Optimal dynamic operation of chemical processes: current opportunities

Transcription

1 Optimal dynamic operation of chemical processes: current opportunities James B. Rawlings Department of Chemical and Biological Engineering August 6, 29 Westhollow Research Center Shell Rawlings Current opportunities / 48

2 Outline Introductory overview to Model Predictive Control (MPC) 2 Industrial impact of these ideas 3 Recent developments Distributed MPC Large-scale systems and partial enumeration Tools for controller commissioning Optimizing economics Continuous time MPC 4 Conclusions and future outlook 5 Further Reading Rawlings Current opportunities 2 / 48

3 The model predictive control framework Measurement MH Estimate MPC control Forecast t time Reconcile the past Forecast the future sensors y actuators u Rawlings Current opportunities 3 / 48

4 Predictive control Measurement MH Estimate MPC control Forecast t time Reconcile the past Forecast the future sensors y actuators u min u(t) T y sp g(x, u) 2 Q + u sp u 2 R dt ẋ = f (x, u) x() = x (given) y = g(x, u) Rawlings Current opportunities 4 / 48

5 State estimation Measurement MH Estimate MPC control Forecast t time Reconcile the past Forecast the future sensors y actuators u min x,w(t) T y g(x, u) 2 R + ẋ f (x, u) 2 Q dt ẋ = f (x, u) + w (process noise) y = g(x, u) + v (measurement noise) Rawlings Current opportunities 5 / 48

6 Industrial practice of MPC Planning and Scheduling Steady State Optimization Validation Model Update Reconciliation Two layer structure Steady-state layer RTO optimizes steady state model Optimal setpoints passed to dynamic layer Controller Plant Rawlings Current opportunities 6 / 48

7 Industrial practice of MPC Planning and Scheduling Steady State Optimization Validation Controller Plant Model Update Reconciliation Two layer structure Steady-state layer RTO optimizes steady state model Optimal setpoints passed to dynamic layer Dynamic layer Controller tracks the setpoints Linear MPC (replaces multiloop PID) Rawlings Current opportunities 6 / 48

8 Large industrial success story! Linear MPC and ethylene manufacturing Number of MPC applications in ethylene: 8 to 2 Rawlings Current opportunities 7 / 48

9 Large industrial success story! Linear MPC and ethylene manufacturing Number of MPC applications in ethylene: 8 to 2 Credits 5 to 8 M$/yr (27) Rawlings Current opportunities 7 / 48

10 Large industrial success story! Linear MPC and ethylene manufacturing Number of MPC applications in ethylene: 8 to 2 Credits 5 to 8 M$/yr (27) Achieved primarily by increased on-spec product, decreased energy use Rawlings Current opportunities 7 / 48

11 Large industrial success story! Linear MPC and ethylene manufacturing Number of MPC applications in ethylene: 8 to 2 Credits 5 to 8 M$/yr (27) Achieved primarily by increased on-spec product, decreased energy use Eastman Chemical experience with MPC First MPC implemented in 996 Rawlings Current opportunities 7 / 48

12 Large industrial success story! Linear MPC and ethylene manufacturing Number of MPC applications in ethylene: 8 to 2 Credits 5 to 8 M$/yr (27) Achieved primarily by increased on-spec product, decreased energy use Eastman Chemical experience with MPC First MPC implemented in 996 Currently 55-6 MPC applications of varying complexity Rawlings Current opportunities 7 / 48

13 Large industrial success story! Linear MPC and ethylene manufacturing Number of MPC applications in ethylene: 8 to 2 Credits 5 to 8 M$/yr (27) Achieved primarily by increased on-spec product, decreased energy use Eastman Chemical experience with MPC First MPC implemented in 996 Currently 55-6 MPC applications of varying complexity 3-5 M$/year increased profit due to increased throughput (28) Rawlings Current opportunities 7 / 48

14 Large industrial success story! Linear MPC and ethylene manufacturing Number of MPC applications in ethylene: 8 to 2 Credits 5 to 8 M$/yr (27) Achieved primarily by increased on-spec product, decreased energy use Eastman Chemical experience with MPC First MPC implemented in 996 Currently 55-6 MPC applications of varying complexity 3-5 M$/year increased profit due to increased throughput (28) Praxair experience with MPC Praxair currently has more than 5 MPC installations Rawlings Current opportunities 7 / 48

15 Large industrial success story! Linear MPC and ethylene manufacturing Number of MPC applications in ethylene: 8 to 2 Credits 5 to 8 M$/yr (27) Achieved primarily by increased on-spec product, decreased energy use Eastman Chemical experience with MPC First MPC implemented in 996 Currently 55-6 MPC applications of varying complexity 3-5 M$/year increased profit due to increased throughput (28) Praxair experience with MPC Praxair currently has more than 5 MPC installations 6 M$/year increased profit (28) Rawlings Current opportunities 7 / 48

16 Impact for 3 ethylene plants (Starks and Arrieta, 27) Hydrocarbons AC&O 7 We re Doing it For the Money $6,, $6,, $5,, $5,, $4,, $4,, $3,, $3,, $2,, $2,, $,, $,, $ $ Q 2 3Q 2 Q 2 3Q 2 Q 22 3Q 22 Q 23 3Q 23 Q 24 3Q 24 Q 25 3Q 25 Q26 3Q26 Cumulative Quarterly Advanced Control & Optimization Rawlings Current opportunities 8 / 48

17 Maintaining technology (Starks and Arrieta, 27) Hydrocarbons AC&O Challenges (continued ) 26 Average DMC Service Factors (per plant) DMC Service Factor (%).% 8.% 6.% 4.% 2.%.% A B C D E F G H I J K L M Plant Plants L & M experienced APC personnel changes Advanced Control & Optimization Rawlings Current opportunities 9 / 48

18 Broader industrial impact (Qin and Badgwell, 23) Area Aspen Honeywell Adersa PCL MDC Total Technology Hi-Spec Refining Petrochemicals Chemicals Pulp and Paper Air & Gas Utility Mining/Metallurgy Food Processing Polymer Furnaces Aerospace/Defense Automotive Unclassified Total First App. DMC:985 PCT:984 IDCOM:973 PCL: SMOC: IDCOM-M:987 RMPCT:99 HIECON: OPC:987 Largest App 63x x85-3x2 - Rawlings Current opportunities / 48

19 Are all the problems solved? Some questions to consider Has the application base stopped growing? Rawlings Current opportunities / 48

20 Are all the problems solved? Some questions to consider Has the application base stopped growing? Is the technology mature? Rawlings Current opportunities / 48

21 Are all the problems solved? Some questions to consider Has the application base stopped growing? Is the technology mature? Is the theory complete? Rawlings Current opportunities / 48

22 Are all the problems solved? Some questions to consider Has the application base stopped growing? Is the technology mature? Is the theory complete? Do we have tools to decompose large-scale systems into manageable problems? Rawlings Current opportunities / 48

23 Are all the problems solved? Some questions to consider Has the application base stopped growing? Is the technology mature? Is the theory complete? Do we have tools to decompose large-scale systems into manageable problems? Do we have tools to optimize dynamic economic operation? Rawlings Current opportunities / 48

24 Are all the problems solved? Some questions to consider Has the application base stopped growing? Is the technology mature? Is the theory complete? Do we have tools to decompose large-scale systems into manageable problems? Do we have tools to optimize dynamic economic operation? Do we have tools to commission and maintain the controllers? Rawlings Current opportunities / 48

25 Are all the problems solved? Some questions to consider Has the application base stopped growing? Is the technology mature? Is the theory complete? Do we have tools to decompose large-scale systems into manageable problems? Do we have tools to optimize dynamic economic operation? Do we have tools to commission and maintain the controllers? Can we handle fast systems? Rawlings Current opportunities / 48

26 Has the application base stopped growing? European Commission DG information Society & Media Monitoring and control: today's market, its evolution till 22 and the impact of ICT on these Workshop: 9 th of October 28 A report & a presentation prepared by: Available for download: EC DG INFSO SMART Rawlings 27/47 Current opportunities 2 / 48

27 A report & a presentation prepared by: Has the application base stopped growing? Available for download: 23 EC DG INFSO SMART 27/47 Monitoring and control: today's market, its evolution till 22 and the impact of ICT on these WORKSHOP: Brussels the 9th of October Worldwide Monitoring & Control Market The worldwide market for Monitoring & Control products and solutions is around 88 billion euros. This represents 8% of total ICT expenditures worldwide. In the field of ICT, this is comparable to: the whole semiconductor industry world revenues; twice the world mobile phone manufacturers revenues. Total world: 87,9 bn 35% Services, with more than 5% of the market value, have the biggest share. The 3 larger sub markets represent together over billion euros, namely: integration, installation & training services with 38 billion euros; control hardware with 36 billion euros; maintenance, repair & overall services with 3 billion euros. 53% 2% Hardware Soft Services Definitions - bundle OS & drivers included in hardware - application development included in services The 3 larger application markets are Vehicles, Process and Manufacturing industries. Europe represents 32 % of the world total market value. A report & a presentation prepared by: Available for download: 24 Rawlings Current opportunities 3 / 48

28 Is the theory complete? 45 4 MPC 35 3 papers year year Rawlings Current opportunities 4 / 48

29 Is the theory complete? 4 2 MPC FB Control Chem Engr papers year year Rawlings Current opportunities 5 / 48

30 Ratio of MPC papers to feedback control papers.4.35 MPC/FB Control Rawlings Current opportunities 6 / 48

31 Distributed MPC Decomposing large-scale systems Rawlings Current opportunities 7 / 48

32 Decomposing large-scale systems Material flow Energy flow Rawlings Current opportunities 8 / 48

33 MPC at the large scale Decentralized Control Traditional approach Wealth of literature from the early 97 s on improved decentralized control a Well-known that poor performance may result if the interconnections are not negligible a (Sandell Jr. et al., 978; Šiljak, 99; Lunze, 992) Rawlings Current opportunities 9 / 48

34 MPC at the large scale Decentralized Control Traditional approach Wealth of literature from the early 97 s on improved decentralized control a Well-known that poor performance may result if the interconnections are not negligible a (Sandell Jr. et al., 978; Šiljak, 99; Lunze, 992) Centralized Control Steady increase in available computational power has provided the opportunity for centralized control Many practitioners view centralized control of large, networked systems as impractical and unrealistic Rawlings Current opportunities 9 / 48

35 Cooperative control as suboptimal MPC Properties established by suboptimal MPC theory Stability: Given a feasible initial condition, adding the stability constraint u i d i x i () gives nominal closed-loop stability for any number of information exchanges (Venkat et al., 26a) and (Venkat et al., 26b) Rawlings Current opportunities 2 / 48

36 Cooperative control as suboptimal MPC Properties established by suboptimal MPC theory Stability: Given a feasible initial condition, adding the stability constraint u i d i x i () gives nominal closed-loop stability for any number of information exchanges Cost decrease: Plant-wide objective is decreased at each iterate (Venkat et al., 26a) and (Venkat et al., 26b) Rawlings Current opportunities 2 / 48

37 Cooperative control as suboptimal MPC Properties established by suboptimal MPC theory Stability: Given a feasible initial condition, adding the stability constraint u i d i x i () gives nominal closed-loop stability for any number of information exchanges Cost decrease: Plant-wide objective is decreased at each iterate Convergence: Cooperative MPC produces centralized control performance at the limit of infinite iterates (Venkat et al., 26a) and (Venkat et al., 26b) Rawlings Current opportunities 2 / 48

38 Distributed MPC publications G. Pannocchia, S. J. Wright, B. T. Stewart, and J. B. Rawlings. Efficient cooperative distributed MPC using partial enumeration. In ADCHEM 29, International Symposium on Advanced Control of Chemical Processes, Istanbul, Turkey, July 2-5, 29. J. B. Rawlings and B. T. Stewart. Coordinating multiple optimization-based controllers: new opportunities and challenges. J. Proc. Cont., 8: , 28. B. T. Stewart, A. N. Venkat, J. B. Rawlings, S. J. Wright, and G. Pannocchia. Cooperative distributed model predictive control. Submitted for publication in Sys. Cont. Let., July 29. A. N. Venkat, J. B. Rawlings, and S. J. Wright. Stability and optimality of distributed, linear MPC. Part : state feedback. Technical Report 26 3, TWMCC, Department of Chemical and Biological Engineering, University of Wisconsin Madison (Available at October 26a. A. N. Venkat, J. B. Rawlings, and S. J. Wright. Stability and optimality of distributed, linear MPC. Part 2: output feedback. Technical Report 26 4, TWMCC, Department of Chemical and Biological Engineering, University of Wisconsin Madison (Available at October 26b. Rawlings Current opportunities 2 / 48

39 Large-scale systems and partial enumeration Unconstrained solution: LQ regulator (Kalman, 96) Constrained solution: MPC u = Kx u = K i x + b i in which i enumerates different possible active sets for the inequality constraints (Bemporad et al., 22) There are 3 mn different active sets u u u u. u 2. u. u u m u Rawlings Current opportunities 22 / 48

40 The active set table u = K i x + b i Rawlings Current opportunities 23 / 48

41 The active set table N = 2 u = K i x + b i i constraint set K i b i {u, u} u 2 {u, } u 3 {u, u} u 4 {, u} K 4 b 4 5 {, } K 5 b 5 6 {, u} K 6 b 6 7 {u, u} u 8 {u, } u 9 {u, u} u Rawlings Current opportunities 23 / 48

42 The active set table u = K i x + b i N = 2 N = 4 i constraint set K i b i {u, u} u 2 {u, } u 3 {u, u} u 4 {, u} K 4 b 4 5 {, } K 5 b 5 6 {, u} K 6 b 6 7 {u, u} u 8 {u, } u 9 {u, u} u i constraint set K i b i {u, u, u, u} u 2 {u, u, u, } u 3 {u, u, u, u} u 4 {,,, u} K 4 b 4 4 {,,, } K 4 b 4 42 {,,, u} K 42 b {u, u, u, u} u 8 {u, u, u, } u 8 {u, u, u, u} u Rawlings Current opportunities 23 / 48

43 The computational challenge of large scale centralized MPC On-line QP solution may not be practical for large-scale applications Rawlings Current opportunities 24 / 48

44 The computational challenge of large scale centralized MPC On-line QP solution may not be practical for large-scale applications Computation time may be too long Rawlings Current opportunities 24 / 48

45 The computational challenge of large scale centralized MPC On-line QP solution may not be practical for large-scale applications Computation time may be too long Full enumeration is not possible! Rawlings Current opportunities 24 / 48

46 The computational challenge of large scale centralized MPC On-line QP solution may not be practical for large-scale applications Computation time may be too long Full enumeration is not possible! Typical large-sized applications: m = 5, N = 5, 9 m = 32, N = 5, 229 Rawlings Current opportunities 24 / 48

47 The computational challenge of large scale centralized MPC On-line QP solution may not be practical for large-scale applications Computation time may be too long Full enumeration is not possible! Typical large-sized applications: m = 5, N = 5, 9 m = 32, N = 5, 229 Consider partial enumeration Rawlings Current opportunities 24 / 48

48 Partial Enumeration Conjecture: In practice, the state might visit relatively few different active constraints! SO... don t do the calculations for all possible active sets. Instead: Initialize by calculating and storing the table only for likely active sets; Given ˆx, search through the table and see if one of them is optimal; If so, use that control; If not, is there a feasible (suboptimal) solution? If so, use this one; If not, solve the MPC QP problem with small N and add this solution to the table; Make room in the table by deleting the entry that was used least recently. Rawlings Current opportunities 25 / 48

49 Initializing the Table Choose a size for the table, and initialize its contents in a training period. Add disturbances during training period to ensure that useful parts of the active-set space are sampled. Larger disturbances tend to cause saturation of some inputs. Saturation causes more active sets to be interesting because of small variations in the timepoint at which a constraint becomes active/inactive. Rawlings Current opportunities 26 / 48

50 Computational Results: Problem Copolymerization reactor: 5 inputs, 8 states, 4 outputs, horizon N = 5. 9 possible active sets. Rawlings Current opportunities 27 / 48

51 Computational Results: Problem Copolymerization reactor: 5 inputs, 8 states, 4 outputs, horizon N = 5. 9 possible active sets. Training period: 288 sample intervals (2 days, -minute intervals), 2 random disturbances applied. Visited a total of 376 active sets during this time. Validation period: 432 sample intervals (3 days), 3 random disturbances. Considered tables of sizes 5,, and 2. Rawlings Current opportunities 27 / 48

52 CPU time and suboptimality ratio of four solvers TAB5 TAB TAB2 qpsol Max time (s) Mean time (s) Active sets visited Suboptimality Rawlings Current opportunities 28 / 48

53 Partial enumeration publications G. Pannocchia, J. B. Rawlings, and S. J. Wright. Fast, large-scale model predictive control by partial enumeration. Automatica, 43:852 86, 27. G. Pannocchia, S. J. Wright, B. T. Stewart, and J. B. Rawlings. Efficient cooperative distributed MPC using partial enumeration. In ADCHEM 29, International Symposium on Advanced Control of Chemical Processes, Istanbul, Turkey, July 2-5, 29. Rawlings Current opportunities 29 / 48

54 Controller Commissioning Obtaining Q and R from Data Xf, Ff X D X X 2 Condenser y Model discretized with t k = k t: 2 3 d dt 6 4 X D. X B 7 5 {z } x(t)» y (t k ) = y 2 = F (x(t), u(t) ) {z} X f,f f» x(t k ) X N X B Reboiler Measurements are only X D, X B at the discretization times y 2 Rawlings Current opportunities 3 / 48

55 Controller Commissioning Obtaining Q and R from Data X D X X 2 Condenser y Model discretized with t k = k t: x k+ = f (x k, u k ) + g(x k, u k )w k [ ] [ ] y y 2 = x k k Xf, Ff w X N X B Reboiler Measurements are only X D, X B at the discretization times Noise w k affects all the states y 2 Rawlings Current opportunities 3 / 48

56 Controller Commissioning Obtaining Q and R from Data X D X X 2 Condenser v y Model discretized with t k = k t: x k+ = f (x k, u k ) + g(x k, u k )w k [ ] [ ] [ ] y v y 2 = x k + v 2 k k Xf, Ff w v 2 y 2 X N X B Reboiler Measurements are only X D, X B at the discretization times Noise w k affects all the states Noise v k corrupts the measurements Rawlings Current opportunities 3 / 48

57 Motivation for Using Autocovariances Xf, Ff X D X X 2 Condenser v y Idea of Autocovariances The state noise w k gets propagated in time The measurement noise v k appears only at the sampling times and is not propagated in time w X N X B Taking autocovariances of data at different time lags gives covariances of w k and v k v 2 Reboiler y 2 Rawlings Current opportunities 3 / 48

58 Motivation for Using Autocovariances Xf, Ff X D X X 2 Condenser v y Idea of Autocovariances The state noise w k gets propagated in time The measurement noise v k appears only at the sampling times and is not propagated in time w X N X B Taking autocovariances of data at different time lags gives covariances of w k and v k v 2 y 2 Reboiler Let w k, v k have zero means and covariances Q and R Rawlings Current opportunities 3 / 48

59 Mathematical Formulation of the ALS Linear State-Space Model: x k+ = Ax k + Gw k w k N(, Q) y k = Cx k + v k v k N(, R) Model (A, C, G) known from the linearization, finite set of measurements: {y,..., y k } given. Only unknowns are noises w k and v k. Rawlings Current opportunities 32 / 48

60 Mathematical Formulation of the ALS Linear State-Space Model: x k+ = Ax k + Gw k w k N(, Q) y k = Cx k + v k v k N(, R) Model (A, C, G) known from the linearization, finite set of measurements: {y,..., y k } given. Only unknowns are noises w k and v k. y k = Cx k y k+ = CAx k + CGw k y k+2 = CA 2 x k +CAGw k +CGw k+ Rawlings Current opportunities 32 / 48

61 Mathematical Formulation of the ALS Linear State-Space Model: x k+ = Ax k + Gw k w k N(, Q) y k = Cx k + v k v k N(, R) Model (A, C, G) known from the linearization, finite set of measurements: {y,..., y k } given. Only unknowns are noises w k and v k. y k = Cx k + v k y k+ = CAx k + CGw k + v k+ y k+2 = CA 2 x k +CAGw k +CGw k+ +v k+2 Rawlings Current opportunities 32 / 48

62 Mathematical Formulation of the ALS Linear State-Space Model: x k+ = Ax k + Gw k w k N(, Q) y k = Cx k + v k v k N(, R) Model (A, C, G) known from the linearization, finite set of measurements: {y,..., y k } given. Only unknowns are noises w k and v k. y k = Cx k + v k y k+ = CAx k + CGw k + v k+ E[y k y k ] = R y k+2 = CA 2 x k +CAGw k +CGw k+ +v k+2 Rawlings Current opportunities 32 / 48

63 Mathematical Formulation of the ALS Linear State-Space Model: x k+ = Ax k + Gw k w k N(, Q) y k = Cx k + v k v k N(, R) Model (A, C, G) known from the linearization, finite set of measurements: {y,..., y k } given. Only unknowns are noises w k and v k. y k = Cx k + v k y k+ = CAx k + CGw k + v k+ y k+2 = CA 2 x k +CAGw k +CGw k+ +v k+2 E[y k y k ] = R E[y k+2 y k+ ] = CAGQG C Rawlings Current opportunities 32 / 48

64 The Autocovariance Least-Squares (ALS) Problem Skipping a lot of algebra, we can write: Autocovariance Least Squares [ ] Φ = min Q,R A (Q)s N ˆb (R) s 2 A least-squares problem in a vector of unknowns, Q, R Rawlings Current opportunities 33 / 48

65 The Autocovariance Least-Squares (ALS) Problem Skipping a lot of algebra, we can write: Autocovariance Least Squares [ ] Φ = min Q,R A (Q)s N ˆb (R) s 2 A least-squares problem in a vector of unknowns, Q, R 2 Form A N from known system matrices Rawlings Current opportunities 33 / 48

66 The Autocovariance Least-Squares (ALS) Problem Skipping a lot of algebra, we can write: Autocovariance Least Squares [ ] Φ = min Q,R A (Q)s N ˆb (R) s 2 A least-squares problem in a vector of unknowns, Q, R 2 Form A N from known system matrices 3 ˆb is a vector containing the estimated correlations from data ˆb = y k y k T. k= y k+n y k s Rawlings Current opportunities 33 / 48

67 State estimation using (Q, R) from ALS Efficient software developed for ALS Rawlings Current opportunities 34 / 48

68 State estimation using (Q, R) from ALS Efficient software developed for ALS Data requirements for (Q, R) identification are reasonable Rawlings Current opportunities 34 / 48

69 State estimation using (Q, R) from ALS Efficient software developed for ALS Data requirements for (Q, R) identification are reasonable Industrial testing underway Rawlings Current opportunities 34 / 48

70 State estimation using (Q, R) from ALS Efficient software developed for ALS Data requirements for (Q, R) identification are reasonable Industrial testing underway Preliminary results from Eastman chemical cracking furnace and Aspentech evaluation are encouraging Rawlings Current opportunities 34 / 48

71 Distributed MPC publications B. J. Odelson, M. R. Rajamani, and J. B. Rawlings. A new autocovariance least-squares method for estimating noise covariances. Automatica, 42(2): 33 38, February 26. M. R. Rajamani and J. B. Rawlings. Estimation of the disturbance structure from data using semidefinite programming and optimal weighting. Automatica, 45: 42 48, 29. M. R. Rajamani, J. B. Rawlings, and S. J. Qin. Achieving state estimation equivalence for misassigned disturbances in offset-free model predictive control. AIChE J., 55(2):396 47, February 29. Rawlings Current opportunities 35 / 48

72 Optimizing economics: Current industrial practice Planning and Scheduling Steady State Optimization Validation Model Update Reconciliation Two layer structure Drawbacks Controller Plant Rawlings Current opportunities 36 / 48

73 Optimizing economics: Current industrial practice Planning and Scheduling Steady State Optimization Validation Controller Plant Model Update Reconciliation Two layer structure Drawbacks Inconsistent models Re-identify linear model as setpoint changes Time scale separation may not hold Economics unavailable in dynamic layer Rawlings Current opportunities 36 / 48

74 Optimizing economics: what s desirable? Profit -4-2 Input (u) State (x) Rawlings Current opportunities 37 / 48

75 Optimizing economics: what s desirable? Profit -4-2 Input (u) State (x) Rawlings Current opportunities 37 / 48

76 Economics MPC publications D. Angeli, R. Amrit, and J. B. Rawlings. Receding horizon cost optimization for overly constrained nonlinear plants. In Proceedings of the Conference on Decision and Control, Shanghai, China, December 29. J. B. Rawlings and R. Amrit. Optimizing process economic performance using model predictive control. In L. Magni, D. M. Raimondo, and F. Allgöwer, editors, Nonlinear Model Predictive Control, volume 384 of Lecture Notes in Control and Information Sciences, pages 9 38, Berlin, 29. Springer. L. Würth, J. B. Rawlings, and W. Marquardt. Economic dynamic real-time optimization and nonlinear model-predictive control on infinite horizons. In ADCHEM 29, International Symposium on Advanced Control of Chemical Processes, Istanbul, Turkey, July 2-5, 29. Rawlings Current opportunities 38 / 48

77 Continuous-time system and input parameterization We consider linear time-invariant continuous-time systems ẋ = Ax + Bu () V (x, u) = 2 ( x Qx + u Ru ) dt (2) u s u s s 2 u 2 2 Kx t t t 2 t 3 t N Rawlings Current opportunities 39 / 48

78 Relative error between CT and DT systems 2 P Π 2/ P ZOH FFOH PWLH Rawlings Current opportunities 4 / 48

79 Open-loop unstable system 7 u y t CLQR Sat-LQR t Decision Time t/t N Rawlings Current opportunities 4 / 48

80 System with slow and fast modes u 4 y t CLQR Sat-LQR t Decision Time t/t N Rawlings Current opportunities 42 / 48

81 Computation time for infinite horizon CT solution FFOH-CLQR Algorithm µ V II (x ) V (x ) V (x ) N CPU-time (ms) Rawlings Current opportunities 43 / 48

82 Continuous time MPC publications G. Pannocchia, J. B. Rawlings, D. Q. Mayne, and W. Marquardt. On computing the solutions to the continuous time constrained linear quadratic regulator, July 29a. Accepted for publication in IEEE Trans. Auto. Cont. G. Pannocchia, J. B. Rawlings, D. Q. Mayne, and W. Marquardt. Computation of the infinite horizon continuous time constrained linear quadratic regulator. In ADCHEM 29, International Symposium on Advanced Control of Chemical Processes, Istanbul, Turkey, July 2-5, 29b. Rawlings Current opportunities 44 / 48

83 Conclusions Optimal dynamic operation of chemical processes has undergone a total transformation in the last 2 years. Rawlings Current opportunities 45 / 48

84 Conclusions Optimal dynamic operation of chemical processes has undergone a total transformation in the last 2 years. Both in theory and in practice. Rawlings Current opportunities 45 / 48

85 Conclusions Optimal dynamic operation of chemical processes has undergone a total transformation in the last 2 years. Both in theory and in practice. The currently available theory splits the problem into state estimation and regulation. Rawlings Current opportunities 45 / 48

86 Conclusions Optimal dynamic operation of chemical processes has undergone a total transformation in the last 2 years. Both in theory and in practice. The currently available theory splits the problem into state estimation and regulation. Both are posed and solved as online optimization problems. Rawlings Current opportunities 45 / 48

87 Conclusions Optimal dynamic operation of chemical processes has undergone a total transformation in the last 2 years. Both in theory and in practice. The currently available theory splits the problem into state estimation and regulation. Both are posed and solved as online optimization problems. Basic properties have been established. Rawlings Current opportunities 45 / 48

88 Conclusions Optimal dynamic operation of chemical processes has undergone a total transformation in the last 2 years. Both in theory and in practice. The currently available theory splits the problem into state estimation and regulation. Both are posed and solved as online optimization problems. Basic properties have been established. The economic impact of the technology is large. Rawlings Current opportunities 45 / 48

89 Future Outlook Distributed MPC. Rawlings Current opportunities 46 / 48

90 Future Outlook Distributed MPC. Basic properties of cooperative MPC have been established. Ready for industrial testing. Rawlings Current opportunities 46 / 48

91 Future Outlook Distributed MPC. Basic properties of cooperative MPC have been established. Ready for industrial testing. 2 Partial enumeration. Rawlings Current opportunities 46 / 48

92 Future Outlook Distributed MPC. Basic properties of cooperative MPC have been established. Ready for industrial testing. 2 Partial enumeration. Tested on large examples. Ready for industrial testing. Rawlings Current opportunities 46 / 48

93 Future Outlook Distributed MPC. Basic properties of cooperative MPC have been established. Ready for industrial testing. 2 Partial enumeration. Tested on large examples. Ready for industrial testing. 3 Controller commissioning. Rawlings Current opportunities 46 / 48

94 Future Outlook Distributed MPC. Basic properties of cooperative MPC have been established. Ready for industrial testing. 2 Partial enumeration. Tested on large examples. Ready for industrial testing. 3 Controller commissioning. ALS properties established and efficient prototype software developed. Has had some industrial testing. Rawlings Current opportunities 46 / 48

95 Future Outlook Distributed MPC. Basic properties of cooperative MPC have been established. Ready for industrial testing. 2 Partial enumeration. Tested on large examples. Ready for industrial testing. 3 Controller commissioning. ALS properties established and efficient prototype software developed. Has had some industrial testing. 4 Optimizing dynamic economic objective. Rawlings Current opportunities 46 / 48

96 Future Outlook Distributed MPC. Basic properties of cooperative MPC have been established. Ready for industrial testing. 2 Partial enumeration. Tested on large examples. Ready for industrial testing. 3 Controller commissioning. ALS properties established and efficient prototype software developed. Has had some industrial testing. 4 Optimizing dynamic economic objective. Early stages. Theory finished for linear models and convex cost functions. Extensions to nonlinear models and nonconvex cost functions under study. Rawlings Current opportunities 46 / 48

97 Future Outlook Distributed MPC. Basic properties of cooperative MPC have been established. Ready for industrial testing. 2 Partial enumeration. Tested on large examples. Ready for industrial testing. 3 Controller commissioning. ALS properties established and efficient prototype software developed. Has had some industrial testing. 4 Optimizing dynamic economic objective. Early stages. Theory finished for linear models and convex cost functions. Extensions to nonlinear models and nonconvex cost functions under study. 5 Continuous time MPC. Rawlings Current opportunities 46 / 48

98 Future Outlook Distributed MPC. Basic properties of cooperative MPC have been established. Ready for industrial testing. 2 Partial enumeration. Tested on large examples. Ready for industrial testing. 3 Controller commissioning. ALS properties established and efficient prototype software developed. Has had some industrial testing. 4 Optimizing dynamic economic objective. Early stages. Theory finished for linear models and convex cost functions. Extensions to nonlinear models and nonconvex cost functions under study. 5 Continuous time MPC. Algorithm developed and tested on examples. Theory for convergence and closed-loop stability under development now. Rawlings Current opportunities 46 / 48

99 Acknowledgments Don Bartusiak, ExxonMobil Tom Badgwell, Aspentech Jim Downs, Eastman Chemical Larry Megan, Praxair Rahul Bindlish, Dow Financial support from NSF #CTS-82536, and Texas Wisconsin California Control Consortium (TWCCC) members Rawlings Current opportunities 47 / 48

100 Further reading A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos. The explicit linear quadratic regulator for constrained systems. Automatica, 38():3 2, 22. R. E. Kalman. Contributions to the theory of optimal control. Bull. Soc. Math. Mex., 5:2 9, 96. J. Lunze. Feedback Control of Large Scale Systems. Prentice-Hall, London, U.K., 992. S. J. Qin and T. A. Badgwell. A survey of industrial model predictive control technology. Control Eng. Prac., (7): , 23. N. R. Sandell Jr., P. Varaiya, M. Athans, and M. Safonov. Survey of decentralized control methods for larger scale systems. IEEE Trans. Auto. Cont., 23(2):8 28, 978. D. D. Šiljak. Decentralized Control of Complex Systems. Academic Press, London, 99. ISBN D. M. Starks and E. Arrieta. Maintaining AC&O applications, sustaining the gain. In Proceedings of National AIChE Spring Meeting, Houston, Texas, April 27. Rawlings Current opportunities 48 / 48