Learning Koopman Operators with Control Using Bi-level Optimization

TL;DR

This paper introduces a bi-level optimization approach for learning Koopman operators with control, improving long-term prediction accuracy and stability in nonlinear dynamical systems for robotic applications.

Contribution

It proposes a novel bi-level optimization framework that jointly learns the Koopman embedding and dynamics with long-term constraints, enhancing robustness and predictive performance.

Findings

More accurate long-term system predictions

Enhanced stability over traditional methods

Applicable to various robotic systems

Abstract

The accurate modeling and control of nonlinear dynamical effects are crucial for numerous robotic systems. The Koopman formalism emerges as a valuable tool for linear control design in nonlinear systems within unknown environments. However, it still remains a challenging task to learn the Koopman operator with control from data, and in particular, the simultaneous identification of the Koopman linear dynamics and the mapping between the physical and Koopman states. Conventionally, the simultaneous learning of the dynamics and mapping is achieved via single-level optimization based on one-step or multi-step discrete-time predictions, but the learned model may lack model robustness, training efficiency, and/or long-term predictive accuracy. This paper presents a bi-level optimization framework that jointly learns the Koopman embedding mapping and Koopman dynamics with exact long-term dynamical constraints. Our formulation allows back-propagation in standard learning framework and the use of state-of-the-art optimizers, yielding more accurate and stable system prediction in long-time horizon over various applications compared to conventional methods.