The above Kalman filter formalism is under the assumptions that the system-noise process and the measurement-noise process are uncorrelated and that they are all Gaussian white noise sequences. These assumptions are adequate in solving the problems addressed in this monograph. In the case that noise processes are correlated or they are not white (i.e., colored), the reader is referred to [4] for the derivation of the Kalman filter equations. The numerical instability of Kalman filter implementation is well known. Several techniques are developed to overcome those problems, such as square-root filtering and U-D factorization. See [14] for a thorough discussion.

There exist many other methods to solve the parameter estimation problem: general minimization procedures, weighted least-squares method, and the Bayesian decision-theoretic approach. In the appendix to this chapter, we review briefly several least-squares techniques. We choose the Kalman filter approach as our main tool to solve the parameter estimation problem. This is for the following reasons:

- the Kalman filter takes explicitly into account the measurement uncertainties,
- the Kalman filter takes measurements into account incrementally (recursivity),
- the Kalman filter is a simple and efficient procedure to solve the problem (computational tractability),
- the Kalman filter can take into account
*a priori*information, if any.

The linearization of a nonlinear model leads to small errors in the
estimates, which in general can be neglected, especially if the
relative accuracy is better than 10% [16, 5]. However, as pointed
by Maybank [13], the extended Kalman filter seriously *
underestimates* covariance.
Furthermore, if the
current estimate is very different from the true
one, the first-order approximation, (22
and 23), is not good anymore, and the final estimate
given by the filter may be significantly different from the true one.
One approach to reduce the effect of nonlinearities is to apply
iteratively the Kalman filter (called the *
iterated extended Kalman filter*).

Thu Feb 8 11:42:20 MET 1996