1 PLANTWIDE CONTROL How to design the control system for a complete plant in a systematic manner Sigurd Skogestad Department of Chemical Engineering Norwegian University of Science and Tecnology (NTNU) Trondheim, Norway Singapore / Petronas / Petrobras, March 2010, India 2010

2 Plantwide control intro course: Contents Overview of plantwide control Selection of primary controlled variables based on economic : The link 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

3 c s = y 1s MPC PID y 2s RTO u (valves) Follow path (+ look after other variables) CV=y 1 (+ u) ; MV=y 2s Stabilize + avoid drift CV=y 2 ; MV=u Min J (economics); MV=y 1s OBJECTIVE Dealing with complexity Main simplification: Hierarchical decomposition Process control The controlled variables (CVs) interconnect the layers

4 Summary and references The following paper summarizes the procedure: –S. Skogestad, ``Control structure design for complete chemical plants'', Computers and Chemical Engineering, 28 (1-2), (2004).``Control structure design for complete chemical plants'', There are many approaches to plantwide control as discussed in the following review paper: –T. Larsson and S. Skogestad, ``Plantwide control: A review and a new design procedure'' Modeling, Identification and Control, 21, (2000).``Plantwide control: A review and a new design procedure''

5 S. Skogestad ``Plantwide control: the search for the self-optimizing control structure'', J. Proc. Control, 10, (2000).``Plantwide control: the search for the self-optimizing control structure'', S. Skogestad, ``Self-optimizing control: the missing link between steady-state optimization and control'', Comp.Chem.Engng., 24, (2000).``Self-optimizing control: the missing link between steady-state optimization and control'', I.J. Halvorsen, M. Serra and S. Skogestad, ``Evaluation of self-optimising control structures for an integrated Petlyuk distillation column'', Hung. J. of Ind.Chem., 28, (2000).``Evaluation of self-optimising control structures for an integrated Petlyuk distillation column'', T. Larsson, K. Hestetun, E. Hovland, and S. Skogestad, ``Self-Optimizing Control of a Large-Scale Plant: The Tennessee Eastman Process'', Ind. Eng. Chem. Res., 40 (22), (2001).``Self-Optimizing Control of a Large-Scale Plant: The Tennessee Eastman Process'', K.L. Wu, C.C. Yu, W.L. Luyben and S. Skogestad, ``Reactor/separator processes with recycles-2. Design for composition control'', Comp. Chem. Engng., 27 (3), (2003).``Reactor/separator processes with recycles-2. Design for composition control'', T. Larsson, M.S. Govatsmark, S. Skogestad, and C.C. Yu, ``Control structure selection for reactor, separator and recycle processes'', Ind. Eng. Chem. Res., 42 (6), (2003).``Control structure selection for reactor, separator and recycle processes'', A. Faanes and S. Skogestad, ``Buffer Tank Design for Acceptable Control Performance'', Ind. Eng. Chem. Res., 42 (10), (2003).``Buffer Tank Design for Acceptable Control Performance'', I.J. Halvorsen, S. Skogestad, J.C. Morud and V. Alstad, ``Optimal selection of controlled variables'', Ind. Eng. Chem. Res., 42 (14), (2003).``Optimal selection of controlled variables'', A. Faanes and S. Skogestad, ``pH-neutralization: integrated process and control design'', Computers and Chemical Engineering, 28 (8), (2004).``pH-neutralization: integrated process and control design'', S. Skogestad, ``Near-optimal operation by self-optimizing control: From process control to marathon running and business systems'', Computers and Chemical Engineering, 29 (1), (2004).``Near-optimal operation by self-optimizing control: From process control to marathon running and business systems'', E.S. Hori, S. Skogestad and V. Alstad, ``Perfect steady-state indirect control'', Ind.Eng.Chem.Res, 44 (4), (2005).``Perfect steady-state indirect control'', M.S. Govatsmark and S. Skogestad, ``Selection of controlled variables and robust setpoints'', Ind.Eng.Chem.Res, 44 (7), (2005).``Selection of controlled variables and robust setpoints'', V. Alstad and S. Skogestad, ``Null Space Method for Selecting Optimal Measurement Combinations as Controlled Variables'', Ind.Eng.Chem.Res, 46 (3), (2007).``Null Space Method for Selecting Optimal Measurement Combinations as Controlled Variables'', S. Skogestad, ``The dos and don'ts of distillation columns control'', Chemical Engineering Research and Design (Trans IChemE, Part A), 85 (A1), (2007).``The dos and don'ts of distillation columns control'', E.S. Hori and S. Skogestad, ``Selection of control structure and temperature location for two-product distillation columns'', Chemical Engineering Research and Design (Trans IChemE, Part A), 85 (A3), (2007).``Selection of control structure and temperature location for two-product distillation columns'', A.C.B. Araujo, M. Govatsmark and S. Skogestad, ``Application of plantwide control to the HDA process. I Steady-state and self- optimizing control'', Control Engineering Practice, 15, (2007).``Application of plantwide control to the HDA process. I Steady-state and self- optimizing control'', A.C.B. Araujo, E.S. Hori and S. Skogestad, ``Application of plantwide control to the HDA process. Part II Regulatory control'', Ind.Eng.Chem.Res, 46 (15), (2007).``Application of plantwide control to the HDA process. Part II Regulatory control'', V. Kariwala, S. Skogestad and J.F. Forbes, ``Reply to ``Further Theoretical results on Relative Gain Array for Norn-Bounded Uncertain systems'''' Ind.Eng.Chem.Res, 46 (24), 8290 (2007).``Reply to ``Further Theoretical results on Relative Gain Array for Norn-Bounded Uncertain systems'''' V. Lersbamrungsuk, T. Srinophakun, S. Narasimhan and S. Skogestad, ``Control structure design for optimal operation of heat exchanger networks'', AIChE J., 54 (1), (2008). DOI /aic.11366``Control structure design for optimal operation of heat exchanger networks'', T. Lid and S. Skogestad, ``Scaled steady state models for effective on-line applications'', Computers and Chemical Engineering, 32, (2008). T. Lid and S. Skogestad, ``Data reconciliation and optimal operation of a catalytic naphtha reformer'', Journal of Process Control, 18, (2008).``Scaled steady state models for effective on-line applications'', ``Data reconciliation and optimal operation of a catalytic naphtha reformer'', E.M.B. Aske, S. Strand and S. Skogestad, ``Coordinator MPC for maximizing plant throughput'', Computers and Chemical Engineering, 32, (2008).``Coordinator MPC for maximizing plant throughput'', A. Araujo and S. Skogestad, ``Control structure design for the ammonia synthesis process'', Computers and Chemical Engineering, 32 (12), (2008).``Control structure design for the ammonia synthesis process'', E.S. Hori and S. Skogestad, ``Selection of controlled variables: Maximum gain rule and combination of measurements'', Ind.Eng.Chem.Res, 47 (23), (2008).``Selection of controlled variables: Maximum gain rule and combination of measurements'', V. Alstad, S. Skogestad and E.S. Hori, ``Optimal measurement combinations as controlled variables'', Journal of Process Control, 19, (2009)``Optimal measurement combinations as controlled variables'', E.M.B. Aske and S. Skogestad, ``Consistent inventory control'', Ind.Eng.Chem.Res, 48 (44), (2009).``Consistent inventory control'',

6 Outline Control structure design (plantwide control) A procedure for control structure design I Top Down Step 1: Define operational objective (cost) and constraints Step 2: Identify degrees of freedom and optimizate for disturbances Step 3: What to control ? (primary CV’s) (self-optimizing control) Step 4: Where set the production rate? (Inventory control) 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

7 Main message 1. Control for economics (Top-down steady-state arguments) –Primary controlled variables c = y 1 : Control active constraints For remaining unconstrained degrees of freedom: Look for “self-optimizing” variables 2. Control for stabilization (Bottom-up; regulatory PID control) –Secondary controlled variables y 2 (“inner cascade loops”) Control variables which otherwise may “drift” Both cases: Control “sensitive” variables (with a large gain)!

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

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

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

11 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.

12 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

13 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 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

14 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!

15 Main objectives control system 1.Stabilization 2.Implementation of acceptable (near-optimal) operation ARE THESE OBJECTIVES CONFLICTING? Usually NOT –Different time scales Stabilization fast time scale –Stabilization doesn’t “use up” any degrees of freedom Reference value (setpoint) available for layer above But it “uses up” part of the time window (frequency range)

16 c s = y 1s MPC PID y 2s RTO u (valves) Follow path (+ look after other variables) CV=y 1 (+ u) ; MV=y 2s Stabilize + avoid drift CV=y 2 ; MV=u Min J (economics); MV=y 1s OBJECTIVE Dealing with complexity Main simplification: Hierarchical decomposition Process control The controlled variables (CVs) interconnect the layers

17 Example: Bicycle riding Note: design starts from the bottom Regulatory control: –First need to learn to stabilize the bicycle CV = y 2 = tilt of bike MV = body position Supervisory control: –Then need to follow the road. CV = y 1 = distance from right hand side MV=y 2s –Usually a constant setpoint policy is OK, e.g. y 1s =0.5 m Optimization: –Which road should you follow? –Temporary (discrete) changes in y 1s Hierarchical decomposition

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 Note: The regulatory should be independent of changes in overall objectives

19 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. Process industry: –Before : Many different “advanced” controllers, including feedforward, decouplers, overrides, cascades, selectors, Smith Predictors, etc. –Trend: Model predictive control (MPC) used as unifying tool. Structural issue: –What primary variables c=y 1 should we control?

20 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

21 Summary: The three layers Optimization layer (RTO; steady-state nonlinear model): Identifies active constraints and computes optimal setpoints for primary controlled variables (y 1 ). Supervisory control (MPC; linear model with constraints): Follow setpoints for y 1 (usually constant) by adjusting setpoints for secondary variables (MV=y 2s ) Look after other variables (e.g., avoid saturation for MV’s used in regulatory layer) Regulatory control (PID): Stabilizes the plant and avoids drift, in addition to following setpoints for y 2. MV=valves (u). Problem definition and overall control objectives (y 1, y 2 ) starts from the top. Design starts from the bottom. A good example is bicycle riding: Regulatory control: First you need to learn how to stabilize the bicycle (y 2 ) Supervisory control: Then you need to follow the road. Usually a constant setpoint policy is OK, for example, stay y 1s =0.5 m from the right hand side of the road (in this case the "magic" self-optimizing variable self-optimizing variable is y1=distance to right hand side of road)self-optimizing variable Optimization: Which road (route) should you follow?

22 Control structure design procedure I Top Down Step 1: Define operational objectives (optimal operation) –Cost function J (to be minimized) –Operational constraints Step 2: Identify degrees of freedom (MVs) and optimize for expected disturbances –Identify regions of active constraints Step 3: Select primary controlled variables c=y 1 (CVs) Step 4: Where set the production rate? (Inventory control) II Bottom Up Step 5: Regulatory / stabilizing control (PID layer) –What more to control (y 2 ; local CVs)? –Pairing of inputs and outputs Step 6: Supervisory control (MPC layer) Step 7: Real-time optimization (Do we need it?) Understanding and using this procedure is the most important part of this course!!!! y1y1 y2y2 Process MVs

23 Step 1. Define optimal operation (economics) What are we going to use our degrees of freedom u (MVs) for? Define scalar cost function J(u,x,d) –u: degrees of freedom (usually steady-state) –d: disturbances –x: states (internal variables) Typical cost function: Optimize operation with respect to u for given d (usually steady-state): min u J(u,x,d) subject to: Model equations: f(u,x,d) = 0 Operational constraints: g(u,x,d) < 0 J = cost feed + cost energy – value products

24 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

25 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 what to control”) Control: Operate at optimal trade-off (not obvious what to control to achieve this)

26 Comments optimal operation Do not forget to include feedrate as a degree of freedom!! –For LNG plant it may be optimal to have max. compressor power or max. compressor speed, and adjust feedrate of LNG –For paper machine it may be optimal to have max. drying and adjust the feedrate of paper (speed of the paper machine) to meet spec! Control at bottleneck –see later: “Where to set the production rate”

27 Step 2: (a) Identify degrees of freedom and (b) optimize for expected disturbances Optimization: Identify regions of active constraints Time consuming! 3 3 unconstrained degrees of freedom -> Find 3 CVs Control 3 active constraints

28 Active constraint regions for two distillation columns in series

29 Step 2a: Degrees of freedom (DOFs) for operation NOT as simple as one may think! To find all operational (dynamic) degrees of freedom: Count valves! (N valves ) “Valves” also includes adjustable compressor power, etc. Anything we can manipulate! BUT: not all these have a (steady-state) effect on the economics

30 Optimizer (RTO) PROCESS Supervisory controller (MPC) Regulatory controller (PID) H1H1 H2H2 CV 1 CV2 s y nyny d Stabilized process CV1 s CV 2 Physical inputs (valves) Optimally constant valves TPM unused valves Degrees of freedom for optimization (usually steady-state DOFs), MVopt = CV1s Degrees of freedom for supervisory control, MV1=CV2s + unused valves Physical degrees of freedom for stabilizing control, MV2 = valves (dynamic process inputs)

31 Steady-state degrees of freedom (DOFs) IMPORTANT! DETERMINES THE NUMBER OF VARIABLES TO CONTROL! No. of primary CVs = No. of steady-state DOFs CV = controlled variable (c) Methods to obtain no. of steady-state degrees of freedom (N ss ): 1.Equation-counting N ss = no. of variables – no. of equations/specifications Very difficult in practice 2.Valve-counting (easier!) N ss = N valves – N 0ss – N specs N 0ss = variables with no steady-state effect 3.Potential number for some units (useful for checking!) 4.Correct answer: Will eventually it when we perform optimization

32 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)

33 N valves = 6, N 0y = 2, N DOF,SS = 6 -2 = 4 (including feed and pressure as DOFs) Typical Distillation column N 0y : no. controlled variables (liquid levels) with no steady-state effect With given feed and pressure: NEED TO IDENTIFY 2 more CV’s - Typical: Top and btm composition

34 Heat-integrated distillation process

35 Heat-integrated distillation process

36 Heat exchanger with bypasses

37 Heat exchanger with bypasses

38 Steady-state degrees of freedom (N ss ): 3. Potential 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/speed) adiabatic flash tank: 0 * liquid phase reactor: 1 (holdup reactant) gas phase reactor: 0 * heat exchanger: 1 (bypass or flow) column (e.g. distillation) excluding heat exchangers: 0 * + no. of sidestreams pressure * : add 1DOF at each extra place you set pressure (using an extra valve, compressor or pump), e.g. in adiabatic flash tank, gas phase reactor or absorption column *Pressure is normally assumed to be given by the surrounding process and is then not a degree of freedom Ref: Araujo, Govatsmark and Skogestad (2007) Extension to closed cycles: Jensen and Skogestad (2009) Real number may be less, for example, if there is no bypass valve

39 Heat exchanger with bypasses

40 “Potential number”, N ss = 0 (distillation) + 1 (feed) + 2*1 (heat exchangers) + 1 (split) = 4 With given feed and pressure: N’ ss = 4 – 2 = 2 Distillation column (with feed and pressure as DOFs) split

41 Heat-integrated distillation process

42 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 )

43 HDA process: steady-state degrees of freedom Conclusion: 14 steady-state DOFs Assume given column pressures feed:1.2 hex: 3, 4, 6 splitter 5, 7 compressor: 8 distillation: 2 each column

44 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

45 QUIZ 1.Degrees of freedom (Dynamic, steady-state)? 2.Expected active constraints? 3.Proposed control structure? Heating Cooling Feed 51%A, 49%B Purge (mostly A, some B, trace C) Liquid Product (C) Flash Gas phase process (e.g. ammonia, methanol)

46 Step 3: 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 we adjust the degrees of freedom (u)? What should we control?

47 Solution I (“obvious”): Optimal feedforward Problem: UNREALISTIC! 1.Lack of measurements of d 2.Sensitive to model error

48 Solution II (”obvious”): Optimizing control Estimate d from measurements y and recompute u opt (d) Problem: COMPLICATED! Requires detailed model and description of uncertainty y

49 Solution III (in practice): FEEDBACK with hierarchical decomposition When disturbance d: Degrees of freedom (u) are updated indirectly to keep CVs at setpoints y CVs: link optimization and control layers

50 “Self-Optimizing Control” = Solution III with constant setpoints Self-optimizing control: Constant setpoints give acceptable loss y Issue: What should we control?

51 Formal Definition Self-optimizing control is said to occur when we can achieve an acceptable loss (in comparison with truly optimal operation) with constant setpoint values for the controlled variables without the need to reoptimize when disturbances occur. Reference: S. Skogestad, “Plantwide control: The search for the self-optimizing control structure'', Journal of Process Control, 10, (2000). Acceptable loss ) self-optimizing control Controller Process d u(d)u(d) c = f(y) cscs e n cmcm

52 How does self-optimizing control (solution III) work? When disturbances d occur, controlled variable c deviates from setpoint c s Feedback controller changes degree of freedom u to u FB (d) to keep c at c s Near-optimal operation / acceptable loss (self-optimizing control) is achieved if –u FB (d) ≈ u opt (d) –or more generally, J(u FB (d)) ≈ J(u opt (d)) Of course, variation of u FB (d) is different for different CVs c. We need to look for variables, for which J(u FB (d)) ≈ J(u opt (d)) or Loss = J(u FB (d)) - J(u opt (d)) is small

53 Remarks “self-optimizing control” 1. Old idea (Morari et al., 1980): “We want to find a function c of the process variables which when held constant, leads automatically to the optimal adjustments of the manipulated variables, and with it, the optimal operating conditions.” 2. “Self-optimizing control” = acceptable steady-state behavior (loss) with constant CVs. is similar to “Self-regulation” = acceptable dynamic behavior with constant MVs. 3. The ideal self-optimizing variable c is the gradient (c =  J/  u = J u ) –Keep gradient at zero for all disturbances (c = J u =0) –Problem: no measurement of gradient

54 Step 3. What should we control (c)? Simple examples What should we control?

55 –Cost to be minimized, J=T –One degree of freedom (u=power) –What should we control? Optimal operation - Runner Optimal operation of runner

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

57 2. Optimal operation of Marathon runner, J=T Optimal operation - Runner Self-optimizing control: Marathon (40 km)

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

59 Solution 2 Marathon – Feedback (Self-optimizing control) –What should we control? Optimal operation - Runner

60 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 Optimal operation - Runner Self-optimizing control: Marathon (40 km)

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

62 Example: Cake Baking Objective: Nice tasting cake with good texture u 1 = Heat input u 2 = Final time d 1 = oven specifications d 2 = oven door opening d 3 = ambient temperature d 4 = initial temperature y 1 = oven temperature y 2 = cake temperature y 3 = cake color Measurements Disturbances Degrees of Freedom

63 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)

64 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

65 Summary Step 3. What should we control (c)? c = H y y – available measurements (including u’s) H – selection of combination matrix What should we control? Equivalently: What should H be? 1.Control active constraints! 2.Unconstrained variables: Control self-optimizing variables!

66 1. CONTROL ACTIVE CONSTRAINTS!

67 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: 0 · D, B, L etc. · max Column capacity (flooding): V · V max, etc. value products cost energy (heating+ cooling) cost feed Example: Optimal operation distillation

68 Expected active constraints distillation 1.Valuable product: Purity spec. always active (if we get paid for impurity) –Reason: Amount of valuable product (D or B) should always be maximized Avoid product “give-away” (“Sell water as methanol”) Also saves energy Control implications valuable product: Control purity at spec. 2. “Cheap” product. May want to over-purify! Trade-off: –Yes, increased recovery of valuable product (less loss) –No, costs energy May give unconstrained optimum valuable product methanol + max. 5% water cheap product (byproduct) water + max. 2% methanol + water

69 Active constraint regions distillation No pay for impurity (p D =x*P D0 ): Can have no active constraints F pV Infeas. xDmax xBmax, Vmax (-1) No constraints. Overpurify D and B (2) xDmax (1) xDmax, xBmax (0) Vmax, (1) Vmax, xDmax (0) Region below here does not exist if pD is constant (“expensive product purity constraint xDmax always active”) pV=0

70 a)If constraint can be violated dynamically (only average matters) Required Back-off = “bias” (steady-state measurement error for c) b)If constraint cannot be violated dynamically (“hard constraint”) Required Back-off = “bias” + maximum dynamic control error J opt Back-off Loss c ≥ c constraint c J 1. CONTROL ACTIVE CONSTRAINTS! Active input constraints: Just set at MAX or MIN Active output constraints: Need back-off Want tight control of hard output constraints to reduce the back-off “Squeeze and shift”

71 Example. Optimal operation = max. throughput. Want tight bottleneck control to reduce backoff! Time Back-off = Lost production Rule for control of hard output constraints: “Squeeze and shift”! Reduce variance (“Squeeze”) and “shift” setpoint c s to reduce backoff

72 Hard Constraints: «SQUEEZE AND SHIFT» © Richalet SHIFT SQUEEZE

73 SUMMARY ACTIVE CONSTRAINTS –c constraint = value of active constraint –Implementation of active constraints is usually “obvious”, but may need “back-off” (safety limit) for hard output constraints C s = C constraint - backoff –Want tight control of hard output constraints to reduce the back-off “Squeeze and shift”

74 QUIZ (again) 1.Degrees of freedom (Dynamic, steady-state)? 2.Expected active constraints? (1. Feed given; 2. Feed free) 3.Proposed control structure? Heating Cooling Feed 51%A, 49%B Purge (mostly A, some B, trace C) Liquid Product (C) Flash Gas phase process (e.g. ammonia, methanol)

75 2. UNCONSTRAINED VARIABLES: - WHAT MORE SHOULD WE CONTROL? - WHAT ARE GOOD “SELF-OPTIMIZING” VARIABLES? Intuition: “Dominant variables” (Shinnar) Is there any systematic procedure? A. Sensitive variables: “Max. gain rule” (Gain= Minimum singular value) B. “Brute force” loss evaluation C. Optimal linear combination of measurements, c = Hy