1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 21: A Bayesian Approach to the Kalman Filter Derivation

2. University of Colorado Boulder  Homework 6 Due Friday  No lecture quiz this week 2

3. University of Colorado Boulder 3 Homework 6 – Common Question

4. University of Colorado Boulder  What are the dimensions of the Htilde matrix?  Since the observations are generated via a single ground station, what is the partial w.r.t. to the other stations?  Need to add logic to your code to properly select the non-zero columns for the ground station partials! 4

5. University of Colorado Boulder 5 The Kalman Filter – A Bayesian Approach Ho and Lee, “A Bayesian Approach to Problems in Stochastic Estimation and Control”, IEEE Transactions on Automatic Control, DOI: 10.1109/TAC.1964.1105763

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8. University of Colorado Boulder  We start with a previous state PDF at some time t k-1:  Assume a linear description of the dynamics: 8

9. University of Colorado Boulder  If we map the (Gaussian) previous-state PDF through a set of linear equations, what is the output? 9

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11. University of Colorado Boulder  A linear relationship between the state and the observations, i.e.,:  All input PDFs are independent and Gaussian: 11

12. University of Colorado Boulder  As you will show in HW7: 12

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15. University of Colorado Boulder  Do we know anything about the PDF of ε ?  Do we know if ε is independent of x ? 15

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19. University of Colorado Boulder  We have a solution, but it is not “elegant”  Can we manipulate the terms in the exponent to look like something a little more familiar? (Perhaps a Gaussian…)  We can, but we need a couple of tricks… 19

20. University of Colorado Boulder 20  Schur Identity (Appendix B, Theorem 4):

21. University of Colorado Boulder 21  We need to “complete the square”:  After applying those tricks and about 1-2 pages of linear algebra…

22. University of Colorado Boulder  We have the Kalman filter as derived using Bayes theorem! 22

23. University of Colorado Boulder  In this derivation, what did we assume? 23

24. University of Colorado Boulder  Since the Kalman and the Batch processor are mathematically equivalent, then the batch can also be derived via Bayes theorem, right? ◦ Yes! (See book section 4.5)  Both proofs/arguments work, but this important derivation of the Kalman filter was not included in the book 24