  Recursive least squares (RLS) is derived as the recursive minimizer of the
least-squares cost function. Moreover, it is well known that RLS is a special
case of the Kalman filter. This work presents the Kalman filter least squares
(KFLS) cost function, whose recursive minimizer gives the Kalman filter. KFLS
is an extension of generalized forgetting recursive least squares (GF-RLS), a
general framework which contains various extensions of RLS from the literature
as special cases. This then implies that extensions of RLS are also special
cases of the Kalman filter. Motivated by this connection, we propose an
algorithm that combines extensions of RLS with the Kalman filter, resulting in
a new class of adaptive Kalman filters. A numerical example shows that one such
adaptive Kalman filter provides improved state estimation for a
mass-spring-damper with intermittent, unmodeled collisions. This example
suggests that such adaptive Kalman filtering may provide potential benefits for
systems with non-classical disturbances.
