On Koopman and dynamic mode decompositions for application to dynamic data with low spatial dimension

Dynamic Mode Decomposition (DMD) has attracted a fair amount of attention in recent years. Applications of DMD have ranged from fluid mechanics, thermal dynamics in a building, power systems, and so on. Connections to the so-called Koopman operator have been highlighted, and variants of DMD have been called the Koopman Mode Decomposition (KMD). In many applications, these techniques are used as an attempt to derive a reduced-order system or to identify coherent dynamic structures characterized by modes oscillating with single frequencies. Therefore, they become highly relevant to control applications via the data-derived system modeling. However, in many utilizations other than within computational fluid dynamics, the spatial dimension, here defined as the number of measurement locations, of dynamic data becomes low, or even singular in the case of just one measurement sensor. Thus, the question arises on how well the different variants of DMD/KMD suggested in the literature work in this setting. In this paper, we apply three different DMD/KMD algorithms to experimental data and evaluate how well they identify modes in terms of their frequency, and show how the sampling frequency and temporal length of the data window affect the decompositions. We interpret the results with support from a new unified interpretation of the algorithms. It is shown that the so-called vector Prony analysis works well for a small spatial dimension, where standard DMD is not suitable.