Recursive linear estimation in Krein spaces - Part I. Theory

We develop a self-contained theory for linear estimation in Krein spaces. The theory is based on simple concepts such as projections and matrix factorizations, and leads to an interesting connection between Krein space projection and the computation of the stationary points of certain second order (or quadratic) forms. We use the innovations process to obtain a rather general recursive linear estimation algorithm, which when specialized to a state space model yields a Krein space generalization of the celebrated Kalman filter with applications in several areas such as H^∞-filtering and control, game problems, risk sensitive control, and adaptive filtering.