Connections between $H_2$ optimal filters and unknown input observers : performance limitations of $H_2$ optimal filtering

For linear time invariant continuous-time systems with either unknown or white noise input, two well-known filtering problems are considered. These are the unknown input observer problem and the Kalman filtering problem. Most of the available literature on Kalman filtering considers the so-called regular filtering problem. We consider here the general singular filtering problem. We show that such a Kalman filtering problem for a given system can be transformed to the unknown input observer problem for an auxiliary system constructed from the data of the given system. Such transformations between these two filtering problems enable us to study various properties of Kalman filtering, including existence and uniqueness of Kalman filters, computation of performance indices of Kalman filtering, and performance limitations of Kalman filtering as related to the structural properties of the given system.