Online Tracking with Predictions for Nonlinear Systems with Koopman Linear Embedding

Chih-Fan Pai, Xu Shang, Jiachen Qian, Yang Zheng
Proceedings of The 8th Annual Learning for Dynamics and Control Conference, PMLR 331:1575-1600, 2026.

Abstract

We study the problem of online tracking in unknown nonlinear dynamical systems, where only short-horizon predictions of future target states are available. This setting arises in practical scenarios where full future information and exact system dynamics are unavailable. We focus on a class of nonlinear systems that admit a Koopman linear embedding, enabling the dynamics to evolve linearly in a lifted space. Exploiting this structure, we analyze a model-free predictive tracking algorithm based on Willems’ fundamental lemma, which imposes dynamic constraints using only past data within a receding-horizon control framework. We show that, for Koopman-linearizable systems, the cumulative cost and dynamic regret of the nonlinear tracking problem coincide with those of the lifted linear counterpart. Moreover, we prove that the dynamic regret of our algorithm decays exponentially with the prediction horizon, as validated by numerical experiments.

Cite this Paper

BibTeX
@InProceedings{pmlr-v331-pai26a, title = {Online Tracking with Predictions for Nonlinear Systems with Koopman Linear Embedding}, author = {Pai, Chih-Fan and Shang, Xu and Qian, Jiachen and Zheng, Yang}, booktitle = {Proceedings of The 8th Annual Learning for Dynamics and Control Conference}, pages = {1575--1600}, year = {2026}, editor = {Sukhatme, Gaurav and Lindemann, Lars and Tu, Stephen and Wierman, Adam and Atanasov, Nikolay}, volume = {331}, series = {Proceedings of Machine Learning Research}, month = {17--19 Jun}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v331/main/assets/pai26a/pai26a.pdf}, url = {https://proceedings.mlr.press/v331/pai26a.html}, abstract = {We study the problem of online tracking in unknown nonlinear dynamical systems, where only short-horizon predictions of future target states are available. This setting arises in practical scenarios where full future information and exact system dynamics are unavailable. We focus on a class of nonlinear systems that admit a Koopman linear embedding, enabling the dynamics to evolve linearly in a lifted space. Exploiting this structure, we analyze a model-free predictive tracking algorithm based on Willems’ fundamental lemma, which imposes dynamic constraints using only past data within a receding-horizon control framework. We show that, for Koopman-linearizable systems, the cumulative cost and dynamic regret of the nonlinear tracking problem coincide with those of the lifted linear counterpart. Moreover, we prove that the dynamic regret of our algorithm decays exponentially with the prediction horizon, as validated by numerical experiments.} }
Endnote
%0 Conference Paper %T Online Tracking with Predictions for Nonlinear Systems with Koopman Linear Embedding %A Chih-Fan Pai %A Xu Shang %A Jiachen Qian %A Yang Zheng %B Proceedings of The 8th Annual Learning for Dynamics and Control Conference %C Proceedings of Machine Learning Research %D 2026 %E Gaurav Sukhatme %E Lars Lindemann %E Stephen Tu %E Adam Wierman %E Nikolay Atanasov %F pmlr-v331-pai26a %I PMLR %P 1575--1600 %U https://proceedings.mlr.press/v331/pai26a.html %V 331 %X We study the problem of online tracking in unknown nonlinear dynamical systems, where only short-horizon predictions of future target states are available. This setting arises in practical scenarios where full future information and exact system dynamics are unavailable. We focus on a class of nonlinear systems that admit a Koopman linear embedding, enabling the dynamics to evolve linearly in a lifted space. Exploiting this structure, we analyze a model-free predictive tracking algorithm based on Willems’ fundamental lemma, which imposes dynamic constraints using only past data within a receding-horizon control framework. We show that, for Koopman-linearizable systems, the cumulative cost and dynamic regret of the nonlinear tracking problem coincide with those of the lifted linear counterpart. Moreover, we prove that the dynamic regret of our algorithm decays exponentially with the prediction horizon, as validated by numerical experiments.
APA
Pai, C., Shang, X., Qian, J. & Zheng, Y.. (2026). Online Tracking with Predictions for Nonlinear Systems with Koopman Linear Embedding. Proceedings of The 8th Annual Learning for Dynamics and Control Conference, in Proceedings of Machine Learning Research 331:1575-1600 Available from https://proceedings.mlr.press/v331/pai26a.html.

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