1. 1 Outline About Trondheim and myself Control structure design (plantwide control) A procedure for control structure design I Top Down Step 1: Degrees of freedom Step 2: Operational objectives (optimal operation) Step 3: What to control ? (self-optimizing control) Step 4: Where set production rate? II Bottom Up Step 5: Regulatory control: What more to control ? Step 6: Supervisory control Step 7: Real-time optimization Case studies

2. 2 Optimal operation (economics) What are we going to use our degrees of freedom for? Define scalar cost function J(u 0,x,d) –u 0 : degrees of freedom –d: disturbances –x: states (internal variables) Typical cost function: Optimal operation for given d: min uss J(u ss,x,d) subject to: Model equations: f(u ss,x,d) = 0 Operational constraints: g(u ss,x,d) < 0 J = cost feed + cost energy – value products

3. 3 Optimal operation distillation column Distillation at steady state with given p and F: N=2 DOFs, e.g. L and V Cost to be minimized (economics) J = - P where P= p D D + p B B – p F F – p V V Constraints Purity D: For example x D, impurity · max Purity B: For example, x B, impurity · max Flow constraints: min · D, B, L etc. · max Column capacity (flooding): V · V max, etc. Pressure: 1) p given, 2) p free: p min · p · p max Feed: 1) F given 2) F free: F · F max Optimal operation: Minimize J with respect to steady-state DOFs value products cost energy (heating+ cooling) cost feed

4. 4 Example: Paper machine drying section ~10m water recycle water up to 30 m/s (100 km/h) (~20 seconds)

5. 5 Paper machine: Overall operational objectives Degrees of freedom (inputs) drying section –Steam flow each drum (about 100) –Air inflow and outflow (2) Objective: Minimize cost (energy) subject to satisfying operational constraints –Humidity paper ≤10% (active constraint: controlled!) –Air outflow T < dew point – 10C (active – not always controlled) –ΔT along dryer (especially inlet) < bound (active?) –Remaining DOFs: minimize cost

6. 6 Optimal operation 1.Given feed Amount of products is then usually indirectly given and J = cost energy. Optimal operation is then usually unconstrained: 2.Feed free Products usually much more valuable than feed + energy costs small. Optimal operation is then usually constrained: minimize J = cost feed + cost energy – value products “maximize efficiency (energy)” “maximize production” Two main cases (modes) depending on marked conditions: Control: Operate at bottleneck (“obvious”) Control: Operate at optimal trade-off (not obvious how to do and what to control)

7. 7 Comments optimal operation Do not forget to include feedrate as a degree of freedom!! –For paper machine it may be optimal to have max. drying and adjust the speed of the paper machine! Control at bottleneck –see later: “Where to set the production rate”

8. 8 Outline About Trondheim and myself Control structure design (plantwide control) A procedure for control structure design I Top Down Step 1: Degrees of freedom Step 2: Operational objectives (optimal operation) Step 3: What to control ? (self-optimizing control) Step 4: Where set production rate? II Bottom Up Step 5: Regulatory control: What more to control ? Step 6: Supervisory control Step 7: Real-time optimization Case studies

9. 9 Step 3. What should we control (c)? (primary controlled variables y 1 =c) Outline Implementation of optimal operation Self-optimizing control Uncertainty (d and n) Example: Marathon runner Methods for finding the “magic” self-optimizing variables: A. Large gain: Minimum singular value rule B. “Brute force” loss evaluation C. Optimal combination of measurements Example: Recycle process Summary

10. 10 Implementation of optimal operation Optimal operation for given d * : min u J(u,x,d) subject to: Model equations: f(u,x,d) = 0 Operational constraints: g(u,x,d) < 0 → u opt (d * ) Problem: Usally cannot keep u opt constant because disturbances d change How should be adjust the degrees of freedom (u)?

11. 11 Problem: Too complicated (requires detailed model and description of uncertainty) Implementation of optimal operation (Cannot keep u 0opt constant) ”Obvious” solution: Optimizing control Estimate d from measurements and recompute u opt (d)

12. 12 In practice: Hierarchical decomposition with separate layers What should we control?

13. 13 Self-optimizing control: When constant setpoints is OK Constant setpoint

14. 14 Unconstrained variables: Self-optimizing control Self-optimizing control: Constant setpoints c s give ”near-optimal operation” (= acceptable loss L for expected disturbances d and implementation errors n) Acceptable loss ) self-optimizing control

15. 15 What c’s should we control? Optimal solution is usually at constraints, that is, most of the degrees of freedom are used to satisfy “active constraints”, g(u,d) = 0 CONTROL ACTIVE CONSTRAINTS! –c s = value of active constraint –Implementation of active constraints is usually simple. WHAT MORE SHOULD WE CONTROL? –Find “self-optimizing” variables c for remaining unconstrained degrees of freedom u.

16. 16 What should we control? – Sprinter Optimal operation of Sprinter (100 m), J=T –One input: ”power/speed” –Active constraint control: Maximum speed (”no thinking required”)

17. 17 What should we control? – Marathon Optimal operation of Marathon runner, J=T –No active constraints –Any self-optimizing variable c (to control at constant setpoint)?

18. 18 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

19. 19 Further examples self-optimizing control Marathon runner Central bank Cake baking Business systems (KPIs) Investment portifolio Biology Chemical process plants: Optimal blending of gasoline Define optimal operation (J) and look for ”magic” variable (c) which when kept constant gives acceptable loss (self- optimizing control)

20. 20 More on 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 BREAK

21. 21 Summary so far: Active constrains and unconstrained variables Optimal operation: Minimize J with respect to DOFs General: Optimal solution with N DOFs: –N active: DOFs used to satisfy “active” constraints ( · is =) –N u = N – N active. remaining unconstrained variables Often: N u is zero or small It is “obvious” how to control the active constraints Difficult issue: What should we use the remaining N u degrees of for, that is what should we control?

22. 22 Recall: Optimal operation distillation column Distillation at steady state with given p and F: N=2 DOFs, e.g. L and V Cost to be minimized (economics) J = - P where P= p D D + p B B – p F F – p V V Constraints Purity D: For example x D, impurity · max Purity B: For example, x B, impurity · max Flow constraints: min · D, B, L etc. · max Column capacity (flooding): V · V max, etc. Pressure: 1) p given, 2) p free: p min · p · p max Feed: 1) F given 2) F free: F · F max Optimal operation: Minimize J with respect to steady-state DOFs value products cost energy (heating+ cooling) cost feed

23. 23 Solution: Optimal operation distillation Cost to be minimized J = - P where P= p D D + p B B – p F F – p V V N=2 steady-state degrees of freedom Active constraints distillation: –Purity spec. valuable product is always active (“avoid give- away of valuable product”). –Purity spec. “cheap” product may not be active (may want to overpurify to avoid loss of valuable product – but costs energy) Three cases: 1.N active =2: Two active constraints (for example, x D, impurity = max. x B, impurity = max, “TWO-POINT” COMPOSITION CONTROL) 2.N active =1: One constraint active (1 remaining DOF) 3.N active =0: No constraints active (2 remaining DOFs) Can happen if no purity specifications (e.g. byproducts or recycle) Problem : WHAT SHOULD WE CONTROL (TO SATISFY THE UNCONSTRAINED DOFs )? Solution: Often compositions but not always!

24. 24 Unconstrained variables: What should we control? Intuition: “Dominant variables” (Shinnar) Is there any systematic procedure?

25. 25 What should we control? Systematic procedure Systematic: Minimize cost J(u,d * ) w.r.t. DOFs u. 1.Control active constraints (constant setpoint is optimal) 2.Remaining unconstrained DOFs: Control “self-optimizing” variables c for which constant setpoints c s = c opt (d * ) give small (economic) loss Loss = J - J opt (d) when disturbances d ≠ d * occur c = ? (economics) y 2 = ? (stabilization)

26. 26 The difficult unconstrained variables Cost J Selected controlled variable (remaining unconstrained) c opt J opt c

27. 27 Example: Tennessee Eastman plant J c = Purge rate Nominal optimum setpoint is infeasible with disturbance 2 Oopss.. bends backwards Conclusion: Do not use purge rate as controlled variable

28. 28 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

29. 29 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 d n LOSS

30. 30 Effect of implementation error on cost (“problem 2”) BADGood

31. 31 Example sharp optimum. High-purity distillation : c = Temperature top of column Temperature T top Water (L) - acetic acid (H) Max 100 ppm acetic acid 100 C: 100% water 100.01C: 100 ppm 99.99 C: Infeasible

32. 32 Candidate controlled variables We are looking for some “magic” variables c to control..... What properties do they have?’ Intuitively 1: Should have small optimal range delta c opt –since we are going to keep them constant! Intuitively 2: Should have small “implementation error” n “Intuitively” 3: Should be sensitive to inputs u (remaining unconstrained degrees of freedom), that is, the gain G 0 from u to c should be large –G 0 : (unscaled) gain from u to c –large gain gives flat optimum in c –Charlie Moore (1980’s): Maximize minimum singular value when selecting temperature locations for distillation Will show shortly: Can combine everything into the “maximum gain rule”: –Maximize scaled gain G = G o / span(c) Unconstrained degrees of freedom: span(c)

33. 33 Optimizer Controller that adjusts u to keep c m = c s Plant cscs c m =c+n u c n d u c J c s =c opt u opt n Unconstrained degrees of freedom: Justification for “intuitively 2 and 3” Want the slope (= gain G 0 from u to c) large – corresponds to flat optimum in c Want small n

34. 34 Mathematic local analysis (Proof of “maximum gain rule”) u cost J u opt

35. 35 Minimum singular value of scaled gain Maximum gain rule (Skogestad and Postlethwaite, 1996): Look for variables that maximize the scaled gain  (G) (minimum singular value of the appropriately scaled steady-state gain matrix G from u to c)  (G) is called the Morari Resiliency index (MRI) by Luyben Detailed proof: I.J. Halvorsen, S. Skogestad, J.C. Morud and V. Alstad, ``Optimal selection of controlled variables'', Ind. Eng. Chem. Res., 42 (14), 3273-3284 (2003).

36. 36 Maximum gain rule for scalar system Unconstrained degrees of freedom: J uu : Hessian for effect of u’s on cost Problem: J uu can be difficult to obtain Fortunate for scalar system: J uu does not matter

37. 37 Maximum gain rule in words Select controlled variables c for which the gain G 0 (=“controllable range”) is large compared to its span (=sum of optimal variation and control error)

38. 38 B. “Brute-force” procedure for selecting (primary) controlled variables (Skogestad, 2000) Step 1 Determine DOFs for optimization Step 2 Definition of optimal operation J (cost and constraints) Step 3 Identification of important disturbances Step 4 Optimization (nominally and with disturbances) Step 5 Identification of candidate controlled variables (use active constraint control) Step 6 Evaluation of loss with constant setpoints for alternative controlled variables Step 7 Evaluation and selection (including controllability analysis) Case studies: Tenneessee-Eastman, Propane-propylene splitter, recycle process, heat-integrated distillation

39. 39 Unconstrained degrees of freedom: C. Optimal measurement combination (Alstad, 2002)

40. 40 Unconstrained degrees of freedom: C. Optimal measurement combination (Alstad, 2002) Basis: Want optimal value of c independent of disturbances ) –  c opt = 0 ¢  d Find optimal solution as a function of d: u opt (d), y opt (d) Linearize this relationship:  y opt = F  d F – sensitivity matrix Want: To achieve this for all values of  d: Always possible if Optimal when we disregard implementation error (n)

41. 41 Alstad-method continued To handle implementation error: Use “sensitive” measurements, with information about all independent variables (u and d)

42. 42 Summary unconstrained degrees of freedom: Looking for “magic” variables to keep at constant setpoints. How can we find them systematically? Candidates A. Start with: Maximum gain (minimum singular value) rule: B. Then: “Brute force evaluation” of most promising alternatives. Evaluate loss when the candidate variables c are kept constant. In particular, may be problem with feasibility C. More general candidates: Find optimal linear combination (matrix H):

43. 43 Toy Example

44. 44 Toy Example

45. 45 Toy Example

46. 46 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

47. 47 Recycle plant: Optimal operation mTmT 1 remaining unconstrained degree of freedom

48. 48 Control of recycle plant: Conventional structure (“Two-point”: x D ) LC XC LC XC LC xBxB xDxD Control active constraints (M r =max and x B =0.015) + x D

49. 49 Luyben rule Luyben rule (to avoid snowballing): “Fix a stream in the recycle loop” (F or D)

50. 50 Luyben rule: D constant Luyben rule (to avoid snowballing): “Fix a stream in the recycle loop” (F or D) LC XC

51. 51 A. Maximum gain rule: Steady-state gain Luyben rule: Not promising economically Conventional: Looks good

52. 52 How did we find the gains in the Table? 1.Find nominal optimum 2.Find (unscaled) gain G 0 from input to candidate outputs:  c = G 0  u. In this case only a single unconstrained input (DOF). Choose at u=L Obtain gain G 0 numerically by making a small perturbation in u=L while adjusting the other inputs such that the active constraints are constant (bottom composition fixed in this case) 3.Find the span for each candidate variable For each disturbance d i make a typical change and reoptimize to obtain the optimal ranges  c opt (d i ) For each candidate output obtain (estimate) the control error (noise) n span(c) =  i |  c opt (d i )| + n 4.Obtain the scaled gain, G = G 0 / span(c) IMPORTANT!

53. 53 B. “Brute force” loss evaluation: Disturbance in F 0 Loss with nominally optimal setpoints for M r, x B and c Luyben rule: Conventional

54. 54 B. “Brute force” loss evaluation: Implementation error Loss with nominally optimal setpoints for M r, x B and c Luyben rule:

55. 55 C. Optimal measurement combination 1 unconstrained variable (#c = 1) 1 (important) disturbance: F 0 (#d = 1) “Optimal” combination requires 2 “measurements” (#y = #u + #d = 2) –For example, c = h 1 L + h 2 F BUT: Not much to be gained compared to control of single variable (e.g. L/F or x D )

56. 56 Conclusion: Control of recycle plant Active constraint M r = M rmax Active constraint x B = x Bmin L/F constant: Easier than “two-point” control Assumption: Minimize energy (V) Self-optimizing

57. 57 Recycle systems: Do not recommend Luyben’s rule of fixing a flow in each recycle loop (even to avoid “snowballing”)

58. 58 Summary: Self-optimizing Control Self-optimizing control is when acceptable operation can be achieved using constant set points (c s ) for the controlled variables c (without the need to re-optimizing when disturbances occur). c=c s

59. 59 Summary: Procedure selection controlled variables 1.Define economics and operational constraints 2.Identify degrees of freedom and important disturbances 3.Optimize for various disturbances 4.Identify (and control) active constraints (off-line calculations) May vary depending on operating region. For each operating region do step 5: 5.Identify “self-optimizing” controlled variables for remaining degrees of freedom 1.(A) Identify promising (single) measurements from “maximize gain rule” (gain = minimum singular value) (C) Possibly consider measurement combinations if no promising 2.(B) “Brute force” evaluation of loss for promising alternatives Necessary because “maximum gain rule” is local. In particular: Look out for feasibility problems. 3.Controllability evaluation for promising alternatives

60. 60 Summary ”self-optimizing” control Operation of most real system: Constant setpoint policy (c = c s ) –Central bank –Business systems: KPI’s –Biological systems –Chemical processes Goal: Find controlled variables c such that constant setpoint policy gives acceptable operation in spite of uncertainty ) Self-optimizing control Method A: Maximize  (G) Method B: Evaluate loss L = J - J opt Method C: Optimal linear measurement combination:  c = H  y where HF=0

61. 61 Outline Control structure design (plantwide control) A procedure for control structure design I Top Down Step 1: Degrees of freedom Step 2: Operational objectives (optimal operation) Step 3: What to control ? (self-optimzing control) Step 4: Where set production rate? II Bottom Up Step 5: Regulatory control: What more to control ? Step 6: Supervisory control Step 7: Real-time optimization Case studies