Transcript Lecture 21
ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 21: A Bayesian Approach to the Kalman Filter Derivation University of Colorado Boulder Homework 6 Due Friday No lecture quiz this week University of Colorado Boulder 2 Homework 6 – Common Question University of Colorado Boulder 3 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! University of Colorado Boulder 4 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 University of Colorado Boulder 5 University of Colorado Boulder 6 University of Colorado Boulder 7 We start with a previous state PDF at some time tk-1: Assume a linear description of the dynamics: University of Colorado Boulder 8 If we map the (Gaussian) previous-state PDF through a set of linear equations, what is the output? University of Colorado Boulder 9 University of Colorado Boulder 10 A linear relationship between the state and the observations, i.e.,: All input PDFs are independent and Gaussian: University of Colorado Boulder 11 As you will show in HW7: University of Colorado Boulder 12 University of Colorado Boulder 13 University of Colorado Boulder 14 Do we know anything about the PDF of ε ? Do we know if ε is independent of x ? University of Colorado Boulder 15 University of Colorado Boulder 16 University of Colorado Boulder 17 University of Colorado Boulder 18 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… University of Colorado Boulder 19 Schur Identity (Appendix B, Theorem 4): University of Colorado Boulder 20 We need to “complete the square”: After applying those tricks and about 1-2 pages of linear algebra… University of Colorado Boulder 21 We have the Kalman filter as derived using Bayes theorem! University of Colorado Boulder 22 In this derivation, what did we assume? University of Colorado Boulder 23 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 University of Colorado Boulder 24