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A Framework for Integrating Model PredictiveControllers to Control Large-Scale Systems

Aswin N. Venkat1, James B. Rawlings1

and Stephen J. Wright2

1Department of Chemical and Biological Engineering2Department of Computer SciencesUniversity of Wisconsin–Madison

Future Directions in Systems and ControlAIChE Annual Meeting

October 31, 2005

Venkat,Rawlings and Wright (UW-Madison) Plantwide control with distributed MPC AIChE 2005 1 / 26

Outline

1 Introduction

2 Results for Distributed MPCModelsCommunicationCooperation

3 Conclusions

4 Future Directions


Introduction

Increase reliability of critical infrastructures

Typically consist of networks of interconnected/interacting subsystemsChemical plants, electrical power grids, water distribution networks, etc.

R

x ipaR

Trim Cooler

Ext

ract

orFlash

F

xF

SyS

Bx ipaB

Fw

xFs

Make up CO2

Subsystem 1

TS

Tsh

StrippingColumn

Subsystem 2

Subsystem 3

Reboiler

x ipaD

L

V


Introduction



CONTROL AREA 2 CONTROL AREA 3

P23tie

CONTROL AREA 1CONTROL AREA 4

P34tie

P12tie


Introduction



Reach 8

Gate 1

4 km 4 km 3 km 2 km 2 km

Reach 1

Gate 8

Q1

3 km3 km7 km


Introduction

Traditional approach: decentralized control

Wealth of literature from the 1970’s on improved decentralized controlWell known that poor performance may result if the interconnectionsare not negligible

Steady increase in available computational power has provided theopportunity for centralized control

Most practitioners view centralized control of large, networkedsystems as impractical and unrealistic

Centralized control law grows exponentially with system sizeDifficult to tailor a centralized controller to meet operational objectives

A divide and conquer strategy is essential for control of large,networked systems (Ho [2005])


Introduction

Integrating subsystem-based MPCs

1990’s: linear MPC became a dominant advanced control technology

Notion of Nash equilibrium and Pareto optimality in multi-agentgames (Basar and Olsder [1999])

Potential benefits and requirements of cross-integration within theMPC framework (Kulhavy et al. [2001], Havlena and Lu [2005])

Available distributed MPC formulations in the literature(Camponogara et al. [2002], Jia and Krogh [2002], Keviczky et al.[2004])

Nominal properties (feasibility, optimality, closed-loop stability) havenot all been established for any single distributed MPC framework


Modeling for Distributed MPCDecentralized, interaction models

xii(k + 1) = Aiixii(k) + Biiui(k)

Decentralized Model

(Aii, Bii, Cii)

yi(k)ui(k)

xii(k + 1) = Aiixii(k) + Biiui(k)

Decentralized Model

(Aii, Bii, Cii)

(local subsystem inputs)

Interaction Model

xij(k + 1) = Aijxij(k) + Bijuj(k)

(Aij, Bij, Cij)

yi(k) =∑

j Cijxij(k)

uj 6=i(k)

(external subsystem inputs)

+ +


Distributed MPCFormulations and Assumptions

Formulations for distributed MPC

Communication-based MPC

Cooperation-based MPC

Assumptions

All MPC cost functions are positive definite, quadratic

Each subsystem represented by a linear, state-space model

All interaction models are stable

Local input inequality constraints (e.g., input bounds)


Nomenclature: Consider Two Interacting Units

Objective functions Φ1(u1, u2), Φ2(u1, u2)

and Φ = w1Φ1 + w2Φ2

decision variables for units u1, u2

Decentralized Control minu1

Φ1(u1) minu2

Φ2(u2)

Communication-based Control minu1

Φ1(u1, u2) minu2

Φ2(u1, u2)

(Nash equilibrium)

Cooperation-based Control minu1

Φ(u1, u2) minu2

Φ(u1, u2)

(Pareto optimal)

Centralized Control minu1,u2

Φ(u1, u2)

(Pareto optimal)


Communication-based MPC1

Exchange of state and inputtrajectory informationbetween MPCsSubsystems’ MPCoptimizations solved untilstate and input trajectoriesconvergeFirst move in each convergedinput trajectory injected intothe plant

MPC 1 Prediction MPC 2 Prediction

Prediction horizonPrediction horizon

Process-process interactions

State trajectorySetpoint trajectory

Process

MPC 1 MPC 2

Process

u1

Controlled input trajectory

y2

u2

y1

1 2

1Similar schemes proposed by Jia and Krogh [2001], Camponogara et al. [2002]Venkat,Rawlings and Wright (UW-Madison) Plantwide control with distributed MPC AIChE 2005 9 / 26

Geometry of Communication-based MPCStable Nash Equilibrium

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

u1

u2

Φ2(u)

Φ1(u)

b

a

pd 0

1

2

3

n


Geometry of Communication-based MPCNash Equilibrium does not exist

-2

-1

0

1

2

3

4

5

-3 -2 -1 0 1 2 3 4 5

u1

u2

Φ2(u)

Φ1(u)

b

ap

d0

1

2


Communication-based MPC

Distillation column of Ogunnaike and Ray [1994]

Outputs T21,T7; Inputs L,VTwo SISO MPCsIntentionally choose bad pairing:

MPC-1 : T21 − VMPC-2 : T7 − L

Stable Nash Equilibrium exists butcommunication-based MPC isunstable

An unreliable plantwide control strategy

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0 50 100 150 200Time (sec)

T21

setpointcent-MPC

comm-MPC

-1.5

-1

-0.5

0

0.5

1

1.5

0 50 100 150 200Time (sec)

V

cent-MPCcomm-MPC


Feasible Cooperation-based MPC (FC-MPC)

Tasks involved

Model interconnections between subsystemsExchange input trajectories among interconnected subsystems’ MPCsReplace local objectives by a suitable global objective e.g.,

Φ =∑

i

wiΦi wi > 0,

M∑i=1

wi = 1

Eliminate the state variables using the model equality constraints

Each MPC solves an optimization problem in the local input variables


Feasible Cooperation-based MPC (FC-MPC)

Properties

1 All iterates are plantwide feasible2 The sequence of cost functions is a non-increasing function of the

iteration number

Also bounded below, hence convergent

3 The sequence of iterates converges to an optimal limit point(centralized MPC solution)


Geometry of Feasible Cooperation-based MPC

-2

-1

0

1

2

3

4

5

-3 -2 -1 0 1 2 3 4 5

u1

u2

Φ2(u)

Φ1(u)

b

ap

d0

1

2


Geometry of Feasible Cooperation-based MPC

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-3 -2 -1 0 1 2

u1

u2

Φ2(u)

Φ1(u)

b

ap

d0

1

2


Closed-loop Properties of FC-MPC with State Estimation

Distributed MPC control law

Subsystem states (decentralized+interaction) estimated using asteady-state Kalman filter

First input move in the last calculated input trajectory of eachsubsystem’s FC-MPC injected into the plant

Properties

Nominal closed-loop stability under intermediate termination

Disturbance scenarios that destabilize FC-MPC also destabilizecentralized MPC


Performance of FC-MPCDistillation column of Ogunnaike and Ray [1994]


MPC-1 : T21 − VMPC-2 : T7 − L

Communication-based MPC cannotfix this kind of bad design choice

FC-MPC can fix this kind of baddesign choice

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0 50 100 150 200Time (sec)

T21

setpointcent-MPC

comm-MPC

-1.5

-1

-0.5

0

0.5

1

1.5

0 50 100 150 200Time (sec)

V

cent-MPCcomm-MPC


Performance of FC-MPCDistillation column of Ogunnaike and Ray [1994]


MPC-1 : T21 − VMPC-2 : T7 − L

Communication-based MPC cannotfix this kind of bad design choiceFC-MPC can fix this kind of baddesign choice

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0 50 100 150 200Time (sec)

T21

setpointcent-MPC

comm-MPCFC-MPC (1 iterate)

-1.5

-1

-0.5

0

0.5

1

1.5

0 50 100 150 200Time (sec)

V

cent-MPCcomm-MPC

FC-MPC (1 iterate)


Integrated Styrene Polymerization Plants

Plant 1

End use grade fraction: (1− β)

V

L D

B

Frecy, Cmr, Tr

Fm1, cm1

, T1

Plant 2

Fs0, cs, Tf0

Fc2, Tc2

Fs2, cs2

, Tf2

Fm2, cm2

, Tf2

Fc0, Tc0

Cp, Cmbot, Cinitbot

Finit2, ci2, Tf2

Finit0, ci0, Tf0

Fm0, cm0

, Tf0

Fm3, cm3

, T2

Two MPCs, one for each plant

Plant 1:

Manipulate Finit0 to controlT1

Produces grade A (lowergrade) of polymer

Plant 2:

Two units–polymerizationreactor and separatorMPC manipulates Finit2 ,Frecy and V to control T2,Cmr and Cp

Produces grade B (highergrade) of polymer


Integrated Styrene Polymerization PlantsPerformance of different MPC frameworks

-10

-8

-6

-4

-2

0

0 10 20 30 40 50 60Time (hrs)

T1

setpointCentralized MPC

Decentralized MPC

-8

-7

-6

-5

-4

-3

-2

-1

0

0 10 20 30 40 50 60Time (hrs)

T2


Decentralized MPC

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 10 20 30 40 50 60Time (hrs)

Finit0

Centralized MPCDecentralized MPC

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40 50 60Time (hrs)

Finit2

Centralized MPCDecentralized MPC


Integrated Styrene Polymerization PlantsPerformance of different MPC frameworks

-10

-8

-6

-4

-2

0

0 10 20 30 40 50 60Time (hrs)

T1


Decentralized MPCFC-MPC (1 iterate)

-8

-7

-6

-5

-4

-3

-2

-1

0

0 10 20 30 40 50 60Time (hrs)

T2


Decentralized MPCFC-MPC (1 iterate)

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 10 20 30 40 50 60Time (hrs)

Finit0

Centralized MPCDecentralized MPCFC-MPC (1 iterate)

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40 50 60Time (hrs)

Finit2

Centralized MPCDecentralized MPCFC-MPC (1 iterate)


Integrated Styrene Polymerization Plants

Controller performance measure

Λcost(k) =1

k

k∑j=0

M∑i=1

L [xi (j), ui (j)]

Performance comparisonΛcost Performance loss

(w.r.t centralized MPC)

Centralized-MPC 18.84 -Decentralized-MPC 1608 8400%FC-MPC (1 iterate) 18.94 0.54%FC-MPC (5 iterates) 18.84 0%


Conclusions

In this talk, we have shown some first results on

sharing information and combining objectives in multiple MPCs.a distributed MPC methodology with guaranteed feasibility, stability,and optimality properties.

Wealth of unexplored issues and approaches remain.

Large potential for societal impact.


Future Directions

Develop methods for identification of significant interactions fromclosed-loop data (minimal modeling).Develop methods to reduce the information sharing betweensubsystems’ MPCs.Develop strategies that update and adapt the models when the plantchanges.Extend these ideas to nonlinear models.Test and implement the approach on industrial applications.


Further Reading I

J. Antwerp and R. Braatz. Model predictive control of large scale processes. J. Proc.Control, 10:1–8, 2000.

T. Basar and G. J. Olsder. Dynamic Noncooperative Game Theory. SIAM, Philadelphia,1999.

R. Bartlett, L. Biegler, J. Backstrom, and V. Gopal. Quadratic programming algorithmsfor large-scale model predictive control. J. Proc. Cont., 12(7):775–795, 2002.

A. Bemporad, M. Morari, V. Dua, and E. Pistikopoulos. The explicit linear quadraticregulator for constrained systems. Automatica, 38(1):3–20, 2002.

E. Camponogara, D. Jia, B. H. Krogh, and S. Talukdar. Distributed model predictivecontrol. IEEE Ctl. Sys. Mag., pages 44–52, February 2002.

J. E. Cohen. Cooperation and self interest: Pareto-inefficiency of Nash equilibria in finiterandom games. Proc. Natl. Acad. Sci. USA, 95:9724–9731, 1998.

V. Havlena and J. Lu. A distributed automation framework for plant-wide control,optimisation, scheduling and planning. In Proceedings of the 16th IFAC WorldCongress, Prague, Czech Republic, July 2005.

Y.-C. Ho. On Centralized Optimal Control. IEEE Trans. Auto. Cont., 50(4):537–538,2005.


Further Reading II

D. Jia and B. H. Krogh. Distributed model predictive control. In Proceedings of theAmerican Control Conference, Arlington, Virginia, June 2001.

D. Jia and B. H. Krogh. Min-max feedback model predictive control for distributedcontrol with communication. In Proceedings of the American Control Conference,Anchorage,Alaska, May 2002.

T. Keviczky, F. Borelli, and G. J. Balas. A study on decentralized receding horizoncontrol for decoupled systems. In Proceedings of the American Control Conference,Boston, Massachusetts, July 2004.

R. Kulhavy, J. Lu, and T. Samad. Emerging technologies for enterprise optimization inthe process industries. In J. B. Rawlings, B. A. Ogunnaike, and J. W. Eaton, editors,Chemical Process Control–VI: Sixth International Conference on Chemical ProcessControl, pages 352–363, Tucson, Arizona, January 2001. AIChE Symposium Series,Volume 98, Number 326.

S. Li and T. Basar. Distributed algorithms for the computation of noncooperativeequilibria. Automatica, 23(4):523–533, 1987.

J. Lu. Challenging control problems and emerging technologies in enterpriseoptimization. Control Eng. Prac., 11(8):847–858, August 2003.

J. Lunze. Feedback Control of Large Scale Systems. Prentice-Hall, London, U.K, 1992.


Further Reading III

D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert. Constrained modelpredictive control: Stability and optimality. Automatica, 36(6):789–814, 2000.

M. Morari and J. H. Lee. Model predictive control: past, present and future. InProceedings of joint 6th international symposium on process systems engineering(PSE ’97) and 30th European symposium on computer aided process systemsengineering (ESCAPE 7), Trondheim, Norway, 1997.

B. A. Ogunnaike and W. H. Ray. Process Dynamics, Modeling, and Control. OxfordUniversity Press, New York, 1994.

S. J. Qin and T. A. Badgwell. A survey of industrial model predictive controltechnology. Control Eng. Prac., 11(7):733–764, 2003.

J. B. Rawlings and K. R. Muske. Stability of constrained receding horizon control. IEEETrans. Auto. Cont., 38(10):1512–1516, October 1993.

N. R. Sandell-Jr., P. Varaiya, M. Athans, and M. Safonov. Survey of decentralizedcontrol methods for larger scale systems. IEEE Trans. Auto. Cont., 23(2):108–128,1978.

D. Siljak. Decentralized Control of Complex Systems. Academic Press, London, 1991.

M. Sznaier and M. J. Damborg. Heuristically enhanced feedback control of constraineddiscrete-time linear systems. Automatica, 26(3):521–532, 1990.


Further Reading IV

R. E. Young, R. D. Bartusiak, and R. W. Fontaine. Evolution of an industrial nonlinearmodel predictive controller. In J. B. Rawlings, B. A. Ogunnaike, and J. W. Eaton,editors, Chemical Process Control–VI: Sixth International Conference on ChemicalProcess Control, pages 342–351, Tucson, Arizona, January 2001. AIChE SymposiumSeries, Volume 98, Number 326.


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