1. STATISTICAL ORBIT DETERMINATION Coordinate Systems and Time Kalman Filtering
ASEN 5070
LECTURE 21
10/16/09

5. Arc sec

9. Numerical Considerations for the Kalman Filter
Time and measurement update
Of the estimation error covariance
matrix

10. Numerical Considerations for the Kalman Filter

11. Equivalence of Joseph and Conventional Formulation of the Measurement Update of P
Need to show that
First derive the Joseph formulation
subst.
subtract from both sides and rearrange

12. Equivalence of Joseph and Conventional Formulation of the Measurement Update of P
where
since is independent of

13. Equivalence of Joseph and Conventional Formulation of the Measurement Update of P
Next, show that this is the same as the conventional Kalman
subst. for

14. Example Illustrating Numerical Instability of Sequential (Kalman) Filter (see 4.7.1)
The following example problem from Bierman (1977) illustrates the numerical characteristics of several algorithms we have studied to date. Consider the problem of estimating and from scalar measurements and
Where and are uncorrelated zero mean random variables with unit variance. If the did not have the above traits we could perform a whitening transformation so that they do. In matrix form

15. Example Illustrating Numerical Instability of Sequential (Kalman) Filter
The a priori covariance associated with our knowledge of is assumed to be
Where and The quantity is assumed to be
small enough such that computer round-off produces
This estimation problem is well posed. The observation provides an estimate of which, when coupled with should accurately determine However, when the various data processing algorithms are applied several diverse results are obtained.

16. Example Illustrating Numerical Instability of Sequential (Kalman) Filter
Let the gain and estimation error covariance associated with be denoted as and respectively. Similarly the measurement is associated with and respectively. Note that this is a parameter (constant) estimation problem.
Hence,
We will process the observation one at a time. Hence,

17. Example Illustrating Numerical Instability of Sequential (Kalman) Filter
Note: Since we will process the observations one at a time, the matrix inversion in Eq (4.7.21a) is a scalar inversion. The exact solution is
where
The estimation error covariance associated with processing the first data point is

18. Example Illustrating Numerical Instability of Sequential (Kalman) Filter
where
Processing the second observation:
where

19. Example Illustrating Numerical Instability of Sequential (Kalman) Filter
The exact solution for is given by:
The conventional Kalman filter yields (using )

20. Example Illustrating Numerical Instability of Sequential (Kalman) Filter
Note that is no longer positive definite. The diagonals of a matrix must be
positive. However does exist since
Compute P2 using the Conventional Kalman
Now is not positive definite ( ) nor does it have positive terms on the diagonal. In fact, the conventional Kalman filter has failed for these observations.

21. Example Illustrating Numerical Instability of Sequential (Kalman) Filter
The Joseph formulation yields
The Batch Processor yields

22. Example Illustrating Numerical Instability of Sequential (Kalman) Filter
P2 for the batch
To order , the exact solution for is

23. Example Illustrating Numerical Instability of Sequential (Kalman) Filter
Summary of Results
Exact to order
Conventional Kalman
Joseph
Batch