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Online Tracking with Predictions for Nonlinear Systems with Koopman Linear Embedding

Chih-Fan Pai, Xu Shang, Jiachen Qian, Yang Zheng

TL;DR

This work analyzes a model-free predictive tracking algorithm based on Willems' fundamental lemma, and shows that, for Koopman-linearizable systems, the cumulative cost and dynamic regret of the nonlinear tracking problem coincide with those of the lifted linear counterpart.

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.

Online Tracking with Predictions for Nonlinear Systems with Koopman Linear Embedding

TL;DR

This work analyzes a model-free predictive tracking algorithm based on Willems' fundamental lemma, and shows that, for Koopman-linearizable systems, the cumulative cost and dynamic regret of the nonlinear tracking problem coincide with those of the lifted linear counterpart.

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.
Paper Structure (28 sections, 29 theorems, 83 equations, 2 figures, 1 algorithm)

This paper contains 28 sections, 29 theorems, 83 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. From Nonlinear to Linear via Koopman Embedding
  4. Equivalence in the lifted space
  5. Optimal offline policy
  6. Predictive Tracking for Koopman-linearizable Systems
  7. Predictive control with known dynamics
  8. Data-driven predictive control with unknown dynamics
  9. Performance Guarantees for Predictive Tracking
  10. Dynamic regret bound
  11. Dynamic regret analysis
  12. Numerical Experiments
  13. Conclusion
  14. The Optimal Noncausal Policy for Linear Quadratic Tracking (LQT)
  15. Equivalence between LQT and LQR with disturbance
  16. ...and 13 more sections

Key Result

Lemma 3.1

Let $x_1=\psi(z_1)$. The following statements hold:

Figures (2)

  • Figure 1: Performance of Algorithm \ref{['alg:MPC']} with \ref{['eq:DDPC']} under varying prediction horizon $W$: tracking and reference trajectories (left), dynamic regret for different $R$ (middle) and different $M$ (right).
  • Figure 2: Tracking performance of the two-wheel robot under varying prediction horizons: $W=6$ (left), $W=9$ (middle), and $W=12$ (right). The red curve denotes the reference trajectory, and the blue curve shows the robot's actual path.

Theorems & Definitions (51)

  • Example 2.1: brunton2016koopman
  • Lemma 3.1
  • proof
  • Corollary 3.1: Equivalence of dynamic regrets
  • Definition 3.1: Riccati recursion
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 4.1
  • Corollary 4.1
  • Definition 4.1: Lifted excitation
  • ...and 41 more