Abstract
In the setting of energy efficient building operation, an optimal boundary control problem governed by a linear parabolic advection-diffusion equation is considered together with bilateral control and state constraints. To keep the temperature in a prescribed range with the less possible heating cost, an economic model predictive control (MPC) strategy is applied. To speed-up the MPC method, a reduced-order approximation based on proper orthogonal decomposition (POD) is utilized. A-posteriori error analysis ensures the quality of the POD models. A numerical test illustrates the efficiency of the proposed strategy.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
A. Alla, S. Volkwein, Asymptotic stability of POD based model predictive control for a semilinear parabolic PDE. Adv. Comput. Math. 41, 1073–1102 (2015)
P. Benner, A. Cohen, M. Ohlberger, K. Willcox, Model Reduction and Approximation: Theory and Algorithms (SIAM Publications, Philadelphia, 2017)
R. Dautray, J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Evolution Problems I, vol. 5 (Springer, Berlin, 2000)
J. Ghiglieri, S. Ulbrich, Optimal flow control based on POD and MPC and an application to the cancellation of Tollmien-Schlichting waves. Optim. Softw. 29, 1042–1074 (2014)
E. Grimm, M. Gubisch, S. Volkwein, Numerical analysis of optimality-system POD for constrained optimal control. Lect. Notes Comput. Sci. Eng. 105, 297–317 (2015)
L. Grüne, J. Pannek, Nonlinear Model Predictive Control:Theory and Algorithms, 2nd edn. (Springer, London, 2016)
M. Gubisch, S. Volkwein, Proper orthogonal decomposition for linear-quadratic optimal control, in Model Reduction and Approximation: Theory and Algorithms, chap. 1 (SIAM Publications, Philadelphia, 2017)
M. Gubisch, I. Neitzel, S. Volkwein, A-posteriori error estimation of discrete POD models for PDE-constrained optimal controls, in Model Reduction of Parametrized Systems, ed. by P. Benner, M. Ohlberger, A.T. Patera, G. Rozza, K. Urban. Springer-Series Modeling, Simulation and Applications, vol. 17, 213–234 (2017)
M. Hintermüller, K. Ito, K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13, 865–888 (2003)
P. Holmes, J.L. Lumley, G. Berkooz, C.W. Rowley. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics, 2nd edn. (Cambridge University Press, Cambridge, 2012)
K. Krumbiegel, A. Rösch, A virtual control concept for state constrained optimal control problems. Comput. Optim. Appl. 43, 213–233 (2009)
L. Mechelli, S. Volkwein, POD-based economic optimal control of heat-convection phenomena (2017, Submitted). Preprint download at: http://nbn-resolving.de/urn:nbn:de:bsz:352-2--au6ei3apyzpv0
J.B. Rawlings, D.Q. Mayne. Model Predictive Control: Theory and Design (Nob Hill Publishing, Madison, 2009)
F. Tröltzsch, Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints. SIAM J. Optim. 22, 616–635 (2005)
F. Tröltzsch, S. Volkwein, POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44, 83–115 (2009)
Acknowledgments
L. Mechelli gratefully acknowledges support by the German Research Foundation DFG grant Reduced-Order Methods for Nonlinear Model Predictive Control.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Mechelli, L., Volkwein, S. (2019). POD-Based Economic Model Predictive Control for Heat-Convection Phenomena. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_61
Download citation
DOI: https://doi.org/10.1007/978-3-319-96415-7_61
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-96414-0
Online ISBN: 978-3-319-96415-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)