Partially-observable non-linear system identification

Abstract

In the paper "Koopman-based lifting techniques for nonlinear systems identification" [1], Mauroy and Goncalves present a data-driven framework for approximately identifying a given nonlinear system whose associated vector field is linear in unknown weight parameters. Informally, the framework presented in the aforementioned paper identifies these weight parameters by identifying a finite-dimensional approximation of the infinite-dimensional linear Koopman operator in the space of observables, instead of identifying the nonlinear system in the state space. It does so using trajectories of equally spaced in time data points, originating from the nonlinear system. However, the two methods presented by Mauroy and Goncalves that follow this framework rely on full observability of each data point in the given trajectories, which is rarely the case in real-world systems. Due to Takens’ Embedding Theorem, one can - in some cases - use trajectories of data points that are only partially observable to approximately construct fully observable data points, originating from a system that is an embedding of the given nonlinear system. By exploiting Takens’ Embedding Theorem, we present in this thesis a novel method for constructing approximations of trajectories of fully observable data points that originate from a given nonlinear system, assuming that we only have access to trajectories of partially observable data points that also originate from this nonlinear system. The resulting fully observable trajectories will approximately be equally spaced in time also, so that they may fit as input for at least one of the methods presented in the aforementioned paper, and therefore possibly provide us with a combined method for computing the weight parameters from partially observable data.

[1] Mauroy, A. & Goncalves, J. (2019). "Koopman-based lifting techniques for nonlinear systems identification." https://arxiv.org/pdf/1709.02003