  State estimation is critical to control systems, especially when the states
cannot be directly measured. This paper presents an approximate optimal filter,
which enables to use policy iteration technique to obtain the steady-state gain
in linear Gaussian time-invariant systems. This design transforms the optimal
filtering problem with minimum mean square error into an optimal control
problem, called Approximate Optimal Filtering (AOF) problem. The equivalence
holds given certain conditions about initial state distributions and policy
formats, in which the system state is the estimation error, control input is
the filter gain, and control objective function is the accumulated estimation
error. We present a policy iteration algorithm to solve the AOF problem in
steady-state. A classic vehicle state estimation problem finally evaluates the
approximate filter. The results show that the policy converges to the
steady-state Kalman gain, and its accuracy is within 2 %.
