Online Tracking with Predictions for Nonlinear Systems with Koopman Linear Embedding

TL;DR

This paper develops a model-free predictive tracking algorithm for unknown nonlinear systems that can be linearized via Koopman embeddings, demonstrating exponential regret decay and matching linear system performance.

Contribution

It introduces a novel online tracking method leveraging Koopman linear embeddings and Willems' lemma, with proven exponential regret decay in nonlinear systems.

Findings

Regret decays exponentially with prediction horizon.

Performance matches that of linear systems in Koopman-embeddable cases.

Validated through numerical experiments.

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.