Learning deep Koopman operators with convex stability constraints

TL;DR

This paper introduces a new method for ensuring the stability of learned Koopman operators in discrete-time systems using convex constraints and a control barrier function-based optimization, improving flexibility and performance.

Contribution

We propose a novel stability condition for Koopman matrices and a gradient descent method that enforces stability constraints during learning, enhancing control applications.

Findings

Achieves near state-of-the-art performance in stability enforcement.

Provides a flexible optimization framework for stability in Koopman learning.

Demonstrates effectiveness through comparative experiments.

Abstract

In this paper, we present a novel sufficient condition for the stability of discrete-time linear systems that can be represented as a set of piecewise linear constraints, which make them suitable for quadratic programming optimization problems. More specifically, we tackle the problem of imposing asymptotic stability to a Koopman matrix learned from data during iterative gradient descent optimization processes. We show that this sufficient condition can be decoupled by rows of the system matrix, and propose a control barrier function-based projected gradient descent to enforce gradual evolution towards the stability set by running an optimization-in-the-loop during the iterative learning process. We compare the performance of our algorithm with other two recent approaches in the literature, and show that we get close to state-of-the-art performance while providing the added flexibility of allowing the optimization problem to be further customized for specific applications.