Learning Stable Koopman Embeddings for Identification and Control

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.