Approximations to a particular class of ill-conditioned Riccati matrices

The solution of the Riccati equation corresponding to the Kalman filter, with a single perfect measurement and possibly one or more noisy measurements, requires the numerical integration of a nonlinear matrix differential equation whose dimension is one less than that of the system state vector. A similar requirement exists for the linear regulator problem when one of the controls has no penalty and the others, if they exist are penalized. One would think that similar requirements would exist when the single measurement is almost perfect and the single control is only slightly penalized; however, not only is the dimension of the matrix differential equations increased (it is equal to the dimension of the state), the small numbers make the solution ill-conditioned. Approximations to these ill-conditioned Riccati matrices are derived along with measures of the slight degradations in the performances of the estimators and regulators resulting from their use.