Probabilistic ODE solvers for integration error-aware numerical optimal control

Amon Lahr, Filip Tronarp, Nathanael Bosch, Jonathan Schmidt, Philipp Hennig, Melanie N. Zeilinger
Proceedings of the 6th Annual Learning for Dynamics & Control Conference, PMLR 242:1018-1032, 2024.

Abstract

Appropriate time discretization is crucial for real-time applications of numerical optimal control, such as nonlinear model predictive control. However, if the discretization error strongly depends on the applied control input, meeting accuracy and sampling time requirements simultaneously can be challenging using classical discretization methods. In particular, neither fixed-grid nor adaptive-grid discretizations may be suitable, when they suffer from large integration error or exceed the prescribed sampling time, respectively. In this work, we take a first step at closing this gap by utilizing probabilistic numerical integrators to approximate the solution of the initial value problem, as well as the computational uncertainty associated with it, inside the optimal control problem (OCP). By taking the viewpoint of probabilistic numerics and propagating the numerical uncertainty in the cost, the OCP is reformulated such that the optimal input reduces the computational uncertainty insofar as it is beneficial for the control objective. The proposed approach is illustrated using a numerical example, and potential benefits and limitations are discussed.

Cite this Paper

BibTeX
@InProceedings{pmlr-v242-lahr24a, title = {Probabilistic {ODE} solvers for integration error-aware numerical optimal control}, author = {Lahr, Amon and Tronarp, Filip and Bosch, Nathanael and Schmidt, Jonathan and Hennig, Philipp and Zeilinger, Melanie N.}, booktitle = {Proceedings of the 6th Annual Learning for Dynamics & Control Conference}, pages = {1018--1032}, year = {2024}, editor = {Abate, Alessandro and Cannon, Mark and Margellos, Kostas and Papachristodoulou, Antonis}, volume = {242}, series = {Proceedings of Machine Learning Research}, month = {15--17 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v242/lahr24a/lahr24a.pdf}, url = {https://proceedings.mlr.press/v242/lahr24a.html}, abstract = {Appropriate time discretization is crucial for real-time applications of numerical optimal control, such as nonlinear model predictive control. However, if the discretization error strongly depends on the applied control input, meeting accuracy and sampling time requirements simultaneously can be challenging using classical discretization methods. In particular, neither fixed-grid nor adaptive-grid discretizations may be suitable, when they suffer from large integration error or exceed the prescribed sampling time, respectively. In this work, we take a first step at closing this gap by utilizing probabilistic numerical integrators to approximate the solution of the initial value problem, as well as the computational uncertainty associated with it, inside the optimal control problem (OCP). By taking the viewpoint of probabilistic numerics and propagating the numerical uncertainty in the cost, the OCP is reformulated such that the optimal input reduces the computational uncertainty insofar as it is beneficial for the control objective. The proposed approach is illustrated using a numerical example, and potential benefits and limitations are discussed.} }
Endnote
%0 Conference Paper %T Probabilistic ODE solvers for integration error-aware numerical optimal control %A Amon Lahr %A Filip Tronarp %A Nathanael Bosch %A Jonathan Schmidt %A Philipp Hennig %A Melanie N. Zeilinger %B Proceedings of the 6th Annual Learning for Dynamics & Control Conference %C Proceedings of Machine Learning Research %D 2024 %E Alessandro Abate %E Mark Cannon %E Kostas Margellos %E Antonis Papachristodoulou %F pmlr-v242-lahr24a %I PMLR %P 1018--1032 %U https://proceedings.mlr.press/v242/lahr24a.html %V 242 %X Appropriate time discretization is crucial for real-time applications of numerical optimal control, such as nonlinear model predictive control. However, if the discretization error strongly depends on the applied control input, meeting accuracy and sampling time requirements simultaneously can be challenging using classical discretization methods. In particular, neither fixed-grid nor adaptive-grid discretizations may be suitable, when they suffer from large integration error or exceed the prescribed sampling time, respectively. In this work, we take a first step at closing this gap by utilizing probabilistic numerical integrators to approximate the solution of the initial value problem, as well as the computational uncertainty associated with it, inside the optimal control problem (OCP). By taking the viewpoint of probabilistic numerics and propagating the numerical uncertainty in the cost, the OCP is reformulated such that the optimal input reduces the computational uncertainty insofar as it is beneficial for the control objective. The proposed approach is illustrated using a numerical example, and potential benefits and limitations are discussed.
APA
Lahr, A., Tronarp, F., Bosch, N., Schmidt, J., Hennig, P. & Zeilinger, M.N.. (2024). Probabilistic ODE solvers for integration error-aware numerical optimal control. Proceedings of the 6th Annual Learning for Dynamics & Control Conference, in Proceedings of Machine Learning Research 242:1018-1032 Available from https://proceedings.mlr.press/v242/lahr24a.html.

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