Ensemble Observability of Linear Systems

TL;DR

This paper investigates the problem of reconstructing probability distributions of initial states in linear systems from output distributions over time, proposing two complementary methods based on inverse problems and moment reconstruction.

Contribution

It introduces two novel approaches for ensemble observability in linear systems, linking inverse problems with tensor systems and moments reconstruction.

Findings

Established a connection between ensemble observability and tomography.

Developed a tensor systems framework for moments dynamics.

Demonstrated the duality of geometric and systems theoretic viewpoints.

Abstract

We address the observability problem for ensembles that are described by probability distributions. The problem is to reconstruct a probability distribution of the initial state from the time-evolution of the probability distribution of the output under a classical finite-dimensional linear system. We present two solutions to this problem, one based on formulating the problem as an inverse problem and the other one based on reconstructing all the moments of the distribution. The first approach leads us to a connection between the reconstruction problem and mathematical tomography problems. In the second approach we use the framework of tensor systems to describe the dynamics of the moments which leads to a more systems theoretic treatment of the reconstruction problem. Furthermore we show that both frameworks are inherently related. The appeal of having two dual view points, the first being more geometric and the second one being more systems theoretic, is illuminated in several examples of theoretical or practical importance.