Learning to control switching nonlinear systems with Koopman operator regression

2026-07-14 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Robotics & Autonomous Systems, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

The paper presents a method for identifying and controlling switching nonlinear systems by employing Koopman operator regression within a reproducing kernel Hilbert space (RKHS). This approach estimates unknown dynamics from finite samples, yielding a linear switching predictive model where control variables dictate switches. The learned dynamics are then utilized in an infinite-horizon optimal control problem, which is solved using model predictive control (MPC) with a time-varying stage cost. Theoretical analysis establishes learning rates for Koopman dynamics approximation, demonstrating an O(n^{-1/6}) one-step-ahead prediction error rate, and quantifies the sub-optimality of the MPC strategy for both exact and learned Koopman dynamics. Numerical simulations on the Duffing oscillator empirically corroborate these theoretical findings and the data-driven MPC's performance.

Key takeaway

For Machine Learning Engineers developing control systems for complex nonlinear dynamics, this research offers a robust data-driven approach. You should consider implementing Koopman operator regression in a reproducing kernel Hilbert space to linearize switching systems, especially when dealing with finite control actions. This method allows you to leverage model predictive control with quantifiable performance bounds, improving stability and prediction accuracy by increasing data samples and predictive horizon.

Key insights

Koopman operator regression in RKHS enables data-driven control of switching nonlinear systems with quantifiable performance.

Principles

• Nonlinear dynamics can be linearized into a family of Koopman operators for finite control sets.
• Nonparametric regression in RKHS provides convergence guarantees for Koopman operator learning.
• MPC's sub-optimality gap decays exponentially with predictive horizon length.

Method

Unknown nonlinear dynamics are estimated using Koopman operator regression in an RKHS from finite samples. This yields a linear switching predictive model, which is then used in an infinite-horizon optimal control problem solved by model predictive control.

In practice

• Apply random Fourier features for Gaussian kernel lifting in data-driven control.
• Use time-varying stage costs in MPC for systems with finite control sets.
• Increase training samples (n) and predictive horizon (T) to improve MPC performance.

Topics

• Koopman Operator
• Model Predictive Control
• Reproducing Kernel Hilbert Spaces
• Nonlinear Control Systems
• Switching Dynamical Systems
• Duffing Oscillator

Code references

• caedoard/switching-koopman-mpc

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.