1. ASEN 5070: Statistical Orbit Determination I Fall 2014
Professor Brandon A. Jones
Lecture 25: Potter Algorithm and
Decomposition Methods

2. Announcements/Reminders
Homework 8 Due Friday (10/31)
Lecture Quizzes
Due by 5pm Today
Next one due by 5pm 10/31
Exam 2 – Friday, November 7

3. Announcements/Reminders

4. Xkcd Comic (http://xkcd.com/1132/)

5. Potter Algorithm (continued)

6. Covariance Condition Number
The condition number of P may be described by
With p significant digits, there are estimation difficulties as
If we can’t change the condition number, is there something else we can do?

7. Square-Root Formulation
For W above, the condition number is
Is there something we can do to instead operate on W ?

8. Time Update for W (one method)

9. Derivation so far…

10. Potter Algorithm Assumptions
We must process the observations one at a time
If we have multiple observations at a single time, this requires that R be diagonal.
What can we do if the observations at a single time have a non-zero correlation?

11. Potter Square-Root Filter Derivation

12. Potter Square-Root Filter Derivation

13. Potter Square-Root Filter Derivation

14. Potter Measurement Update
Process the observations one at a time
Repeat if multiple observations available at a single time
More computationally expensive than Kalman, but more accurate
W after the measurement update is not triangular! (Important for some algorithms)
Motivates the derivation of the triangular square-root method (pp )

15. How do we get W ?
If we are given P as a priori information, how do we get W ?
If P is diagonal, this is trivial:
Great, but what if it isn’t diagonal?
Cholesky decomposition

16. Cholesky Decomposition

17. Square-Root Methods Provides improved numeric stability
Method defined by Atilde
Potter algorithm assumed the processing of one measurement at a time

18. How do we get W ?
If we are given P as a priori information, how do we get W ?
If P is diagonal, this is easy:
Great, but what if it isn’t diagonal?
Cholesky decomposition

19. Cholesky Decomposition
Cholesky Decomposition of p.d. matrix:
MATLAB:

20. Solution Algorithm Algorithm found in book Eq. 5.2.6
Implementations readily available in most high-level languages:
MATLAB:
Be sure to check the documentation for default behavior (lower or upper)

21. Cholesky-Based Least Squares

22. Weighted LS w/ A Priori Recall the weighted least squares:
Instead, we will write:
M is the information matrix

23. Solution via Inversion
Usually, we solve via matrix inversion
If the number of estimated parameters is large, then this is expensive and possibly inaccurate
Estimate gravity field of degree 360
n ≈ 129,600

24. Solution via Cholesky Decomposition
Instead, let’s write the equations in terms of the Cholesky decomposition
R here is not the obs. error covariance matrix!

25. Solve for z Using Forward Substitution
Eq in the Book

26. Solve for x Using Backward Substitution
Eq in the Book

27. Covariance Matrix Solution
We may also solve for the covariance matrix using the Cholesky decomposition

28. Covariance Matrix Solution
Using this directly still requires an n×n matrix inversion!
Eq provides a simple algorithm to get S by leveraging

29. Covariance Matrix Solution
Eq :

30. SVD-Based Least Squares (not in book)

31. Singular Value Decomposition (SVD)
The SVD of any real m×n matrix H is

32. Pseudoinverse via SVD

33. Pseudoinverse via SVD It turns out that we can solve the linear system
using the pseudoinverse given by the SVD

34. LS Solution via SVD For the linear system the solution
minimizes the least squares cost function

35. Improved Conditioning with SVD
Recall that for the normal solution,
This squares the condition number of H !
Instead, SVD operates on H, thereby improving solution accuracy

36. State Estimate Covariance via SVD
The covariance matrix P with R the identity matrix is:
Home Practice Exercise: Derive the equation for P above

37. Advantages/Disadvantages of SVD
Solving the LS problem via SVD provides one of (if not the most) numerically stable solutions
Also a square-root method (does not square the condition number of H )
Generating the SVD is more computationally intensive than most methods