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Approximating the Koopman operator from data is numerically challenging whenmany lifting functions are considered. Even low-dimensional systems can yieldunstable or ill-conditioned results in a high-dimensional lifted space. In thispaper, Extended Dynamic Mode Decomposition (DMD) and DMD with control, twomethods for approximating the Koopman operator, are reformulated as convexoptimization problems with linear matrix inequality constraints. Asymptoticstability constraints and system norm regularizers are then incorporated asmethods to improve the numerical conditioning of the Koopman operator.Specifically, the H-infinity norm is used to penalize the input-output gain ofthe Koopman system. Weighting functions are then applied to penalize the systemgain at specific frequencies. These constraints and regularizers introducebilinear matrix inequality constraints to the regression problem, which arehandled by solving a sequence of convex optimization problems. Experimentalresults using data from an aircraft fatigue structural test rig and a softrobot arm highlight the advantages of the proposed regression methods.

System Norm Regularization Methods for Koopman Operator Approximation