  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.
