An introduction to extended dynamic mode decomposition: Estimation of the Koopman operator and outputs

Nibodh Boddupalli

Published 2021-08-03Version 1

Many physical systems exhibit phenomena with unknown governing dynamics. These systems can be high dimensional and often partially observed. Recently, an emerging set of operator-theoretic tools have gained traction, centered on discovering linear representations or approximations of nonlinear dynamical systems in function spaces. Originally derived for Hamiltonian systems, popular numerical and theoretical techniques for Koopman operator theory enable input-output perspective, and spectral analysis of nonlinear systems. One of the primary interests in the development and further exploration of Koopman operator framework is its relations to the properties of the underlying dynamical systems. For this, we briefly introduce the Koopman operator without delving into its spectral properties. We then present vector and matrix representations of observables and the Koopman operator as a projection onto some finite-dimensional function space without assumptions of its invariance to the action of the Koopman operator or spanning the outputs which is the basis for the extended dynamic mode decomposition (EDMD) algorithm. EDMD is a modal decomposition algorithm which is a popular numerical method for obtaining finite-sections of the Koopman operator that has proved useful for predictions and control in nonlinear systems. We also include output data containing both expected and unknown observables.