Koopman Resolvents of Nonlinear Discrete-Time Systems: Formulation and Identification

The Koopman operator framework is a promising direction of analysis and synthesis of systems with nonlinear dynamics based on (linear) Koopman operators. In this paper, we address the resolvent of a Koopman operator for a nonlinear autonomous discrete-time system, which we call the Koopman resolvent, and its identification problem. First, we show that for the nonlinear system with a scalar-valued output, the
$z$
- transform of the output is represented by the action of Koop-man resolvent. Second, we describe an identification method of the Koopman resolvent directly from time-series data of the output, in which we estimate parameters of the resolvent as well as poles and residues of the z-transform of the output. By combining the so-called frequency-domain Prony method with the Vandermonde-Cauchy form in the Dynamic Mode Decomposition (DMD), we propose the method which we call the frequency-domain DMD, in which all the unknowns can be estimated in the frequency domain.