Coordinating multiple optimization-based controllers: new opportunities and challenges

Transcription

1 Coordinating multiple optimization-based controllers: new opportunities and challenges James B. Rawlings and Brett T. Stewart Department of Chemical and Biological Engineering University of Wisconsin Madison October 15, 2008 Rawlings and Stewart Multiple MPCs: Status and Future 1 / 32

2 Outline 1 Introduction 2 Game theory results 3 Distributed control example 4 Current Challenge: Coupled constraints 5 Conclusions Rawlings and Stewart Multiple MPCs: Status and Future 2 / 32

3 Electrical power distribution Rawlings and Stewart Multiple MPCs: Status and Future 3 / 32

4 Chemical plant integration Material flow Energy flow Rawlings and Stewart Multiple MPCs: Status and Future 4 / 32

5 MPC at the large scale Decentralized Control Most large-scale systems consist of networks of interconnected/interacting subsystems Chemical plants, electrical power grids, water distribution networks,... Rawlings and Stewart Multiple MPCs: Status and Future 5 / 32

6 MPC at the large scale Decentralized Control Most large-scale systems consist of networks of interconnected/interacting subsystems Chemical plants, electrical power grids, water distribution networks,... Traditional approach: Decentralized control Wealth of literature from the early 1970 s on improved decentralized control a Well-known that poor performance may result if the interconnections are not negligible a [Sandell Jr. et al., 1978, Siljak, 1991, Lunze, 1992] Rawlings and Stewart Multiple MPCs: Status and Future 5 / 32

7 MPC at the large scale Centralized Control Steady increase in available computational power has provided the opportunity for centralized control Most practitioners view centralized control of large, networked systems as impractical and unrealistic A divide and conquer strategy is essential for control of large, networked systems [Ho, 2005] Centralized control: A benchmark control framework for comparing and assessing other control formulations Rawlings and Stewart Multiple MPCs: Status and Future 6 / 32

8 Nomenclature: consider two interacting units Objective functions Φ 1 (u 1, u 2 ), Φ 2 (u 1, u 2 ) and Φ(u 1, u 2 ) = w 1 Φ 1 (u 1, u 2 ) + w 2 Φ 2 (u 1, u 2 ) decision variables for units u 1 Ω 1, u 2 Ω 2 Rawlings and Stewart Multiple MPCs: Status and Future 7 / 32

9 Nomenclature: consider two interacting units Objective functions Φ 1 (u 1, u 2 ), Φ 2 (u 1, u 2 ) and Φ(u 1, u 2 ) = w 1 Φ 1 (u 1, u 2 ) + w 2 Φ 2 (u 1, u 2 ) decision variables for units u 1 Ω 1, u 2 Ω 2 Decentralized Control min Φ1 (u 1 ) u 1 Ω 1 min Φ2 (u 2 ) u 2 Ω 2 Rawlings and Stewart Multiple MPCs: Status and Future 7 / 32

10 Nomenclature: consider two interacting units Objective functions Φ 1 (u 1, u 2 ), Φ 2 (u 1, u 2 ) and Φ(u 1, u 2 ) = w 1 Φ 1 (u 1, u 2 ) + w 2 Φ 2 (u 1, u 2 ) decision variables for units u 1 Ω 1, u 2 Ω 2 Decentralized Control Noncooperative Control (Nash equilibrium) min Φ1 (u 1 ) u 1 Ω 1 min Φ 1 (u 1, u 2 ) u 1 Ω 1 min Φ2 (u 2 ) u 2 Ω 2 min Φ 2 (u 1, u 2 ) u 2 Ω 2 Rawlings and Stewart Multiple MPCs: Status and Future 7 / 32

11 Nomenclature: consider two interacting units Objective functions Φ 1 (u 1, u 2 ), Φ 2 (u 1, u 2 ) and Φ(u 1, u 2 ) = w 1 Φ 1 (u 1, u 2 ) + w 2 Φ 2 (u 1, u 2 ) decision variables for units u 1 Ω 1, u 2 Ω 2 Decentralized Control Noncooperative Control (Nash equilibrium) Cooperative Control (Pareto optimal) min Φ1 (u 1 ) u 1 Ω 1 min Φ 1 (u 1, u 2 ) u 1 Ω 1 min Φ(u 1, u 2 ) u 1 Ω 1 min Φ2 (u 2 ) u 2 Ω 2 min Φ 2 (u 1, u 2 ) u 2 Ω 2 min Φ(u 1, u 2 ) u 2 Ω 2 Rawlings and Stewart Multiple MPCs: Status and Future 7 / 32

12 Nomenclature: consider two interacting units Objective functions Φ 1 (u 1, u 2 ), Φ 2 (u 1, u 2 ) and Φ(u 1, u 2 ) = w 1 Φ 1 (u 1, u 2 ) + w 2 Φ 2 (u 1, u 2 ) decision variables for units u 1 Ω 1, u 2 Ω 2 Decentralized Control Noncooperative Control (Nash equilibrium) Cooperative Control (Pareto optimal) Centralized Control (Pareto optimal) min Φ1 (u 1 ) u 1 Ω 1 min Φ 1 (u 1, u 2 ) u 1 Ω 1 min Φ(u 1, u 2 ) u 1 Ω 1 min Φ2 (u 2 ) u 2 Ω 2 min Φ 2 (u 1, u 2 ) u 2 Ω 2 min Φ(u 1, u 2 ) u 2 Ω 2 min Φ(u 1, u 2 ) u 1,u 2 Ω 1 Ω 2 Rawlings and Stewart Multiple MPCs: Status and Future 7 / 32

13 Noninteracting systems 2 1 b Φ 2 (u) n, d, p u 2 0 a -1 Φ 1 (u) u 1 Rawlings and Stewart Multiple MPCs: Status and Future 8 / 32

14 Weakly interacting systems 0.5 Φ 2 (u) 0 b p n, d -0.5 u 2-1 a Φ 1 (u) u 1 Rawlings and Stewart Multiple MPCs: Status and Future 9 / 32

15 Moderately interacting systems Φ 1 (u) 1 Φ 2 (u) a u b p d n u 1 Rawlings and Stewart Multiple MPCs: Status and Future 10 / 32

16 Strongly interacting (conflicting) systems Φ 1 (u) Φ 2 (u) p a u 2 0 b d u 1 Rawlings and Stewart Multiple MPCs: Status and Future 11 / 32

17 Strongly interacting (conflicting) systems n u Φ 2 (u) Φ 1 (u) u 1 Rawlings and Stewart Multiple MPCs: Status and Future 12 / 32

18 Geometry of cooperative vs. noncooperative MPC 10 p n 5 a u 2 0 Φ 1 (u) 0 Φ 2 (u) 1 b u 1 Rawlings and Stewart Multiple MPCs: Status and Future 13 / 32

19 Geometry of cooperative vs. noncooperative MPC 10 p n 5 a u 2 0 Φ 1 (u) 1 Φ 2 (u) 0 b u 1 Rawlings and Stewart Multiple MPCs: Status and Future 13 / 32

20 Two reactors with separation and recycle D, x Ad, x Bd MPC 3 MPC 1 MPC 2 F purge F 0, x A0 F 1, x A1 H r Hm F m, x Am, x Bm H b Q F r, x Ar, x Br A B B C A B B C F b, x Ab, x Bb, T Rawlings and Stewart Multiple MPCs: Status and Future 14 / 32

21 Two reactors with separation and recycle H m Time setpoint Cent Ncoop Coop (1 iterate) H b Time setpoint Cent Ncoop Coop (1 iterate) F 1 0 D Time Cent Ncoop Coop (1 iterate) Time Cent Ncoop Coop (1 iterate) Rawlings and Stewart Multiple MPCs: Status and Future 15 / 32

22 Two reactors with separation and recycle Performance comparison Cost ( 10 2 ) Performance loss Centralized MPC Decentralized MPC Noncooperative MPC Cooperative MPC (1 iterate) % Cooperative MPC (10 iterates) % Rawlings and Stewart Multiple MPCs: Status and Future 16 / 32

23 Two reactors with separation and recycle Zero-offset control in the presence of non-zero mean disturbances and plant-model mismatch Rawlings and Stewart Multiple MPCs: Status and Future 17 / 32

24 Two reactors with separation and recycle Zero-offset control in the presence of non-zero mean disturbances and plant-model mismatch Several formulations possible For simplicity, integrating disturbances assumed to be local Under mild assumptions, zero-offset control in the distributed MPC framework can be established Disturbance models that give zero-offset performance under decentralized MPC also give zero-offset performance in the FC-MPC framework Rawlings and Stewart Multiple MPCs: Status and Future 17 / 32

25 Two reactors with separation and recycle d k D, x Ad, x Bd MPC 3 MPC 1 MPC 2 F purge F 0, x A0 F 1, x A1 H r Hm F m, x Am, x Bm H b Q F r, x Ar, x Br A B B C A B B C F b, x Ab, x Bb, T Rawlings and Stewart Multiple MPCs: Status and Future 18 / 32

26 Two reactors with separation and recycle H m Disturbance affects the system from time = 30 to time = Time setpoint Cent Decent Coop (1 iterate) H b Time setpoint Cent Decent Coop (1 iterate) F 1 0 D Time Time Rawlings and Cent Coop (1 iterate) Multiple MPCs: Status and Future Cent Coop (1 iterate)

27 Two reactors with separation and recycle Performance comparison Cost ( 10 2 ) Performance loss Centralized MPC Decentralized MPC - Noncooperative MPC - Cooperative MPC (1 iterate) % Cooperative MPC (10 iterates) % Rawlings and Stewart Multiple MPCs: Status and Future 20 / 32

28 Current Challenge: Coupled Constraints D, x Ad, x Bd MPC 3 MPC 1 MPC 2 F purge F 0, x A0 F 1, x A1 H b H r A B B C Hm F r, x Ar, x Br Q 1 Q 2 A B B C F m, x Am, x Bm Q 3 F b, x Ab, x Bb, T Steam distribution between MPC controllers Q 1 + Q 2 + Q 3 Q T Rawlings and Stewart Multiple MPCs: Status and Future 21 / 32

29 Geometry of Coupled Constraints Q1 + Q 2 Q T Feasible region cannot be separated into Cartesian product of subspaces Υ = (Ω 1 Ω M ) Υ Rawlings and Stewart Multiple MPCs: Status and Future 22 / 32

30 Geometry of Coupled Constraints Coupled constraints give suboptimal points of attraction u 2 u 1 Φ(u 1,u 2 ) u 0 Rawlings and Stewart Multiple MPCs: Status and Future 23 / 32

31 Geometry of Coupled Constraints Coupled constraints give suboptimal points of attraction u 2 u 1 Φ(u 1,u 2 ) u u 0 Rawlings and Stewart Multiple MPCs: Status and Future 23 / 32

32 Geometry of Resource Manager ˆΩ 1 û 1 û 2 ˆΩ 2 Υ Managing constraints feasibly Different inner box constraints Different points of attraction û ˆΩ Rawlings and Stewart Multiple MPCs: Status and Future 24 / 32

33 Resource Manager Resource Manager Problem min ˆΩ M i s.t. { } w i min Φ(u 1,..., u i,..., u M ), u i ˆΩ i ˆΩ = (ˆΩ 1 ˆΩ M ) Υ ˆΩ i local decoupled subspace Υ coupled feasible region Resource manager finds optimal decoupled subspace to pass to each subsystem Rawlings and Stewart Multiple MPCs: Status and Future 25 / 32

34 Resource Manager Example Model A 1 = A 2 = 0.5I R 1 = R 2 = 2I B 11 = B 22 = I B 12 = B 21 = 0.5I Q 1 = 2I Q 2 = I C 11 = C 22 = [I I ] Constraints u 1 0 u 2 0 u 1 + u 2 1 Rawlings and Stewart Multiple MPCs: Status and Future 26 / 32

35 Resource Manager Example Centralized Cooperative Coop. w/rm 1.5 u s u s1 Rawlings and Stewart Multiple MPCs: Status and Future 27 / 32

36 Resource Manager Example y Time Centralized Cooperative Coop. w/rm u Time Rawlings and Stewart Multiple MPCs: Status and Future 28 / 32

37 Resource Manager Example Performance comparison Cost Performance loss Centralized MPC Cooperative MPC (1 iterate) % Coop. + Resource Manager (1 iterate) % Rawlings and Stewart Multiple MPCs: Status and Future 29 / 32

38 Conclusions Distributed MPC can be split into two types based on game theory Noncooperative MPC is unreliable and can produce closed-loop instability Cooperative MPC gives nominal closed-loop stability for any number of iterations Rawlings and Stewart Multiple MPCs: Status and Future 30 / 32

39 Conclusions Distributed MPC can be split into two types based on game theory Noncooperative MPC is unreliable and can produce closed-loop instability Cooperative MPC gives nominal closed-loop stability for any number of iterations A local state estimator can be used with the distributed state regulator Distributed target calculation can be used instead of a centralized target calculation for large-scale systems Rawlings and Stewart Multiple MPCs: Status and Future 30 / 32

40 Conclusions Distributed MPC can be split into two types based on game theory Noncooperative MPC is unreliable and can produce closed-loop instability Cooperative MPC gives nominal closed-loop stability for any number of iterations A local state estimator can be used with the distributed state regulator Distributed target calculation can be used instead of a centralized target calculation for large-scale systems Coupled constraints can be included in target calculation with use of resource manager Rawlings and Stewart Multiple MPCs: Status and Future 30 / 32

41 Acknowledgments Support from the U.S. National Science Foundation through grant CTS Collaboration with and support from Aspentech, Eastman, ExxonMobil and Shell Global Solutions. Rawlings and Stewart Multiple MPCs: Status and Future 31 / 32

42 Further Reading I Y.-C. Ho. On Centralized Optimal Control. IEEE Trans. Auto. Cont., 50(4): , J. Lunze. Feedback Control of Large Scale Systems. Prentice-Hall, London, U.K., 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): , D. D. Siljak. Decentralized Control of Complex Systems. Academic Press, London, ISBN Rawlings and Stewart Multiple MPCs: Status and Future 32 / 32