1. 1 PLANTWIDE CONTROL Sigurd Skogestad Department of Chemical Engineering Norwegian University of Science and Tecnology (NTNU) Trondheim, Norway 01 April 2006

2. 2 Intro, DOFs, control objectives, self-op. control What to control, Production rate, stabilizing control, distillation example Supervisory control. HDA example

3. 3 Contents Overview of plantwide control Selection of primary controlled variables based on economic : The llink between the optimization (RTO) and the control (MPC; PID) layers - Degrees of freedom - Optimization - Self-optimizing control - Applications - Many examples Where to set the production rate and bottleneck Design of the regulatory control layer ("what more should we control") - stabilization - secondary controlled variables (measurements) - pairing with inputs - controllability analysis - cascade control and time scale separation. Design of supervisory control layer - Decentralized versus centralized (MPC) - Design of decentralized controllers: Sequential and independent design - Pairing and RGA-analysis Summary and case studies

4. 4 Trondheim, Norway

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

6. 6 NTNU, Trondheim

7. 7 Main message 1. Control for economics (Top-down steady-state arguments) –Primary controlled variables c 2. Control for stabilization (Bottom-up; regulatory PID control) –Secondary controlled variables (“inner cascade loops”) Both problems: “Maximum gain rule” useful for selecting controlled variables

8. 8 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 ? (primary CV’s) (self-optimizing control) Step 4: Where set production rate? II Bottom Up Step 5: Regulatory control: What more to control (secondary CV’s) ? Step 6: Supervisory control Step 7: Real-time optimization Case studies

9. 9 Idealized view of control (“Ph.D. control”)

10. 10 Practice: Tennessee Eastman challenge problem (Downs, 1991) (“PID control”)

11. 11 How we design a control system for a complete chemical plant? Where do we start? What should we control? and why? etc.

12. 12 Alan Foss (“Critique of chemical process control theory”, AIChE Journal,1973): The central issue to be resolved... is the determination of control system structure. Which variables should be measured, which inputs should be manipulated and which links should be made between the two sets? There is more than a suspicion that the work of a genius is needed here, for without it the control configuration problem will likely remain in a primitive, hazily stated and wholly unmanageable form. The gap is present indeed, but contrary to the views of many, it is the theoretician who must close it. Carl Nett (1989): Minimize control system complexity subject to the achievement of accuracy specifications in the face of uncertainty.

13. 13 Control structure design Not the tuning and behavior of each control loop, But rather the control philosophy of the overall plant with emphasis on the structural decisions: –Selection of controlled variables (“outputs”) –Selection of manipulated variables (“inputs”) –Selection of (extra) measurements –Selection of control configuration (structure of overall controller that interconnects the controlled, manipulated and measured variables) –Selection of controller type (LQG, H-infinity, PID, decoupler, MPC etc.). That is: Control structure design includes all the decisions we need make to get from ``PID control’’ to “Ph.D” control

14. 14 Process control: “Plantwide control” = “Control structure design for complete chemical plant” Large systems Each plant usually different – modeling expensive Slow processes – no problem with computation time Structural issues important –What to control? –Extra measurements –Pairing of loops

15. 15 Previous work on plantwide control Page Buckley (1964) - Chapter on “Overall process control” (still industrial practice) Greg Shinskey (1967) – process control systems Alan Foss (1973) - control system structure Bill Luyben et al. (1975- ) – case studies ; “snowball effect” George Stephanopoulos and Manfred Morari (1980) – synthesis of control structures for chemical processes Ruel Shinnar (1981- ) - “dominant variables” Jim Downs (1991) - Tennessee Eastman challenge problem Larsson and Skogestad (2000): Review of plantwide control

16. 16 Control structure selection issues are identified as important also in other industries. Professor Gary Balas (Minnesota) at ECC’03 about flight control at Boeing: The most important control issue has always been to select the right controlled variables --- no systematic tools used!

17. 17 Main simplification: Hierarchical structure Need to define objectives and identify main issues for each layer PID RTO MPC

18. 18 Regulatory control (seconds) Purpose: “Stabilize” the plant by controlling selected ‘’secondary’’ variables (y 2 ) such that the plant does not drift too far away from its desired operation Use simple single-loop PI(D) controllers Status: Many loops poorly tuned –Most common setting: K c =1,  I =1 min (default) –Even wrong sign of gain K c ….

19. 19 Regulatory control……... Trend: Can do better! Carefully go through plant and retune important loops using standardized tuning procedure Exists many tuning rules, including Skogestad (SIMC) rules: –K c = (1/k) (  1 / [  c +  ])  I = min (  1, 4[  c +  ]), Typical:  c =  –“Probably the best simple PID tuning rules in the world” © Carlsberg Outstanding structural issue: What loops to close, that is, which variables (y 2 ) to control?

20. 20 Supervisory control (minutes) Purpose: Keep primary controlled variables (c=y 1 ) at desired values, using as degrees of freedom the setpoints y 2s for the regulatory layer. Status: Many different “advanced” controllers, including feedforward, decouplers, overrides, cascades, selectors, Smith Predictors, etc. Issues: –Which variables to control may change due to change of “active constraints” –Interactions and “pairing”

21. 21 Supervisory control…... Trend: Model predictive control (MPC) used as unifying tool. –Linear multivariable models with input constraints –Tuning (modelling) is time-consuming and expensive Issue: When use MPC and when use simpler single-loop decentralized controllers ? –MPC is preferred if active constraints (“bottleneck”) change. –Avoids logic for reconfiguration of loops Outstanding structural issue: –What primary variables c=y 1 to control?

22. 22 Local optimization (hour) Purpose: Minimize cost function J and: –Identify active constraints –Recompute optimal setpoints y 1s for the controlled variables Status: Done manually by clever operators and engineers Trend: Real-time optimization (RTO) based on detailed nonlinear steady-state model Issues: –Optimization not reliable. –Need nonlinear steady-state model –Modelling is time-consuming and expensive

23. 23 Objectives of layers: MV’s and CV’s c s = y 1s MPC PID y 2s RTO u (valves) CV=y 1 ; MV=y 2s CV=y 2 ; MV=u Min J (economics); MV=y 1s

24. 24 Stepwise procedure plantwide control I. TOP-DOWN Step 1. DEGREES OF FREEDOM Step 2. OPERATIONAL OBJECTIVES Step 3. WHAT TO CONTROL? (primary CV’s c=y 1 ) Step 4. PRODUCTION RATE II. BOTTOM-UP (structure control system): Step 5. REGULATORY CONTROL LAYER (PID) “Stabilization” What more to control? (secondary CV’s y 2 ) Step 6. SUPERVISORY CONTROL LAYER (MPC) Decentralization Step 7. OPTIMIZATION LAYER (RTO) Can we do without it?

25. 25 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-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

26. 26 Step 1. Degrees of freedom (DOFs) for operation (N valves ): To find all operational (dynamic) degrees of freedom Count valves! (N valves ) “Valves” also includes adjustable compressor power, etc. Anything we can manipulate!

27. 27 Steady-state degrees of freedom Cost J depends normally only on steady-state DOFs Three methods to obtain steady-state degrees of freedom (N ss ): 1.Equation-counting N ss = no. of variables – no. of equations/specifications Very difficult in practice (not covered here) 2.Valve-counting (easier!) N ss = N valves – N 0ss – N specs N 0ss = variables with no steady-state effect 3.Typical number for some units (useful for checking!)

28. 28 Steady-state degrees of freedom (N ss ): 2. Valve-counting N valves = no. of dynamic (control) DOFs (valves) N ss = N valves – N 0ss – N specs : no. of steady-state DOFs N 0ss = N 0y + N 0,valves : no. of variables with no steady-state effect –N 0,valves : no. purely dynamic control DOFs –N 0y : no. controlled variables (liquid levels) with no steady-state effect N specs : No of equality specifications (e.g., given pressure)

29. 29 N valves = 6, N 0y = 2, N specs = 2, N SS = 6 -2 -2 = 2 Distillation column with given feed and pressure

30. 30 Heat-integrated distillation process

31. 31 Heat-integrated distillation process

32. 32 Heat exchanger with bypasses

33. 33 Heat exchanger with bypasses

34. 34 Steady-state degrees of freedom (N ss ): 3. Typical number for some process units each external feedstream: 1 (feedrate) splitter: n-1 (split fractions) where n is the number of exit streams mixer: 0 compressor, turbine, pump: 1 (work) adiabatic flash tank: 0 * liquid phase reactor: 1 (holdup-volume reactant) gas phase reactor: 0 * heat exchanger: 1 (duty or net area) distillation column excluding heat exchangers: 0 * + number of sidestreams pressure * : add 1DOF at each extra place you set pressure (using an extra valve, compressor or pump!). Could be for adiabatic flash tank, gas phase reactor, distillation column * Pressure is normally assumed to be given by the surrounding process and is then not a degree of freedom

35. 35 Heat exchanger with bypasses

36. 36 “Typical number”, N ss = 0 (distillation) + 2*1 (heat exchangers) = 2 Distillation column with given feed and pressure

37. 37 Heat-integrated distillation process

38. 38 HDA process MixerFEHE FurnacePFR Quench Separator Compressor Cooler Stabilizer Benzene Column Toluene Column H 2 + CH 4 Toluene Benzene CH 4 Diphenyl Purge (H 2 + CH 4 )

39. 39 HDA process: steady-state degrees of freedom 1 2 3 8 7 4 6 5 9 10 11 12 13 14 Conclusion: 14 steady-state DOFs Assume given column pressures feed:1.2 hex: 3, 4, 6 splitter 5, 7 compressor: 8 distillation: rest

40. 40 Check that there are enough manipulated variables (DOFs) - both dynamically and at steady-state (step 2) Otherwise: Need to add equipment –extra heat exchanger –bypass –surge tank

41. 41 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

42. 42 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

43. 43 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

44. 44 Example: Paper machine drying section ~10m water recycle water up to 30 m/s (100 km/h) (~20 seconds)

45. 45 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

46. 46 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)

47. 47 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”

48. 48 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

49. 49 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

50. 50 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)?

51. 51 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)

52. 52 In practice: Hierarchical decomposition with separate layers What should we control?

53. 53 Self-optimizing control: When constant setpoints is OK Constant setpoint

54. 54 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

55. 55 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.

56. 56 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”)

57. 57 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)?

58. 58 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

59. 59 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)

60. 60 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

61. 61 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?

62. 62 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

63. 63 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!

64. 64 Unconstrained variables: What should we control? Intuition: “Dominant variables” (Shinnar) Is there any systematic procedure?

65. 65 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)

66. 66 The difficult unconstrained variables Cost J Selected controlled variable (remaining unconstrained) c opt J opt c

67. 67 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

68. 68 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

69. 69 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

70. 70 Effect of implementation error on cost (“problem 2”) BADGood

71. 71 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

72. 72 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)

73. 73 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

74. 74 Mathematic local analysis (Proof of “maximum gain rule”) u cost J u opt

75. 75 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).

76. 76 Improved minimum singular value rule for ill-conditioned plants G: Scaled gain matrix (as before) J uu : Hessian for effect of u’s on cost Problem: J uu can be difficult to obtain Improved rule has been used successfully for distillation

77. 77 Maximum gain rule for scalar system Unconstrained degrees of freedom:

78. 78 Maximum gain rule in words Select controlled variables c for which their controllable range is large compared to their sum of optimal variation and control error controllable range = range c may reach by varying the inputs (=gain) optimal variation: due to disturbance control error = implementation error n span

79. 79 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

80. 80 Unconstrained degrees of freedom: C. Optimal measurement combination (Alstad, 2002)

81. 81 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)

82. 82 Alstad-method continued To handle implementation error: Use “sensitive” measurements, with information about all independent variables (u and d)

83. 83 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):

84. 84 Toy Example

85. 85 Toy Example

86. 86 Toy Example

87. 87 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

88. 88 Recycle plant: Optimal operation mTmT 1 remaining unconstrained degree of freedom

89. 89 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

90. 90 Luyben rule Luyben rule (to avoid snowballing): “Fix a stream in the recycle loop” (F or D)

91. 91 Luyben rule: D constant Luyben rule (to avoid snowballing): “Fix a stream in the recycle loop” (F or D) LC XC

92. 92 A. Maximum gain rule: Steady-state gain Luyben rule: Not promising economically Conventional: Looks good

93. 93 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!

94. 94 B. “Brute force” loss evaluation: Disturbance in F 0 Loss with nominally optimal setpoints for M r, x B and c Luyben rule: Conventional

95. 95 B. “Brute force” loss evaluation: Implementation error Loss with nominally optimal setpoints for M r, x B and c Luyben rule:

96. 96 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 )

97. 97 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

98. 98 Recycle systems: Do not recommend Luyben’s rule of fixing a flow in each recycle loop (even to avoid “snowballing”)

99. 99 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

100. 10 0 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

101. 10 1 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

102. 10 2 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

103. 10 3 Step 4. Where set production rate? Very important! Determines structure of remaining inventory (level) control system Set production rate at (dynamic) bottleneck Link between Top-down and Bottom-up parts

104. 10 4 Production rate set at inlet : Inventory control in direction of flow

105. 10 5 Production rate set at outlet: Inventory control opposite flow

106. 10 6 Production rate set inside process

107. 10 7 Where set the production rate? Very important decision that determines the structure of the rest of the control system! May also have important economic implications

108. 10 8 Often optimal: Set production rate at bottleneck! "A bottleneck is an extensive variable that prevents an increase in the overall feed rate to the plant" If feed is cheap and available: Optimal to set production rate at bottleneck If the flow for some time is not at its maximum through the bottleneck, then this loss can never be recovered.

109. 10 9 Reactor-recycle process: Given feedrate with production rate set at inlet

110. 11 0 Reactor-recycle process: Want to maximize feedrate: reach bottleneck in column Bottleneck: max. vapor rate in column

111. 11 1 Reactor-recycle process with production rate set at inlet Want to maximize feedrate: reach bottleneck in column Bottleneck: max. vapor rate in column FC V max V V max -V s =Back-off = Loss Alt.1: Loss VsVs

112. 11 2 Alt.2 “long loop” MAX Reactor-recycle process with increased feedrate: Optimal: Set production rate at bottleneck

113. 11 3 Reactor-recycle process with increased feedrate: Optimal: Set production rate at bottleneck MAX Alt.3: reconfigure

114. 11 4 Reactor-recycle process: Given feedrate with production rate set at bottleneck F 0s Alt.3: reconfigure (permanently)

115. 11 5 Can reduce loss BUT: Is generally placed on top of the regulatory control system (including level loops), so it still important where the production rate is set! Alt.4: Multivariable control (MPC)

116. 11 6 Conclusion production rate manipulator Think carefully about where to place it! Difficult to undo later BREAK

117. 11 7 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-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

118. 11 8 II. Bottom-up Determine secondary controlled variables and structure (configuration) of control system (pairing) A good control configuration is insensitive to parameter changes Step 5. REGULATORY CONTROL LAYER 5.1Stabilization (including level control) 5.2Local disturbance rejection (inner cascades) What more to control? (secondary variables) Step 6. SUPERVISORY CONTROL LAYER Decentralized or multivariable control (MPC)? Pairing? Step 7. OPTIMIZATION LAYER (RTO)

119. 11 9 Step 5. Regulatory control layer Purpose: “Stabilize” the plant using local SISO PID controllers Enable manual operation (by operators) Main structural issues: What more should we control? (secondary cv’s, y 2 ) Pairing with manipulated variables (mv’s u 2 ) y 1 = c y 2 = ?

120. 12 0 Regulatory loops GK y 2s u2u2 y2y2 y1y1 Key decision: Choice of y 2 (controlled variable) Also important (since we almost always use single loops in the regulatory control layer): Choice of u 2 (“pairing”)

121. 12 1 Example: Distillation Primary controlled variable: y 1 = c = x D, x B (compositions top, bottom) BUT: Delay in measurement of x + unreliable Regulatory control: For “stabilization” need control of (y 2 ): –Liquid level condenser (M D ) –Liquid level reboiler (M B ) –Pressure (p) –Holdup of light component in column (temperature profile) Unstable (Integrating) + No steady-state effect Disturbs (“destabilizes”) other loops Almost unstable (integrating) TC TsTs T-loop in bottom

122. 12 2 XCXC TC FC ysys y LsLs TsTs L T z XCXC Cascade control distillation With flow loop + T-loop in top

123. 12 3 Degrees of freedom unchanged No degrees of freedom lost by control of secondary (local) variables as setpoints become y 2s replace inputs u 2 as new degrees of freedom GK y 2s u2u2 y2y2 y1y1 Original DOF New DOF Cascade control:

124. 12 4 Hierarchical control: Time scale separation With a “reasonable” time scale separation between the layers (typically by a factor 5 or more in terms of closed-loop response time) we have the following advantages: 1.The stability and performance of the lower (faster) layer (involving y 2 ) is not much influenced by the presence of the upper (slow) layers (involving y 1 ) Reason: The frequency of the “disturbance” from the upper layer is well inside the bandwidth of the lower layers 2.With the lower (faster) layer in place, the stability and performance of the upper (slower) layers do not depend much on the specific controller settings used in the lower layers Reason: The lower layers only effect frequencies outside the bandwidth of the upper layers

125. 12 5 Objectives regulatory control layer 1.Allow for manual operation 2.Simple decentralized (local) PID controllers that can be tuned on-line 3.Take care of “fast” control 4.Track setpoint changes from the layer above 5.Local disturbance rejection 6.Stabilization (mathematical sense) 7.Avoid “drift” (due to disturbances) so system stays in “linear region” –“stabilization” (practical sense) 8.Allow for “slow” control in layer above (supervisory control) 9.Make control problem easy as seen from layer above Implications for selection of y 2 : 1.Control of y 2 “stabilizes the plant” 2.y 2 is easy to control (favorable dynamics)

126. 12 6 1. “Control of y 2 stabilizes the plant” A. “Mathematical stabilization” (e.g. reactor): Unstable mode is “quickly” detected (state observability) in the measurement (y 2 ) and is easily affected (state controllability) by the input (u 2 ). Tool for selecting input/output: Pole vectors –y 2 : Want large element in output pole vector: Instability easily detected relative to noise –u 2 : Want large element in input pole vector: Small input usage required for stabilization B. “Practical extended stabilization” (avoid “drift” due to disturbance sensitivity): Intuitive: y 2 located close to important disturbance Or rather: Controllable range for y 2 is large compared to sum of optimal variation and control error More exact tool: Partial control analysis

127. 12 7 Recall rule for selecting primary controlled variables c: Controlled variables c for which their controllable range is large compared to their sum of optimal variation and control error Control variables y 2 for which their controllable range is large compared to their sum of optimal variation and control error controllable range = range y 2 may reach by varying the inputs optimal variation: due to disturbances control error = implementation error n Restated for secondary controlled variables y 2 : Want small Want large

128. 12 8 What should we control (y 2 )? Rule: Maximize the scaled gain General case: Maximize minimum singular value of scaled G Scalar case: |G s | = |G| / span |G|: gain from independent variable (u 2 ) to candidate controlled variable (y 2 ) –IMPORTANT: The gain |G| should be evaluated at the (bandwidth) frequency of the layer above in the control hierarchy! This can be very different from the steady-state gain used for selecting primary controlled variables (y 1 =c) span (of y 2 ) = optimal variation in y 2 + control error for y 2 –Note optimal variation: This is often the same as the optimal variation used for selecting primary controlled variables (c). –Exception: If we at the “fast” regulatory time scale have some yet unused “slower” inputs (u 1 ) which are constant then we may want find a more suitable optimal variation for the fast time scale.

129. 12 9 Minimize state drift by controlling y 2 Problem in some cases: “optimal variation” for y 2 depends on overall control objectives which may change Therefore: May want to “decouple” tasks of stabilization (y 2 ) and optimal operation (y 1 ) One way of achieving this: Choose y 2 such that “state drift” dw/dd is minimized w = Wx – weighted average of all states d – disturbances Some tools developed: –Optimal measurement combination y 2 =Hy that minimizes state drift (Hori) – see Skogestad and Postlethwaite (Wiley, 2005) p. 418 –Distillation column application: Control average temperature column

130. 13 0 2. “y 2 is easy to control” (controllability) 1.Statics: Want large gain (from u 2 to y 2 ) 2.Main rule: y 2 is easy to measure and located close to available manipulated variable u 2 (“pairing”) 3.Dynamics: Want small effective delay (from u 2 to y 2 ) “effective delay” includes inverse response (RHP-zeros) + high-order lags

131. 13 1 Rules for selecting u 2 (to be paired with y 2 ) 1.Avoid using variable u 2 that may saturate (especially in loops at the bottom of the control hieararchy) Alternatively: Need to use “input resetting” in higher layer Example: Stabilize reactor with bypass flow (e.g. if bypass may saturate, then reset in higher layer using cooling flow) 2.“Pair close”: The controllability, for example in terms a small effective delay from u 2 to y 2, should be good.

132. 13 2 Effective delay and tunings θ = effective delay PI-tunings from “SIMC rule” Use half rule to obtain first-order model –Effective delay θ = “True” delay + inverse response time constant + half of second time constant + all smaller time constants –Time constant τ 1 = original time constant + half of second time constant –NOTE: The first (largest) time constant is NOT important for controllability!

133. 13 3 Summary: Rules for selecting y 2 (and u 2 ) 1.y 2 should be easy to measure 2.Control of y 2 stabilizes the plant 3.y 2 should have good controllability, that is, favorable dynamics for control 4.y 2 should be located “close” to a manipulated input (u 2 ) (follows from rule 3) 5.The (scaled) gain from u 2 to y 2 should be large 6.The effective delay from u 2 to y 2 should be small 7.Avoid using inputs u 2 that may saturate (should generally avoid saturation in lower layers)

134. 13 4 Example regulatory control: Distillation (see separate slides) 5 dynamic DOFs (L,V,D,B,VT) Overall objective: Control compositions (x D and x B ) “Obvious” stabilizing loops: 1.Condenser level (M 1 ) 2.Reboiler level (M 2 ) 3.Pressure E.A. Wolff and S. Skogestad, ``Temperature cascade control of distillation columns'', Ind.Eng.Chem.Res., 35, 475-484, 1996.

135. 13 5 Selecting measurements and inputs for stabilization: Pole vectors Maximum gain rule is good for integrating (drifting) modes For “fast” unstable modes (e.g. reactor): Pole vectors useful for determining which input (valve) and output (measurement) to use for stabilizing unstable modes Assumes input usage (avoiding saturation) may be a problem

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138. 13 8 Example: Tennessee Eastman challenge problem

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145. 14 5 Control configuration elements Control configuration. The restrictions imposed on the overall controller by decomposing it into a set of local controllers (subcontrollers, units, elements, blocks) with predetermined links and with a possibly predetermined design sequence where subcontrollers are designed locally. Control configuration elements: Cascade controllers Decentralized controllers Feedforward elements Decoupling elements

146. 14 6 Cascade control arises when the output from one controller is the input to another. This is broader than the conventional definition of cascade control which is that the output from one controller is the reference command (setpoint) to another. In addition, in cascade control, it is usually assumed that the inner loop K2 is much faster than the outer loop K1. Feedforward elements link measured disturbances to manipulated inputs. Decoupling elements link one set of manipulated inputs (“measurements”) with another set of manipulated inputs. They are used to improve the performance of decentralized control systems, and are often viewed as feedforward elements (although this is not correct when we view the control system as a whole) where the “measured disturbance” is the manipulated input computed by another decentralized controller.

147. 14 7 Why simplified configurations? Fundamental: Save on modelling effort Other: –easy to understand –easy to tune and retune –insensitive to model uncertainty –possible to design for failure tolerance –fewer links –reduced computation load

148. 14 8 Cascade control (conventional; with extra measurement) The reference r 2 is an output from another controller General case (“parallel cascade”) Special common case (“series cascade”)

149. 14 9 Series cascade 1.Disturbances arising within the secondary loop (before y 2 ) are corrected by the secondary controller before they can influence the primary variable y 1 2.Phase lag existing in the secondary part of the process (G 2 ) is reduced by the secondary loop. This improves the speed of response of the primary loop. 3.Gain variations in G 2 are overcome within its own loop. Thus, use cascade control (with an extra secondary measurement y 2 ) when: The disturbance d 2 is significant and G 1 has an effective delay The plant G 2 is uncertain (varies) or n onlinear Design: First design K 2 (“fast loop”) to deal with d 2 Then design K 1 to deal with d 1

150. 15 0 Tuning cascade Use SIMC tuning rules K 2 is designed based on G 2 (which has effective delay  2 ) –then y 2 = T 2 r 2 + S 2 d 2 where S 2 ¼ 0 and T 2 ¼ 1 ¢ e -(  2 +  c2 )s T 2 : gain = 1 and effective delay =  2 +  c2 SIMC-rule:  c2 ¸  2 Time scale separation:  c2 ·  c1 /5 (approximately) K 1 is designed based on G 1 T 2 same as G 1 but with an additional delay  2 +  c2 y 2 = T 2 r 2 + S 2 d 2

151. 15 1 Exercise: Tuning cascade 1.(without cascade, i.e. no feedback from y 2 ). Design a controller based on G 1 G 2.(with cascade) Design K 2 and then K 1

152. 15 2 Tuning cascade control

153. 15 3 Extra inputs Exercise: Explain how “valve position control” fits into this framework. As en example consider a heat exchanger with bypass

154. 15 4 Exercise Exercise: (a)In what order would you tune the controllers? (b)Give a practical example of a process that fits into this block diagram

155. 15 5 Cascade control: y 2 not important in itself, and setpoint (r 2 ) is available for control of y 1 Decentralized control (using sequential design): y 2 important in itself Partial control

156. 15 6 Partial control analysis Primary controlled variable y 1 = c (supervisory control layer) Local control of y 2 using u 2 (regulatory control layer) Setpoint y 2s : new DOF for supervisory control

157. 15 7 Partial control: Distillation Supervisory control: Primary controlled variables y 1 = c = (x D x B ) T Regulatory control: Control of y 2 =T using u 2 = L (original DOF) Setpoint y 2s = T s : new DOF for supervisory control u 1 = V

158. 15 8 Limitations of partial control? Cascade control: Closing of secondary loops does not by itself impose new problems –Theorem 10.2 (SP, 2005). The partially controlled system [P 1 P r1 ] from [u 1 r 2 ] to y 1 has no new RHP-zeros that are not present in the open-loop system [G 11 G 12 ] from [u 1 u 2 ] to y 1 provided r 2 is available for control of y 1 K 2 has no RHP-zeros Decentralized control (sequential design): Can introduce limitations. –Avoid pairing on negative RGA for u 2 /y 2 – otherwise P u likely has a RHP- zero BREAK

159. 15 9 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 ? (primary CV’s) (self-optimizing control) Step 4: Where set production rate? II Bottom Up Step 5: Regulatory control: What more to control (secondary CV’s) ? Step 6: Supervisory control Step 7: Real-time optimization Case studies

160. 16 0 Step 6. Supervisory control layer Purpose: Keep primary controlled outputs c=y 1 at optimal setpoints c s Degrees of freedom: Setpoints y 2s in reg.control layer Main structural issue: Decentralized or multivariable?

161. 16 1 Decentralized control (single-loop controllers) Use for: Noninteracting process and no change in active constraints +Tuning may be done on-line +No or minimal model requirements +Easy to fix and change -Need to determine pairing -Performance loss compared to multivariable control - Complicated logic required for reconfiguration when active constraints move

162. 16 2 Multivariable control (with explicit constraint handling = MPC) Use for: Interacting process and changes in active constraints +Easy handling of feedforward control +Easy handling of changing constraints no need for logic smooth transition -Requires multivariable dynamic model -Tuning may be difficult -Less transparent -“Everything goes down at the same time”

163. 16 3 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-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

164. 16 4 Step 7. Optimization layer (RTO) Purpose: Identify active constraints and compute optimal setpoints (to be implemented by supervisory control layer) Main structural issue: Do we need RTO? (or is process self- optimizing) RTO not needed when –Can “easily” identify change in active constraints (operating region) –For each operating region there exists self-optimizing var

165. 16 5 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-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 Conclusion / References

166. 16 6 Summary: Main steps 1.What should we control (y 1 =c=z)? Must define optimal operation! 2.Where should we set the production rate? At bottleneck 3.What more should we control (y 2 )? Variables that “stabilize” the plant 4.Control of primary variables Decentralized? Multivariable (MPC)?

167. 16 7 Conclusion Procedure plantwide control: I. Top-down analysis to identify degrees of freedom and primary controlled variables (look for self-optimizing variables) II. Bottom-up analysis to determine secondary controlled variables and structure of control system (pairing).

168. 16 8 More examples and case studies HDA process Cooling cycle Distillation (C3-splitter) Blending

169. 16 9 References Halvorsen, I.J, Skogestad, S., Morud, J.C., Alstad, V. (2003), “Optimal selection of controlled variables”, Ind.Eng.Chem.Res., 42, 3273-3284. Larsson, T. and S. Skogestad (2000), “Plantwide control: A review and a new design procedure”, Modeling, Identification and Control, 21, 209-240. Larsson, T., K. Hestetun, E. Hovland and S. Skogestad (2001), “Self-optimizing control of a large-scale plant: The Tennessee Eastman process’’, Ind.Eng.Chem.Res., 40, 4889-4901. Larsson, T., M.S. Govatsmark, S. Skogestad and C.C. Yu (2003), “Control of reactor, separator and recycle process’’, Ind.Eng.Chem.Res., 42, 1225-1234 Skogestad, S. and Postlethwaite, I. (1996, 2005), Multivariable feedback control, Wiley Skogestad, S. (2000). “Plantwide control: The search for the self-optimizing control structure”. J. Proc. Control 10, 487-507. Skogestad, S. (2003), ”Simple analytic rules for model reduction and PID controller tuning”, J. Proc. Control, 13, 291-309. Skogestad, S. (2004), “Control structure design for complete chemical plants”, Computers and Chemical Engineering, 28, 219-234. (Special issue from ESCAPE’12 Symposium, Haag, May 2002). … + more….. See home page of S. Skogestad: http://www.nt.ntnu.no/users/skoge/