Approximate Optimal Filter for Linear Gaussian Time-invariant Systems
This work addresses state estimation for control systems, but it is incremental as it applies an existing policy iteration technique to a known filtering context.
The paper tackled the problem of state estimation in linear Gaussian time-invariant systems by proposing an approximate optimal filter that transforms filtering into a control problem, achieving results within 2% accuracy of the steady-state Kalman gain.
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 %.