  This paper studies the state estimation problem of linear discrete-time
systems with stochastic unknown inputs. The unknown input is a wide-sense
stationary process while no other prior informaton needs to be known. We
propose an autoregressive (AR) model based unknown input realization technique
which allows us to recover the input statistics from the output data by solving
an appropriate least squares problem, then fit an AR model to the recovered
input statistics and construct an innovations model of the unknown inputs using
the eigensystem realization algorithm (ERA). An augmented state system is
constructed and the standard Kalman filter is applied for state estimation. A
reduced order model (ROM) filter is also introduced to reduce the computational
cost of the Kalman filter. Two numerical examples are given to illustrate the
procedure.
