1. STATISTICAL ORBIT DETERMINATION Kalman (sequential) filter
ASEN 5070
LECTURE 18
10/09/09

2. Kalman (Sequential Filter)
Kalman solved the problem:
given , , and Find ,
using
is white noise i.e. it is not correlated with any other random variable (e.g., ) and not correlated with past values of .

3. Alternate derivation of the Kalman (Sequential) Filter algorithm
Begin with Eq (4.4.29)
Given , ,
Where and
Note that this requires an matrix inversion where n is the dimension of the state deviation vector, We can reformulate Eq (4.4.29) into a form requiring the inversion of a matrix, where p is the dimension of the observation deviation vector, Note that in general
Using the Schure identity (Theorem 4 of Appendix B)
Let , ,

4. Alternate derivation of the Kalman (Sequential) Filter algorithm
Then the 1st term in Eq (4.4.29),
, becomes
Using the fact that
we can write as
Hence, Eq (1) can be written as
Note that this involves a matrix inversion. If the observation at each time is a scalar such as range, this will be a scalar inversion. In any case we can process the observations one at a time so that this need only be a scalar inversion.

5. Alternate derivation of the Kalman (Sequential) Filter algorithm
We may simplify this expression further. In Eq (2) define a quantity called the Kalman or Optimal gain as
Hence, from Eq (2)
Substituting Eq (4) into Eq (4.4.29) yields

6. Alternate derivation of the Kalman (Sequential) Filter algorithm
But
(Substitute for )
Factor out
We have shown that

7. Alternate derivation of the Kalman (Sequential) Filter algorithm
Hence, Eq (5) becomes
The Sequential Processing Algorithm is illustrated in Fig of the text. The Kalman or Sequential Filter is generally divided into a time and measurement (or observation) update, i.e.
Time Update
Measurement Update

8. Matrix Dimensions
The Kalman (sequential) filter differs from the batch filter as follows:
It uses in place of .
IT uses in place of
This means that
Is reinitialized at each observation time.
Or if we don’t reinitialize, then
i.e., we must invert
At each stage

9. The time update
Set k=k+1 and return to (1)

10. What is the Role of the Kalman Gain?
Let’s examine a simple case where & are both scalars and we
observe directly i.e. so
Furthermore assume that is a constant so that
Then at time k,

11. What is the Role of the Kalman Gain?
The Kalman is
The measurement update becomes

12. What is the Role of the Kalman Gain?
Hence, the best estimate of is a weighted average of the predicted
estimate and the measurement. If the measurement is very noisy
(inaccurate) relative to the predicted state ( )
Then
and
Hence, the measurement has little effect.
Conversely if
and the predicted estimate has little effect. In general the Kalman gain
provides a relative weighting between the apriori and the tracking data.

13. Problem 43c
Given:
apriori information

14. Problem 43c for all values of i
Find: using the Kalman filter. Use the batch
processor to find
1. Time update at (Note that there is no observations at )

15. Problem 43c
Measurement update:
Kalman Gain

16. Problem 43c
Compute

17. Problem 43c

18. Problem 43c
Using the batch processor to get

19. Problem 43c
Hence
agrees with from
Kalman filter

20. Problem 43c

21. Problem 43c
Map the estimation error covariance matrix from the epoch time, to in one time unit steps and plot the envelope of the one standard deviation error in x1 and x2 and the correlation coefficient between x1 and x2 . Recall that

22. Problem 43c

23. Processing Observations One at a Time
Given a vector of observations at we may process these as
scalar observations by performing the measurement update times and
skipping the time update.
Algorithm
1. Do the time update at
2. Measurement update – we must deal with the matrices

24. Processing Observations One at a Time
2a. Process the 1st element of Compute
where

25. Processing Observations One at a Time
2. Do not do a time update but do a measurement update by processing
the 2nd element of (i.e and are not mapped to )
Compute
etc.
Process pth element of .
Compute

26. Processing Observations One at a Time
let
Time update to and repeat this procedure
Note: must be a diagonal matrix. If not we would apply a whitening
Transformation as described in the next slides.

27. Whitening Transformation
(1)
Factor where is the square root of
multiply Eq. (1) by
is chosen to be upper
triangular
let

28. Whitening Transformation
Now
Hence, the new observation has an error with zero mean and
unit variance. We would now process the new observation and use .