System norm regularization methods for Koopman operator approximation

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 operator, reformulated as convex optimization problems linear matrix inequality constraints. Asymptotic stability constraints system norm regularizers then incorporated to improve numerical conditioning of operator. Specifically, H-infinity used penalize input-output gain system. Weighting applied at specific frequencies. These introduce bilinear regression problem, which handled by solving sequence problems. Experimental using an aircraft fatigue structural test rig soft robot arm highlight advantages proposed methods.