Introduction

So far we have seen how linear Kalman filters are designed and how they perform when the real world can be described by linear differential equations expressed in state-space form and when the measurements are linear functions of the states. In addition, we have seen how extended Kalman filtering can be used when the real world is either described by nonlinear differential equations or measurements are not linear functions of the filter states. With both the linear and extended Kalman filters, no a priori assumption was made concerning additional information that might be available. The only information required to start the extended Kalman filter was the initial state estimates and an initial covariance matrix.

In some problems, such as those involving satellite tracking, a great deal of additional a priori information exists. For example, in satellite tracking we know in advance the approximate orbit of the satellite. In principle, this type of information should make the overall filtering problem easier. In this chapter two examples will be presented showing how a linearized Kalman filter can be built if nominal trajectory information exists for a nonlinear problem. In addition, we will also see what happens to the performance of the linearized Kalman filter if the nominal trajectory information is not accurate.

Theoretical Equations ^[1]

To apply linearized Kalman filtering techniques, it is first necessary to describe the real world by a set of nonlinear differential equations. We have already seen from the chapters on extended Kalman filtering that these equations...