Summary
We review a closely connected family of algorithmic approaches for fast and real–time capable nonlinear model predictive control (NMPC) of dynamic processes described by ordinary differential equations or index-1 differential-algebraic equations. Focusing on active–set based algorithms, we present emerging ideas on adaptive updates of the local quadratic subproblems (QPs) in a multi–level scheme. Structure exploiting approaches for the solution of these QP subproblems are the workhorses of any fast active–set NMPC method. We present linear algebra tailored to the QP block structures that act both as a preprocessing and as block structured factorization methods.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Albersmeyer, D. Beigel, C. Kirches, L. Wirsching, H. Bock, and J. Schlöder. Fast nonlinear model predictive control with an application in automotive engineering. In L. Magni, D. Raimondo, and F. Allgöwer, editors, Lecture Notes in Control and Information Sciences, volume 384, pages 471–480. Springer Verlag Berlin Heidelberg, 2009.
J. Albersmeyer and H. Bock. Sensitivity Generation in an Adaptive BDFMethod. In H. G. Bock, E. Kostina, X. Phu, and R. Rannacher, editors, Modeling, Simulation and Optimization of Complex Processes: Proceedings of the International Conference on High Performance Scientific Computing, March 6– 10, 2006, Hanoi, Vietnam, pages 15–24. Springer Verlag Berlin Heidelberg New York, 2008.
R. Bartlett and L. Biegler. QPSchur: A dual, active set, schur complement method for large-scale and structured convex quadratic programming algorithm. Optimization and Engineering, 7:5–32, 2006.
M. Best. An Algorithm for the Solution of the Parametric Quadratic Programming Problem, chapter 3, pages 57–76. Applied Mathematics and Parallel Computing. Physica-Verlag, Heidelberg, 1996.
H. Bock, M. Diehl, E. Kostina, and J. Schlöder. Constrained Optimal Feedback Control for DAE. In L. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes, and B. van Bloemen Waanders, editors, Real-Time PDE-Constrained Optimization, chapter 1, pages 3–24. SIAM, 2007.
H. Bock, M. Diehl, P. Kühl, E. Kostina, J. Schl¨eder, and L. Wirsching. Numerical methods for efficient and fast nonlinear model predictive control. In R. Findeisen, F. Allgöwer, and L. T. Biegler, editors, Assessment and future directions of Nonlinear Model Predictive Control, volume 358 of Lecture Notes in Control and Information Sciences, pages 163–179. Springer, 2005.
H. Bock and K. Plitt. A Multiple Shooting algorithm for direct solution of optimal control problems. In Proceedings of the 9th IFAC World Congress, pages 243–247, Budapest, 1984. Pergamon Press. Available at http://www.iwr.uniheidelberg.de/groups/agbock/FILES/Bock1984.pdf.
A. Bryson and Y.-C. Ho. Applied Optimal Control. Wiley, New York, 1975.
M. Diehl, H. Bock, J. Schlöder, R. Findeisen, Z. Nagy, and F. Allgöwer. Realtime optimization and nonlinear model predictive control of processes governed by differential-algebraic equations. J. Proc. Contr., 12(4):577–585, 2002.
M. Diehl, H. Ferreau, and N. Haverbeke. Efficient numerical methods for nonlinear mpc and moving horizon estimation. In L. Magni, D. Raimondo, and F. Allgöwer, editors, Nonlinear Model Predictive Control, volume 384 of Springer Lecture Notes in Control and Information Sciences, pages 391–417. Springer- Verlag, Berlin, Heidelberg, New York, 2009.
M. Diehl, P. Kuehl, H. Bock, and J. Schlöder. Schnelle Algorithmen für die Zustands- und Parameterschätzung auf bewegten Horizonten. Automatisierungstechnik, 54(12):602–613, 2006.
H. Ferreau, H. Bock, and M. Diehl. An online active set strategy to overcome the limitations of explicit MPC. International Journal of Robust and Nonlinear Control, 18(8):816–830, 2008.
R. Fletcher. Resolving degeneracy in quadratic programming. Numerical Analysis Report NA/135, University of Dundee, Dundee, Scotland, 1991.
P. Gill, G. Golub, W. Murray, and M. A. Saunders. Methods for modifying matrix factorizations. Mathematics of Computation, 28(126):505–535, 1974.
P. Gill, W. Murray, and M. Saunders. User’s Guide For QPOPT 1.0: A Fortran Package For Quadratic Programming, 1995.
G. Golub and C. van Loan. Matrix Computations. Johns Hopkins University Press, Baltimore, 3rd edition, 1996.
A. Griewank. Evaluating Derivatives, Principles and Techniques of Algorithmic Differentiation. Number 19 in Frontiers in Appl. Math. SIAM, Philadelphia, 2000.
N. Haverbeke, M. Diehl, and B. de Moor. A structure exploiting interior-point method for moving horizon estimation. In Proceedings of the 48th IEEE Conference on Decision and Control (CDC09), pages 1–6, 2009.
C. Kirches, H. Bock, J. Schlöder, and S. Sager. Block structured quadratic programming for the direct multiple shooting method for optimal control. Optimization Methods and Software, 2010. DOI 10.1080/10556781003623891.
C. Kirches, H. Bock, J. Schlöder, and S. Sager. A factorization with update procedures for a KKT matrix arising in direct optimal control. Mathematical Programming Computation, 2010. (submitted). Available Online: http://www.optimization-online.org/DBHTML/2009/11/2456.html.
C. Kirches, S. Sager, H. Bock, and J. Schlöder. Time-optimal control of automobile test drives with gear shifts. Optimal Control Applications and Methods, 31(2):137–153, 2010.
D. Leineweber. Efficient reduced SQP methods for the optimization of chemical processes described by large sparse DAE models, volume 613 of Fortschritt- Berichte VDI Reihe 3, Verfahrenstechnik. VDI Verlag, Düsseldorf, 1999.
D. Leineweber, I. Bauer, A. Schäfer, H. Bock, and J. Schlöder. An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization (Parts I and II). Computers and Chemical Engineering, 27:157– 174, 2003.
D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert. Constrained model predictive control: stability and optimality. Automatica, 26(6):789–814, 2000.
L. Petzold, S. Li, Y. Cao, and R. Serban. Sensitivity analysis of differentialalgebraic equations and partial differential equations. Computers and Chemical Engineering, 30:1553–1559, 2006.
K. Plitt. Ein superlinear konvergentes Mehrzielverfahren zur direkten Berechnung beschränkter optimaler Steuerungen. Diploma thesis, Rheinische Friedrich–Wilhelms–Universität zu Bonn, 1981.
C. V. Rao, J. B. Rawlings, and D. Q. Mayne. Constrained state estimation for nonlinear discrete-time systems: Stability and moving horizon approximations. IEEE Transactions on Automatic Control, 48(2):246–258, 2003.
C. Rao, S. Wright, and J. Rawlings. Application of interior-point methods to model predictive control. Journal of Optimization Theory and Applications, 99:723–757, 1998.
C. Schmid and L. Biegler. Quadratic programming methods for tailored reduced Hessian SQP. Computers & Chemical Engineering, 18(9):817–832, September 1994.
M. Steinbach. Structured interior point SQP methods in optimal control. Zeitschrift für Angewandte Mathematik und Mechanik, 76(S3):59–62, 1996.
L. Wirsching. An SQP algorithm with inexact derivatives for a direct multiple shooting method for optimal control problems. Diploma thesis, Universität Heidelberg, 2006.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer -Verlag Berlin Heidelberg
About this paper
Cite this paper
Kirches, C., Wirsching, L., Sager, S., Bock, H.G. (2010). Efficient Numerics for Nonlinear Model Predictive Control. In: Diehl, M., Glineur, F., Jarlebring, E., Michiels, W. (eds) Recent Advances in Optimization and its Applications in Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12598-0_30
Download citation
DOI: https://doi.org/10.1007/978-3-642-12598-0_30
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-12597-3
Online ISBN: 978-3-642-12598-0
eBook Packages: EngineeringEngineering (R0)