Online State Estimation for Time-Varying Systems

TL;DR

This paper introduces an online algorithm for estimating the state of time-varying systems with limited measurements, using a regularized least-squares approach that updates estimates sequentially and guarantees bounded errors.

Contribution

It proposes a novel online estimation method based on a regularized least-squares framework with closed-form solutions, suitable for systems with fewer measurements than states.

Findings

Algorithm is a linear dynamical system updating with new measurements.

Conditions for contractiveness of the algorithm are established.

Estimation error bounds are derived for different noise models.

Abstract

The paper investigates the problem of estimating the state of a time-varying system with a linear measurement model; in particular, the paper considers the case where the number of measurements available can be smaller than the number of states. In lieu of a batch linear least-squares (LS) approach -- well suited for static networks, where a sufficient number of measurements could be collected to obtain a full-rank design matrix -- the paper proposes an online algorithm to estimate the possibly time-varying state by processing measurements as and when available. The design of the algorithm hinges on a generalized LS cost augmented with a proximal-point-type regularization. With the solution of the regularized LS problem available in closed-form, the online algorithm is written as a linear dynamical system where the state is updated based on the previous estimate and based on the new available measurements. Conditions under which the algorithmic steps are in fact a contractive mapping are shown, and bounds on the estimation error are derived for different noise models. Numerical simulations are provided to corroborate the analytical findings.