System Norm Regularization Methods for Koopman Operator Approximation
This work addresses numerical challenges in Koopman operator approximation for dynamical systems, offering incremental improvements for applications like structural testing and robotics.
The paper tackled the numerical instability in approximating the Koopman operator from data by reformulating Extended DMD and DMD with control as convex optimization problems with stability constraints and system norm regularizers, resulting in improved conditioning demonstrated on aircraft and soft robot datasets.
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.