  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.
