Sampled Observability and State Estimation of Linear Discrete Ensembles

• Published in: IEEE transactions on automatic control Vol. 62; no. 5; pp. 2406 - 2418
• Main Authors: Shen Zeng, Ishii, Hideaki, Allgower, Frank
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.05.2017; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Ensemble observability; Group theory; Irregular sampling; Linear systems; Observability; Observability (systems); sampled data control; Sociology; State estimation; Statistics; Time measurement
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2016.2613478

We consider the problem of reconstructing the initial states of a finite group of structurally identical linear systems in the situation that output measurements of the individual systems are received at discrete time steps and in an anonymized manner: While we do know all output measurements of the individual systems in the group, we do not know which output measurement corresponds to which system. This state estimation problem addresses the essence of state estimation problems for populations, in which the output measurements of the individual systems are given only as statistics. We adopt a measure theoretical approach in which the group is modelled by an LTI system describing the structure of the individual systems and an initial state which is expressed by a discrete measure. In this framework we derive a geometric characterization for the state estimation to admit a unique solution, which combined with a result on the observability of linear systems under irregular sampling, yields a sufficient condition for the sampled observability of discrete ensembles. As a supplement to our theoretical findings, we provide illustrations by means of simulation examples. Furthermore we consider the practical state estimation problem under noisy output measurements.