Learning Stable Models for Prediction and Control

• Published in: IEEE transactions on robotics Vol. 39; no. 3; pp. 2255 - 2275
• Main Authors: Mamakoukas, Giorgos, Abraham, Ian, Murphey, Todd D.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.06.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Analytical models; Basis functions; Control Lyapunov functions; data-driven control; Koopman operator; Learning; Liapunov functions; Mathematical models; Nonlinear dynamical systems; Nonlinear systems; Numerical stability; Operators (mathematics); Predictive models; Robots; stability; Stability analysis
• ISSN: 1552-3098; 1941-0468
• DOI: 10.1109/TRO.2022.3228130

In this article, we demonstrate the benefits of imposing stability on data-driven Koopman operators. The data-driven identification of stable Koopman operators (DISKO) is implemented using an algorithm [1] that computes the nearest stable matrix solution to a least-squares reconstruction error. As a first result, we derive a formula that describes the prediction error of Koopman representations for an arbitrary number of time steps, and which shows that stability constraints can improve the predictive accuracy over long horizons. As a second result, we determine formal conditions on basis functions of Koopman operators needed to satisfy the stability properties of an underlying nonlinear system. As a third result, we derive formal conditions for constructing Lyapunov functions for nonlinear systems out of stable data-driven Koopman operators, which we use to verify stabilizing control from data. Finally, we demonstrate the benefits of DISKO in prediction and control with simulations using a pendulum and a quadrotor and experiments with a pusher-slider system. The paper is complemented with a video: https://sites.google.com/view/learning-stable-koopman .