Abstract
MPC formulations with linear dynamics and quadratic objectives can be solved efficiently by using a primal-dual interior-point framework, with complexity proportional to the length of the horizon. An alternative, which is more able to exploit the similarity of the problems that are solved at each decision point of linear MPC, is to use an active-set approach, in which the MPC problem is viewed as a convex quadratic program that is parametrized by the initial state \(x_{0}\). Another alternative is to identify explicitly polyhedral regions of the space occupied by \(x_{0}\) within which the set of active constraints remains constant, and to pre-calculate solution operators on each of these regions. All these approaches are discussed here.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The LU factorization does not exploit the fact that the coefficient matrix is symmetric. The \(LDL^{T}\) factorization is commonly used for such matrices, but unfortunately the permutations required in this factorization tend to destroy the band structure, so it is not appropriate here.
References
Bemporad, A., Borrelli, F., Morari, M.: Model predictive control based on linear programming—the explicit solution. IEEE Trans. Autom. Control 47(12), 1974–1985 (2002)
Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N.: The explicit linear quadratic regulator for constrained systems. Automatica 38, 3–20 (2002)
Ferreau, H.J.: An online active set strategy for fast solution of parametric quadratic programs with applications to predictive engine control. Ph.D. thesis, Ruprecht-Karls-Universit at Heidelberg Fakult at fur Mathematik und Informatik (2006)
Ferreau, H.J., Bock, H.G., Diehl, M.: An online active set strategy to overcome the limitations of explicit MPC. Int. J. Robust Nonlinear Control 18(8), 816–830 (2008)
Ferreau, H.J., Kirches, C., Potschka, A., Bock, H.G., Diehl, M.: qpOASES: a parametric active-set algorithm for quadratic programming. Math. Program. Comput. 6(4), 327–363 (2014)
Ferreau, H.J., Potschka, A., Kirches, C.: qpOASES webpage. http://www.qpOASES.org/ (2007–2017)
Gertz, E.M., Wright, S.J.: OOQP. http://www.cs.wisc.edu/~swright/ooqp/
Gertz, E.M., Wright, S.J.: Object-oriented software for quadratic programming. ACM Trans. Math. Softw. 29, 58–81 (2003)
Pannocchia, G., Rawlings, J.B., Wright, S.J.: Fast, large-scale model predictive control by partial enumeration. Automatica 43, 852–860 (2007)
Pannocchia, G., Wright, S.J., Rawlings, J.B.: Partial enumeration MPC: robust stability results and application to an unstable CSTR. J. Process Control 21, 1459–1466 (2011)
Rao, C.V., Wright, S.J., Rawlings, J.B.: Application of interior-point methods to model predictive control. J. Optim. Theory Appl. 99, 723–757 (1998)
Tondel, P., Johansen, T.A., Bemporad, A.: Evaluation of piecewise affine control via binary search tree. Automatica 39(5), 945–950 (2003)
Wright, S.J.: Applying new optimization algorithms to model predictive control. In: Kantor, J.C. (ed.) Chemical Process Control-V, AIChE Symposium Series, vol. 93, pp. 147–155. CACHE Publications, Austin (1997)
Wright, S.J.: Primal-Dual Interior-Point Methods. SIAM Publications, Philadelphia (1997)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Wright, S.J. (2019). Efficient Convex Optimization for Linear MPC. In: Raković, S., Levine, W. (eds) Handbook of Model Predictive Control. Control Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-77489-3_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-77489-3_13
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-77488-6
Online ISBN: 978-3-319-77489-3
eBook Packages: EngineeringEngineering (R0)