Sampled observability of discrete heterogeneous ensembles from anonymized output measurements

We consider the problem of reconstructing the initial states of a group of heterogeneous LTI systems in the situation that output measurements of the subsystems are received at discrete time steps and in an anonymized manner. By the latter we refer to the fact that in each time step, the association of the output measurements of the subsystems and the subsystem that produced it is lost. Therefore, the output measurements at a given time step are called output snapshot, as they capture only the group´s outputs as a whole. This setup is motivated by state estimation problems for populations of dynamical systems in which the output measurements of the subsystems are only given as statistics. We give a theoretical result concerning the number of measurement times needed in order to be able to uniquely reconstruct the initial states, and furthermore illustrate a practical method for state estimation under noisy output snapshots.