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

2. 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 ? (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

3. 3 Idealized view of control (“Ph.D. control”)

4. 4 Practice: Tennessee Eastman challenge problem (Downs, 1991) (“PID control”)

5. 5 Idealized view II: Optimizing control

6. 6 Practice II: Hierarchical decomposition with separate layers What should we control?

7. 7 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

8. 8 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?

9. 9 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

10. 10 Step 1. Degrees of freedom (DOFs) m – dynamic (control) degrees of freedom = valves u 0 – steady-state degrees of freedom N m : no. of dynamic (control) DOFs (valves) N u0 = N m - N 0 : no. of steady-state DOFs –N 0 = N 0y + N 0m : no. of variables with no steady-state effect N 0m : no. purely dynamic control DOFs N 0y : no. controlled variables (liquid levels) with no steady-state effect Cost J depends normally only on steady-state DOFs

11. 11 N m =, N 0y =, N u0 = Distillation column with given feed

12. 12 Heat-integrated distillation process

13. 13 Heat exchanger with bypasses

14. 14 Typical number of steady-state degrees of freedom (u 0 ) 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: 1 (0 with fixed pressure) liquid phase reactor: 1 (volume) gas phase reactor: 1 (0 with fixed pressure) heat exchanger: 1 (duty or net area) distillation column excluding heat exchangers: 1 (0 with fixed pressure) + number of sidestreams

15. 15 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 )

16. 16 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

17. 17 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

18. 18 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 u0 J(u 0,x,d) subject to: Model equations: f(u 0,x,d) = 0 Operational constraints: g(u 0,x,d) < 0 J = cost feed + cost energy – value products

19. 19 Optimal operation distillation column Steady-state DOFs (given p and F): 2, for example L/D 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

20. 20 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

21. 21 Step 3. What should we control (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

22. 22 Implementation of optimal operation Optimal operation for given d * : min u0 J(u 0,x,d) subject to: Model equations: f(u 0,x,d) = 0 Operational constraints: g(u 0,x,d) < 0 → u 0opt (d * ) Problem: Cannot keep u 0opt constant because disturbances d change Distillation: Steady-state degrees of freedom u 0 could be L/D and V. Cannot keep these constant

23. 23 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 0opt (d)

24. 24 Implementation of optimal operation (Cannot keep u 0opt constant) Simpler solution: Look for another variable c which is better to keep constant and control c at constant setpoint c s = c opt (d*) Note: u 0 will indirectly change when d changes

25. 25 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 0,d) = 0 CONTROL ACTIVE CONSTRAINTS! –c s = value of active constraint –Implementation of active constraints is usually simple. WHAT MORE SHOULD WE CONTROL? –Find variables c for remaining unconstrained degrees of freedom u.

26. 26 Solution to optimal operation distillation Cost to be minimized J = - P where P= p D D + p B B – p F F – p V V Optimal operation: Minimize J with respect to DOFs General: Optimal solution with N DOFs: –N – N u DOFs used to satisfy “active” constraints ( · is =) –N u remaining unconstrained variables Usually: N u zero or small Distillation at steady state with given p and F: N=2 DOFs. Three cases: 1.N u =0: Two active constraints (for example, x D, impurity = max. x B, impurity = max, “TWO-POINT” CONTROL) 2.N u =1: One constraint active 3.N u =2: No constraints active very unlikely unless there are no purity specifications (e.g. byproducts or recycle)

27. 27 Remaining unconstrained variables: What should we control? (primary controlled variables y 1 =c) Intuition: “Dominant variables” (Shinnar) Systematic: Minimize cost J(u 0,d * ) w.r.t. DOFs u 0. 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)

28. 28 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

29. 29 The difficult unconstrained variables Cost J Selected controlled variable (remaining unconstrained) c opt J opt c

30. 30 Implementation of unconstrained variables is not trivial: How do we deal with uncertainty? 1. Disturbances d (c opt (d) changes) 2. Implementation error n (actual c ≠ c opt ) c s = c opt (d * ) – nominal optimization n c = c s + n d Cost J  J opt (d)

31. 31 Problem no. 1: Disturbance d Cost J Controlled variable c opt (d * ) J opt d*d*d*d* d ≠ d * Loss with constant value for c ) Want c opt independent of d

32. 32 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

33. 33 Problem no. 2: Implementation error n Cost J c s =c opt (d * ) J opt d*d*d*d* Loss due to implementation error for c c = c s + n ) Want n small and ”flat” optimum

34. 34 Effect of implementation error on cost (“problem 2”) BADGood

35. 35 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

36. 36 Summary unconstrained variables: Which variable c to 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

37. 37 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)

38. 38 Self-optimizing Control – Marathon Optimal operation of Marathon runner, J=T –Any self-optimizing variable c (to control at constant setpoint)?

39. 39 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

40. 40 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 (Problem: Impl.error n)

41. 41 Self-optimizing Control – Sprinter Optimal operation of Sprinter (100 m), J=T –Active constraint control: Maximum speed (”no thinking required”)

42. 42 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

43. 43 Mathematical: Local analysis u cost J u opt

44. 44 Unconstrained degrees of freedom: Looking for “magic” variables to keep at constant setpoints. What properties do they have? Skogestad and Postlethwaite (1996): c-c opt should be small: The optimal value of c should be insensitive to disturbances c should be easy to measure and control accurately G -1 should be small, i.e  (G) should be large: The value of c should be sensitive to changes in the steady- state 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. Summarized by maximum gain (min. singular value) rule Avoid problem 1 (d) Avoid problem 2 (n)

45. 45 Unconstrained degrees of freedom: Looking for “magic” variables to keep at constant setpoints. How can we find them systematically? A. Maximum gain (minimum singular value) rule: B. “Brute force”: Consider available measurements y, and evaluate loss when they are kept constant: C. More general: Find optimal linear combination (matrix H):

46. 46 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: A. Maximum gain rule Want the slope (= gain G from u to y) as large as possible

47. 47 Unconstrained degrees of freedom: A. Maximum gain (minimum singular value) rule “Maximum gain rule” (Skogestad and Postlethwaite, 1996): Look for variables c that maximize the minimum singular value  ( G) of the appropriately scaled steady-state gain matrix G from u to c u: unconstrained degrees of freedom Loss Scaling is important: –Scale c such that their expected variation is similar (divide by “span”= optimal variation + noise) –Scale inputs u such that they have similar effect on cost J (J uu unitary)  (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).

48. 48 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

49. 49 What should we control? Rule: Maximize the scaled gain Scalar case. Minimum singular value = gain |G| Maximize scaled gain: |G s | = |G| / span –|G|: gain from independent variable (u) to candidate controlled variable (c) –span (of c) = optimal variation in c + control error for c

50. 50 Generally (more than unconstrained variable): Scaling for “maximum gain (  rule” “Control active constraints” and look at the remaining unconstrained problem Candidate outputs: Divide by span = optimal range + implementation error Candidate inputs: –A unit deviation in each input should have the same effect on the cost function (i.e. J uu should be constant times identity) –Alternatively (often simpler), consider  (J uu 1/2 G)

51. 51 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

52. 52 B. Brute-force procedure Define optimal operation: Minimize cost function J Each candidate variable c: With constant setpoints c s compute loss L for expected disturbances d and implementation errors n Select variable c with smallest loss Acceptable loss ) self-optimizing control

53. 53 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)

54. 54 Alstad-method continued To handle implementation error: Use “sensitive” measurements, with information about all independent variables (u and d)

55. 55 Toy Example

56. 56 Toy Example

57. 57 Toy Example

58. 58 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

59. 59 Recycle plant: Optimal operation mTmT 1 remaining unconstrained degree of freedom

60. 60 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

61. 61 Luyben rule Luyben rule (to avoid snowballing): “Fix a stream in the recycle loop” (F or D)

62. 62 Luyben rule: D constant Luyben rule (to avoid snowballing): “Fix a stream in the recycle loop” (F or D) LC XC

63. 63 A. Maximum gain rule: Steady-state gain Luyben rule: Not promising economically Conventional: Looks good

64. 64 B. “Brute force” loss evaluation: Disturbance in F 0 Loss with nominally optimal setpoints for M r, x B and c Luyben rule: Conventional

65. 65 B. “Brute force” loss evaluation: Implementation error Loss with nominally optimal setpoints for M r, x B and c Luyben rule:

66. 66 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 )

67. 67 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

68. 68 Recycle systems: Do not recommend Luyben’s rule of fixing a flow in each recycle loop (even to avoid “snowballing”)

69. 69 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

70. 70 Selection of controlled variables More examples and exercises HDA process Cooling cycle Distillation (C3-splitter) Blending

71. 71 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

72. 72 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

73. 73 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

74. 74 Production rate set at inlet : Inventory control in direction of flow

75. 75 Production rate set at outlet: Inventory control opposite flow

76. 76 Production rate set inside process

77. 77 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

78. 78 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: 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.

79. 79 Reactor-recycle process: Given feedrate with production rate set at inlet

80. 80 Reactor-recycle process: Given feedrate with production rate set at inlet Bottleneck: max. vapor rate in column

81. 81 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

82. 82 Alt.2 “long loop” MAX Reactor-recycle process with increased feedrate: Optimal: Set production rate at bottleneck

83. 83 Reactor-recycle process with increased feedrate: Optimal: Set production rate at bottleneck MAX Alt.3: reconfigure

84. 84 Reactor-recycle process: Given feedrate with production rate set at bottleneck F 0s Alt.3: reconfigure (permanently)

85. 85 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)

86. 86 Conclusion production rate manipulator Think carefully about where to place it! Difficult to undo later

87. 87 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

88. 88 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)

89. 89 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: –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)

90. 90 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 = ?

91. 91 XCXC TC FC ysys y LsLs TsTs L T z XCXC Cascade control distillation

92. 92 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:

93. 93 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

94. 94 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)

95. 95 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

96. 96 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

97. 97 What should we control (y 2 )? Rule: Maximize the scaled gain Scalar case. Minimum singular value = gain |G| Maximize scaled gain: |G s | = |G| / span –|G|: gain from independent variable (u) to candidate controlled variable (y) –span (of y) = optimal variation in c + control error for y Note optimal variation: This is generally NOT the same as the optimal variation used for selecting primary controlled variables (c) because we at the “fast” regulatory time scale have that the “slower” (unused) inputs (u 1 ) are constant. –For each disturbance find optimal input change (u 2 ) that keeps primary variables c=y 1 closest to setpoint (with u 1 constant), and from this obtain optimal variation (c opt )

98. 98 2. “y 2 is easy to control” Main rule: y 2 is easy to measure and located close to manipulated variable u 2 Statics: Want large gain (from u 2 to y 2 ) Dynamics: Want small effective delay (from u 2 to y 2 )

99. 99 Aside: Effective delay and tunings 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!

100. 10 0 Summary: Rules for selecting y 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 Additional rule for selecting u 2 : Avoid using inputs u 2 that may saturate (should generally avoid saturation in lower layers)

101. 10 1 Exercise level control (10.1 in Seborg) Consider control of liquid condenser level (M1) in a distillation column. –Should we use L or D to control M1? –What is your answer if L/D=5? –What is your answer if it is important to avoid disturbing downstream process?

102. 10 2 XCXC TC FC ysys y LsLs TsTs L T z XCXC LV-configuration preferred in practice for most columns (even when L and V are large)

103. 10 3 Exercise distillation (10.3 in Seborg) Consider a column design to separate methanol and water. The desired distillate composition (xD) is methanol with only 5 ppm of water. Because a composition analyzer is not available, it is proposed to control XD indirectly by measuring and controlling a temperature at one of the following locations (a)the reflux stream (b)the top tray (stage) of the column (c)an intermediate tray in the top section (where?) Discuss the relative advantages and disadvantages of each choice based on both steady-state and dynamic considerations

104. 10 4 More exact: Selecting stage for temperature control (y 2 ) in distillation Consider controlling temperature (y=T) with reflux (u=L) –Avoid V because it may saturate (i.e. reach maximum) G= “Controllable range”: Given by gain  T/  L (“sensitivity”). This will be largest where temperature changes are the largest, that is, toward the feed stage for a binary separation T opt = “Optimal variation” for disturbances (in F, z F, etc.): –This is generally NOT the same as the optimal variation used for selecting primary controlled variables (c) because we have to assume that the “slower” variables (e.g. u 1 =V) are constant. –Here: We want to control product compositions. The optimal variation in T will be small towards the end of the column and largest toward the feed stage T err = “Control error” (=implementation error = static measurement error) – This will typically be 0.5 C and the same on stages Rule: Select location corresponding to max. value of –G / (T opt + T err ) Binary separation: This will have a max. in the middle of the top section and middle of bottom section Multicomponent separation: Will have a max. towards column ends

105. 10 5 Distillation example columnn A. u=L. Which temperature (c) to control? 1.Nominal simulation (gives x0) 2.One simulation: Gain with constant inputs (u) –Make small change in input (L) with the other inputs (V) constant Find gain =  x i /  L 3.From process gains: Obtain “optimal” change for candidate measurements to disturbances, (with input L adjusted to keep both xD and xB close to setpoint, i.e. min ||x D -0.99; xb-0.01||, with V=constant). w=[xB,xD]. Then (- Gy pinv(Gw) Gwd + Gyd) d (Note: Gy=gain, d=F: Gyd=gainf etc.) Find  x i,opt (yopt) for the following disturbances 1.F (from 1 to 1.2; has effect!)yoptf 2.z F from 0.5 to 0.6yoptz 3.q F from 1 to 1.2yoptq 4.x B from 0.01 to 0.02yoptxb=0 (no effect) 4.Control (implementation) error. Assume=0.05 on all stages 5.Find scaled-gain = gain/span where span = abs(yoptf)+abs(yoptz)+abs(yoptq)+0.05 “Maximize gain rule”: Prefer stage where scaled-gain is large >> res=[x0' gain' gainf' gainz' gainq'] res = 0.0100 1.0853 0.5862 1.1195 1.0930 0.0143 1.5396 0.8311 1.5893 1.5506 0.0197 2.1134 1.1397 2.1842 2.1285 0.0267 2.8310 1.5247 2.9303 2.8513 0.0355 3.7170 1.9990 3.8539 3.7437 0.0467 4.7929 2.5734 4.9789 4.8273 0.0605 6.0718 3.2542 6.3205 6.1155 0.0776 7.5510 4.0392 7.8779 7.6055 0.0982 9.2028 4.9126 9.6244 9.2694 0.1229 10.9654 5.8404 11.4975 11.0449 0.1515 12.7374 6.7676 13.3931 12.8301 0.1841 14.3808 7.6202 15.1675 14.4857 0.2202 15.7358 8.3133 16.6530 15.8509 0.2587 16.6486 8.7659 17.6861 16.7709 0.2986 17.0057 8.9194 18.1436 17.1311 0.3385 16.7628 8.7524 17.9737 16.8870 0.3770 15.9568 8.2872 17.2098 16.0758 0.4130 14.6955 7.5832 15.9597 14.8058 0.4455 13.1286 6.7215 14.3776 13.2280 0.4742 11.4148 5.7872 12.6286 11.5021 0.4987 9.6930 4.8542 10.8586 9.7680 0.5265 11.0449 5.4538 12.1996 11.1051 0.5578 12.2975 6.0007 13.4225 12.3410 0.5922 13.3469 6.4473 14.4211 13.3727 0.6290 14.0910 6.7478 15.0929 14.0989 0.6675 14.4487 6.8669 15.3588 14.4397 0.7065 14.3766 6.7871 15.1800 14.3528 0.7449 13.8797 6.5135 14.5676 13.8438 0.7816 13.0098 6.0724 13.5807 12.9654 0.8158 11.8545 5.5060 12.3136 11.8051 0.8469 10.5187 4.8635 10.8764 10.4677 0.8744 9.1066 4.1929 9.3765 9.0566 0.8983 7.7071 3.5346 7.9044 7.6602 0.9187 6.3866 2.9184 6.5261 6.3443 0.9358 5.1876 2.3624 5.2829 5.1505 0.9501 4.1314 1.8756 4.1942 4.1000 0.9617 3.2233 1.4592 3.2631 3.1975 0.9712 2.4574 1.1098 2.4816 2.4368 0.9789 1.8211 0.8208 1.8355 1.8053 0.9851 1.2990 0.5848 1.3077 1.2875 0.9900 0.8747 0.3938 0.8805 0.8670

106. 10 6 >> Gw Gw = 1.0853 0.8747 Gwf = 0.5862 0.3938 >> yoptf = (-gain'*pinv(Gw)*Gwf + gainf')*0.2 >> yoptq = (-gain'*pinv(Gw)*Gwq + gainq')*0.2 >> yoptz = (-gain'*pinv(Gw)*Gwz + gainz')*0.1 have checked with nonlinear simulation. OK! >> [yoptf yoptq yoptz span] ans = 0.0077 0.0014 0.0011 0.0601 0.0108 0.0019 0.0016 0.0644 0.0146 0.0027 0.0025 0.0698 0.0192 0.0036 0.0038 0.0765 0.0246 0.0047 0.0056 0.0849 0.0309 0.0061 0.0082 0.0951 0.0379 0.0077 0.0117 0.1073 0.0456 0.0096 0.0163 0.1215 0.0536 0.0118 0.0222 0.1375 0.0612 0.0141 0.0294 0.1546 0.0678 0.0164 0.0379 0.1720 0.0724 0.0186 0.0474 0.1884 0.0742 0.0204 0.0575 0.2022 0.0726 0.0217 0.0676 0.2119 0.0673 0.0222 0.0769 0.2164 0.0584 0.0220 0.0847 0.2151 0.0467 0.0211 0.0906 0.2085 0.0332 0.0196 0.0945 0.1973 0.0190 0.0177 0.0964 0.1831 0.0052 0.0155 0.0966 0.1673 -0.0076 0.0134 0.0955 0.1665 -0.0242 0.0102 0.0915 0.1758 -0.0412 0.0066 0.0858 0.1837 -0.0578 0.0029 0.0784 0.1892 -0.0728 -0.0008 0.0696 0.1932 -0.0851 -0.0042 0.0596 0.1990 -0.0938 -0.0072 0.0491 0.2001 -0.0984 -0.0095 0.0387 0.1965 -0.0988 -0.0111 0.0288 0.1887 -0.0954 -0.0119 0.0202 0.1775 -0.0891 -0.0120 0.0129 0.1640 -0.0807 -0.0115 0.0072 0.1494 -0.0711 -0.0107 0.0030 0.1347 -0.0610 -0.0095 0.0001 0.1206 -0.0512 -0.0083 -0.0017 0.1112 -0.0419 -0.0070 -0.0027 0.1016 -0.0335 -0.0057 -0.0030 0.0923 -0.0261 -0.0045 -0.0029 0.0835 -0.0197 -0.0035 -0.0025 0.0756 -0.0142 -0.0025 -0.0020 0.0686 -0.0095 -0.0017 -0.0013 0.0626 u=L

107. 10 7 >> span = abs(yoptf)+abs(yoptz)+abs(yoptq)+0.05; >> plot(gain’./span) u=L: max. on stage 15 (below feed stage) In practice (because of dynamics): Will use tray in top (peak at 25)

108. 10 8 variation=yoptz+yoptq+yoptxb+0.05 plot(gain./variation) TOP stage 31 NOT QUITE CORRECT: Using scalings from selection of primary variables

109. 10 9 Another tool 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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112. 11 2 Example: Tennessee Eastman challenge problem

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120. 12 0 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

121. 12 1 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

122. 12 2 Limitations of partial control? Cascade control: Closing of secondary loops does not by itself impose new problems –Theorem 10.2. 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

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129. 12 9 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

130. 13 0 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.

131. 13 1 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

132. 13 2 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”)

133. 13 3 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

134. 13 4 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

135. 13 5 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

136. 13 6 Exercise: Tuning cascade

137. 13 7 Extra inputs Exercise: Explain how “valve position control” fits into this framework. As en example consider a heat exchanger with bypass

138. 13 8 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

139. 13 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

140. 14 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?

141. 14 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

142. 14 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”

143. 14 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

144. 14 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

145. 14 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

146. 14 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)?

147. 14 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).

148. 14 8 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), 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.chembio.ntnu.no/users/skoge/