Some new algorithms for recursive estimation in constant linear systems

Recursive least-squares estimates for processes that can be generated from finite-dimensional linear systems are usually obtained via an
n \times n
matrix Riccati differential equation, where
n
is the dimension of the state space. In general, this requires the solution of
n(n + 1)/2
simultaneous nonlinear differential equations. For constant parameter systems, we present some new algorithms that in several cases require only the solution of less than
2np
or
n(m + p)
simultaneous nonlinear differential equations, where
m
and
p
are the dimensions of the input and observation processes, respectively. These differential equations are said to be of Chandrasekhar type, because they are similar to certain equations introduced in 1948 by the astrophysicist S. Chandrasekhar, to solve finite-interval Wiener-Hopf equations arising in radiative transfer. Our algorithms yield the gain matrix for the Kalman filter directly without having to solve separately for the error-covariance matrix and potentially have other computational benefits. The simple method used to derive them also suggests various extensions, for example, to the solution of nonsymmetric Riccati equations.