System Norm Regularization Methods for Koopman Operator Approximation

TL;DR

This paper introduces convex optimization-based regularization techniques, including system norm constraints, to improve the numerical stability and conditioning of Koopman operator approximations from data.

Contribution

It reformulates Extended DMD and DMD with control as convex problems with stability and norm regularizers, enhancing approximation robustness.

Findings

Improved numerical conditioning of Koopman operator approximations.

Enhanced stability through asymptotic stability constraints.

Effective application to real-world systems like aircraft and robots.

Abstract

Approximating the Koopman operator from data is numerically challenging when many lifting functions are considered. Even low-dimensional systems can yield unstable or ill-conditioned results in a high-dimensional lifted space. In this paper, Extended Dynamic Mode Decomposition (DMD) and DMD with control, two methods for approximating the Koopman operator, are reformulated as convex optimization problems with linear matrix inequality constraints. Asymptotic stability constraints and system norm regularizers are then incorporated as methods to improve the numerical conditioning of the Koopman operator. Specifically, the H-infinity norm is used to penalize the input-output gain of the Koopman system. Weighting functions are then applied to penalize the system gain at specific frequencies. These constraints and regularizers introduce bilinear matrix inequality constraints to the regression problem, which are handled by solving a sequence of convex optimization problems. Experimental results using data from an aircraft fatigue structural test rig and a soft robot arm highlight the advantages of the proposed regression methods.