Extending dynamic mode decomposition to data from multiple outputs

TL;DR

This paper extends the application of dynamic mode decomposition to systems with multiple outputs, enabling better spectral analysis of complex systems through regularized least-squares approximations.

Contribution

It introduces a method to apply extended dynamic mode decomposition to multi-output data, broadening the scope of Koopman-based system analysis.

Findings

Effective approximation of observables from multiple outputs.

Analytic solutions for regularized least-squares problems.

Enhanced spectral analysis capabilities for complex systems.

Abstract

System identification based on Koopman operator theory has grown in popularity recently. Spectral properties of the Koopman operator of a system were proven to relate to properties like invariant sets, stability, periodicity, etc. of the underlying system. Estimation of these spectral objects has become widely accessible with the numerous algorithms developed in recent years. We show how one such algorithm -- extended dynamic mode decomposition -- can be used on data from multiple outputs of a system. These outputs that are functions of state are called observables in the literature and could be known outputs like the state itself or unknown outputs like data from sensors used in systems of biological interest. To this end, we approximate the desired observables and their iterates in time using minimizers of regularized least-squares problems which have analytic solutions with heuristic provisions for expected estimates.