant (uncontrolled system) 1 k=10 time 25

10. 10 G GdGd u d y

11. 11 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

12. 12 G GdGd u d y Feedforward control: Nominal (perfect model)

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

14. 14 G GdGd u d y Feedforward: sensitive to time constant error

15. 15 G GdGd u d y Feedforward: Moderate sensitive to delay (in G or G d )

16. 16 Measurement-based correction = Feedback (FB) control d G GdGd u y C ysys e

17. 17 Feedback PI-control: Nominal case d G GdGd u y C ysys e Input u Output y Feedback generates inverse! Resulting output

18. 18 G GdGd u d y C ysys e Feedback PI control: insensitive to gain error

19. 19 Feedback: insenstive to time constant error G GdGd u d y C ysys e

20. 20 Feedback control: sensitive to time delay G GdGd u d y C ysys e

21. 21 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

22. 22 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

23. 23 Outline About myself Why feedback (and not feedforward) ? II. Self-optimizing feedback control: What should we control? Stabilizing feedback control: Anti-slug control Conclusion

24. 24 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

25. 25 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 u.

26. 26 Optimal operation Cost J Independent variable u (remaining unconstrained) u opt J opt

27. 27 Implementation: How do we deal with uncertainty? 1. Disturbances d 2. Implementation error n u s = u opt (d 0 ) – nominal optimization n u = u s + n d Cost J  J opt (d)

28. 28 Problem no. 1: Disturbance d Cost J Independent variable u u opt (d 0 ) J opt d0d0d0d0 d ≠ d 0 Loss with constant value for u

29. 29 Problem no. 2: Implementation error n Cost J Independent variable u u s =u opt (d 0 ) J opt d0d0d0d0 Loss due to implementation error for u u = u s + n

30. 30 Estimate d and compute new u opt (d) Probem: Complicated and sensitive to uncertainty ”Obvious” solution: Optimizing control = ”Feedforward”

31. 31 Alternative: Feedback implementation Issue: What should we control?

32. 32 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

33. 33 Process control hierarchy y 1 = c ? (economics) PID RTO MPC

34. 34 Self-optimizing Control –Self-optimizing control is when acceptable loss can be achieved using constant set points (c s ) for the controlled variables c (without re- optimizing when disturbances occur). Define loss:

35. 35 Constant setpoint policy: Effect of disturbances (“problem 1”)

36. 36 Effect of implementation error (“problem 2”) BADGood

37. 37 Self-optimizing Control – Marathon Optimal operation of Marathon runner, J=T –Any self-optimizing variable c (to control at constant setpoint)?

38. 38 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

39. 39 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

40. 40 Good candidate controlled variables c (for self-optimizing control) Requirements: The optimal value of c should be insensitive to disturbances (avoid problem 1) c should be easy to measure and control (rest: avoid problem 2) The value of c should be sensitive to changes in the degrees of freedom (Equivalently, J as a function of c should be flat) For cases with more than one unconstrained degrees of freedom, the selected controlled variables should be independent. Singular value rule (Skogestad and Postlethwaite, 1996): Look for variables that maximize the minimum singular value of the appropriately scaled steady-state gain matrix G from u to c

41. 41 Self-optimizing control: Recycle process J = V (minimize energy) N m = 5 3 economic (steady- state) DOFs 1 2 3 4 5 Given feedrate F 0 and column pressure: Constraints: Mr < Mrmax, xB > xBmin = 0.98 DOF = degree of freedom

42. 42 Recycle process: Control active constraints Active constraint M r = M rmax Active constraint x B = x Bmin One unconstrained DOF left for optimization: What more should we control? Remaining DOF:L

43. 43 Recycle process: Loss with constant setpoint, c s Large loss with c = F (Luyben rule) Negligible loss with c =L/F or c = temperature

44. 44 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

45. 45 Outline About myself Why feedback (and not feedforward) ? Self-optimizing feedback control: What should we control? III. Stabilizing feedback control: Anti-slug control Conclusion

46. 46 Application stabilizing feedback control: Anti-slug control Slug (liquid) buildup Two-phase pipe flow (liquid and vapor)

47. 47 Slug cycle (stable limit cycle) Experiments performed by the Multiphase Laboratory, NTNU

48. 48

49. 49 Experimental mini-loop

50. 50 p1p1 p2p2 z Experimental mini-loop Valve opening (z) = 100%

51. 51 p1p1 p2p2 z Experimental mini-loop Valve opening (z) = 25%

52. 52 p1p1 p2p2 z Experimental mini-loop Valve opening (z) = 15%

53. 53 p1p1 p2p2 z Experimental mini-loop: Bifurcation diagram Valve opening z % No slug Slugging

54. 54 Avoid slugging? Design changes Feedforward control? Feedback control?

55. 55 p1p1 p2p2 z Avoid slugging: 1. Close valve (but increases pressure) Valve opening z % No slugging when valve is closed Design change

56. 56 Avoid slugging: 2. Other design changes to avoid slugging p1p1 p2p2 z Design change

57. 57 Minimize effect of slugging: 3. Build large slug-catcher Most common strategy in practice p1p1 p2p2 z Design change

58. 58 Avoid slugging: 4. Feedback control? Valve opening z % Predicted smooth flow: Desirable but open-loop unstable Comparison with simple 3-state model:

59. 59 Avoid slugging: 4. ”Active” feedback control PT PC ref Simple PI-controller p1p1 z

60. 60 Anti slug control: Mini-loop experiments Controller ONController OFF p 1 [bar] z [%]

61. 61 Anti slug control: Full-scale offshore experiments at Hod-Vallhall field (Havre,1999)

62. 62 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

63. 63 Stabilization with topside measurements: Avoid “RHP-zeros by using 2 measurements Model based control (LQG) with 2 top measurements: DP and density ρ T

64. 64 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: Espen Storkaas + Heidi Sivertsen and Ingvald Bårdsen

65. 65 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)