Learning Stable Koopman Embeddings for Identification and Control

TL;DR

The paper tackles learning nonlinear dynamical models with guaranteed stability and stabilizability by developing unconstrained parameterizations of stable Koopman embeddings and their stabilizable extensions. It establishes a rigorous link between discrete-time Koopman conditions and contraction analysis, enabling linear lifting of contracting nonlinear systems and generalized feedback linearizable systems. The authors propose practical learning frameworks that minimize lifted-domain simulation errors while enforcing stability via novel parameterizations of $A$, $ abla$, and left-inverse observables, and demonstrate their effectiveness on stable system identification and imitation learning with empirical benefits over baselines. The approach offers scalable, differentiable training using autodiff, supports data-driven control with regularization for closed-loop stability, and provides error bounds when Koopman invariance is imperfect, showcasing potential for robust data-driven control of nonlinear systems.

Abstract

This paper introduces new model parameterizations for learning discrete-time dynamical systems from data via the Koopman operator and studies their properties. Whereas most existing works on Koopman learning do not take into account the stability or stabilizability of the model -- two fundamental pieces of prior knowledge about a given system to be identified -- in this paper, we propose new classes of Koopman models that have built-in guarantees of these properties. These guarantees are achieved through a novel {\em direct parameterization approach} that leads to {\em unconstrained} optimization problems over their parameter sets. {These results rely on the invertibility of the vector fields for autonomous systems and the generalized feedback linearizability (under smooth feedback), respectively.} To explore the representational flexibility of these model sets, we establish the theoretical connections between the stability of discrete-time Koopman embedding and contraction-based forms of nonlinear stability and stabilizability. The proposed approach is illustrated in applications to stable nonlinear system identification and imitation learning via stabilizable models. Simulation results empirically show that the proposed learning approaches outperform prior methods lacking stability guarantees.

Figures (5)

• Figure 1: The proposed model class M1: Use linear system identification approaches to learn nonlinear models.
• Figure 2: Simulations of SKEL and LKIS models on test data. (Red dotted lines: trajectories from the models; Solid black line: true trajectories with the endpoint denoted by $\star$; Initial conditions are sampled from the square region.)
• Figure 3: Boxplots: (a) Comparison of SKEL with other Koopman learning methods. Outliers were clipped for better visibility of boxes. Number of outliers with NSE $>1$ from left to right: 1 (SKEL), 15 (LKIS), 0 (SOC); (b) Training loss for each method; (c) Normalized simulation error of learned controllers on the test set. From left to right: linear parameterization of $\alpha$ --- $c_1=1$, $c_1=10$ and $c_1=100$, nonlinear parameterization of $\alpha$ --- $c_1=100$, behavioural cloning (BC). Number of clipped outliers from left to right: 4, 2, 1, 0, 5.
• Figure 4: Simulations of the closed loop. (Blue dotted lines: trajectories produced by the controllers; Solid black line: true trajectories with the endpoint denoted by the star.)
• Figure 5: Scatter plot of total time to convergence of the proposed method vs. the PGD algorithm of havens2021imitation in log-log scale, plus lines of best fit.