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Feedback: Still the simplest and best solution

Explore the applications of robustness, self-optimizing control, and stabilization in new operating regimes through feedback systems. Learn about the advantages of feedback over feedforward control and the principles behind self-optimizing feedback control. Discover the importance of active constraints and self-optimizing controlled variables for optimal operation.

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Feedback: Still the simplest and best solution

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  1. Feedback:Still the simplest and best solution Sigurd Skogestad Department of Chemical Engineering Norwegian University of Science and Technology (NTNU) Trondheim Applications to 1) robustness 2) economics (self-optimizing control), and 3) stabilization of new operating regimes

  2. NTNU, Trondheim

  3. Outline • 1. Why feedback (and not feedforward) ? • The feedback amplifier • 2. Self-optimizing control: • How do we link optimization and feedback? • What should we control? • 3. Stabilizing feedback control: • Anti-slug control • Conclusion

  4. Example: AMPLIFIER G Amplifier y r • Want: y(t) = α r(t) • Solution 1 (feedforward): • G = α (adjust amplifier gain) • Very difficult to in practice • Cannot get exact value of α • Cannot easily adjust α online • Do not get same amplification at all frequencies • Problems with distortion and nonlinearity

  5. G Amplifier y r K Measured y Black’s feedback amplifier (1927) Want: y(t) = α r(t) Solution 2 (feedback): K= 1/α (adjustable) Closed-loop response MAGIC! Independent of G, provided GK>> 1

  6. k=10 time 25 Example: disturbance rejection 1 d Gd G u y Plant (uncontrolled system)

  7. 1. Feedforward control (measure d) d Gd G u y ”Perfect” feedforward control: u = - G-1 Gd d Our case: G=Gd → Use u = -d

  8. 1.Feedforward control: Nominal (perfect model) d Gd G u y

  9. d Gd ys e C G u y 2. Feedback control

  10. d Gd ys e C G u y 2. Feedback PI-control: Nominal case Output y Input u Feedback generates inverse! Resulting output

  11. Robustness comparison • Gain error, k = 5, 10 (nominal), 20 • Time constant error, τ = 5, 10 (nominal), 20 • Time delay error, θ = 0 (nominal), 1, 2, 3

  12. Robustness: Gain error, k = 5, 10 (nominal), 20 1. FEEDFORWARD 2. FEEDBACK

  13. Robustness: Time constant error, τ= 5, 10 (nominal), 20 1. FEEDFORWARD 2. FEEDBACK

  14. Robustness: Time delay error, θ = 0 (nominal), 1, 2, 3 1. FEEDFORWARD 2. FEEDBACK

  15. 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/unstable zero Unstable (RHP) zero: Fundamental problem with feedback! Does not help with detailed model + state estimator (Kalman filter)…

  16. Outline • I. Why feedback (and not feedforward) ? • II. Self-optimizing feedback control: • How do we link optimization and feedback? • What should we control? • III. Stabilizing feedback control: Anti-slug control • Conclusion

  17. Optimal operation (economics) • Define scalar cost function J(u0,x,d) • u0: degrees of freedom • d: disturbances • x: states (internal variables) • Optimal operation for given d. Dynamic optimization problem: minu0 J(u0,x,d) subject to: Model: f(u0,x,d) = 0 Constraints: g(u0,x,d) < 0 Here: How do we implement optimal operation?

  18. 1. ”Obvious” solution: Optimizing control =”Feedforward” Estimate d and compute new uopt(d) Probem: Complicated and sensitive to uncertainty

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

  20. RTO y1 = c ? (economics) MPC PID Process control hierarchy

  21. What should we control? • CONTROL ACTIVE CONSTRAINTS! • Optimal solution is usually at constraints, that is, most of the degrees of freedom are used to satisfy “active constraints”, g(u0,d) = 0 • Implementation of active constraints is usually simple. • WHAT MORE SHOULD WE CONTROL? • But what about the remaining unconstrained degrees of freedom? • Look for “self-optimizing” controlled variables!

  22. Self-optimizing Control • Definition Self-optimizing Control • Self-optimizing control is when acceptable operation (=acceptable loss) can be achieved using constant set points (cs)for the controlled variables c (without the need for re-optimizing when disturbances occur). c=cs

  23. Optimal operation – Runner • Cost: J=T • One degree of freedom (u=power) • Optimal operation?

  24. Optimal operation - Runner Solution 1: Optimizing control • Even getting a reasonable model requires > 10 PhD’s  … and the model has to be fitted to each individual…. • Clearly impractical!

  25. Optimal operation - Runner Solution 2 – Feedback(Self-optimizing control) • What should we control?

  26. Optimal operation - Runner Self-optimizing control: Sprinter (100m) • 1. Optimal operation of Sprinter, J=T • Active constraint control: • Maximum speed (”no thinking required”)

  27. Optimal operation - Runner Self-optimizing control: Marathon (40 km) • Optimal operation of Marathon runner, J=T • Any self-optimizing variable c (to control at constant setpoint)? • c1 = distance to leader of race • c2 = speed • c3 = heart rate • c4 = level of lactate in muscles

  28. Optimal operation - Runner Conclusion Marathon runner select one measurement c = heart rate • Simple and robust implementation • Disturbances are indirectly handled by keeping a constant heart rate • May have infrequent adjustment of setpoint (heart rate)

  29. Unconstrained optimum Optimal operation Cost J Jopt copt Controlled variable c

  30. Unconstrained optimum Optimal operation Cost J d Jopt n copt Controlled variable c Two problems: • 1. Optimum moves because of disturbances d: copt(d) • 2. Implementation error, c = copt + n

  31. Good Good BAD Unconstrained optimum Candidate controlled variables c for self-optimizing control Intuitive • The optimal value of c should be insensitive to disturbances (avoid problem 1) • Ideal self-optimizing variable is gradient, c = Ju 2. Optimum should be flat (avoid problem 2 – implementation error). Equivalently: “Want large gain” |G| from u to c

  32. Unconstrained optimum Quantitative steady-state: Maximum gain rule 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 Gsfrom u to c)

  33. Unconstrained optimum Optimal measurement combinations Exact solutions for quadratic optimization problems • Nullspace method. No loss for disturbances (d) 2. Generalized (with noise n): Exact local method: • c = Hy can be considered as linear invariants for the quadratic optimization problem – which can be used for feedback implementation of optimal solution! • Example: Explicit MPC * V. Alstad, S. Skogestad and E.S. Hori, Optimal measurement combinations as controlled variables, Journal of Process Control, 19, 138-148 (2009)

  34. Example: CO2 refrigeration cycle pH • J = Ws (work supplied) • DOF = u (valve opening, z) • Main disturbances: • d1 = TH • d2 = TCs (setpoint) • d3 = UAloss • What should we control?

  35. CO2 refrigeration cycle Step 1. One (remaining) degree of freedom (u=z) Step 2. Objective function. J = Ws (compressor work) Step 3. Optimize operation for disturbances (d1=TC, d2=TH, d3=UA) • Optimum always unconstrained Step 4. Implementation of optimal operation • No good single measurements (all give large losses): • ph, Th, z, … • Nullspace method: Need to combine nu+nd=1+3=4 measurements to have zero disturbance loss • Simpler: Try combining two measurements. Exact local method: • c = h1 ph + h2 Th = ph + k Th; k = -8.53 bar/K • Nonlinear evaluation of loss: OK!

  36. Conclusion CO2 refrigeration cycle Self-optimizing c= “temperature-corrected high pressure”

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

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

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

  40. Experimental mini-loop

  41. z p2 p1 Experimental mini-loopValve opening (z) = 100%

  42. z p2 p1 Experimental mini-loopValve opening (z) = 25%

  43. z p2 p1 Experimental mini-loopValve opening (z) = 15%

  44. z p2 p1 Experimental mini-loop:Bifurcation diagram No slug Valve opening z % Slugging

  45. Avoid slugging? • Operate away from optimal point • Design changes • Feedforward control? • Feedback control?

  46. z p2 p1 Design change Avoid slugging:1. Close valve (but increases pressure) No slugging when valve is closed Valve opening z %

  47. z p2 p1 Design change Avoid slugging:2a. Design change to avoid slugging

  48. z p2 p1 Design change Minimize effect of slugging:2b. Build large slug-catcher • Most common strategy in practice

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

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