線形フィルタリング理論 : H_2からH_∞へ

抄録

This paper reviews the Kalman (or H_2) filter and the recent development of H_∞ filter, with a preliminary remark on earlier mathematical works related to the least-squares estimation. For the linear Gaussian state-space model, it is shown that the Kalman filter is a dynamical system that produces the least-squares estimates of the states given observations and that the Kalman gain is determined by the associated Riccati equation. We then show that the H_∞ filtering problem is a minimax estimation problem that minimizes the energy of estimation errors maximized over all possible disturbances and initial states. A comparison of the Kalman filter and the H_∞ filter is made in frequency domain to illustrate the basic feature of two filters. After a brief introduction of Krein space, we also show that the H_∞ (central) filter is the Kalman filter for a Krein space state-space model with indefinite noise Gramian.