On the Projective Geometry of Kalman Filter

This paper is about the asymptotic behavior of the Kalman filter [11]. The Kalman-Bucy filter merges predictions from a trusted model of the dynamics of the system with incoming measurements in order to get an accurate, real-time estimate of the unknown internal state of the system. The estimation relies on the computation of a positive semidefinite matrix P, the covariance of the estimation error. The difference equation verified by P is a discrete-time algebraic Riccati equation. Kalman showed that, for a linear time-invariant system, under detectability conditions, the Riccati equation converges to a fixed point, which is unique under certain stabilizability conditions ([10], see also [9]). The classical convergence analysis requires several steps, showing that the error covariance is upper bounded, that, with zero initial value, it is monotone increasing, so that it admits a limit, and then proving that the corresponding filter is stable and that the limit is the same for all initial covariances. In [4] Bougerol proposed a more geometric convergence analysis by showing that the discrete-time Riccati iteration is a contraction for the Riemannian metric associated to the cone of positive definite matrices. Other authors elaborated along these lines (see e.g.