1. University of Colorado Boulder ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 10: Batch Processors 1

2. University of Colorado Boulder  Homework 3 Graded ◦ Comments included on D2L ◦ Any questions, talk with us this week  Homework 4 due Today  Homework 5 due next week  Bobby Braun will be a guest speaker next Thursday. ◦ NASA Chief Technologist ◦ Now at Georgia Tech.  Exam on 10/11.  I’ll be out of the state 10/9 – 10/16; we’ll have guest lecturers then. 2

3. University of Colorado Boulder  We’re right on track with the syllabus.  Topics today: ◦ Least Squares ◦ Weighted Least Squares ◦ Minimum Norm ◦ A Priori ◦ Minimum Variance ◦ The Batch Processor ◦ Bayesian 3

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14. University of Colorado Boulder  Given: ◦ Nominal state at some time ◦ Dynamical Models ◦ Observations  Want: ◦ Best estimate of state at some (other) time ◦ Covariance of that state  Optional: ◦ a priori information about state or covariance ◦ Weighting matrix for observations ◦ Statistical information about observations 14

15. University of Colorado Boulder  The state has n parameters ◦ n unknowns at any given time.  There are l observations of any given type.  There are p types of observations (range, range-rate, angles, etc)  We have p x l = m total equations.  Three situations: ◦ n < m : Least Squares ◦ n = m : Deterministic ◦ n > m : Minimum Norm 15

16. University of Colorado Boulder  The state deviation vector that minimizes the least-squares cost function:  Additional Details: ◦ is called the normal matrix ◦ If H is full rank, then this will be positive definite.  If it’s not then we don’t have a least squares estimate!

17. University of Colorado Boulder  The state deviation vector that minimizes the least-squares cost function:  Additional Details: ◦ is related to the variance-covariance matrix ◦ If it exists (i.e., if H T H is invertible), then P k is symmetric and positive definite.

18. University of Colorado Boulder  The Minimum Norm Solution

19. University of Colorado Boulder For the least squares solution to exist m ≥ n and H be of rank n Consider a case with m ≤ n and rank H < n There are more unknowns than linearly independent observations

20. University of Colorado Boulder Option 1: specify any n – m of the n components of x and solve for remaining m components of x using observation equations with = 0 Result: an infinite number of solutions for Option 2: use the minimum norm criterion to uniquely determine Using the generally available nominal/initial guess for x the minimum norm criterion chooses x to minimize the sum of the squares of the difference between X and X* with the constraint that = 0

21. University of Colorado Boulder Recall: Want to minimize the sum of the squares of the difference given = 0 That is

22. University of Colorado Boulder Hence the performance index becomes:

23. University of Colorado Boulder Hence the performance index becomes:

24. University of Colorado Boulder Apply when there are more unknowns than equations or more equations than unknowns

25. University of Colorado Boulder  Weighted Least Squares

26. University of Colorado Boulder  If we have information about how to weigh different observations in the filter 26

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36. University of Colorado Boulder  Least Squares  Weighted Least Squares 36

37. University of Colorado Boulder  Least Squares  Weighted Least Squares 37 Note: These P matrices are not necessarily the covariance matrices. They ARE if W is properly selected.

38. University of Colorado Boulder  Weighted Least Squares with a priori information

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40. University of Colorado Boulder Given

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44. University of Colorado Boulder  Least Squares  Weighted Least Squares  Least Squares with a priori 44

45. University of Colorado Boulder  Minimum Variance Estimate

46. University of Colorado Boulder  We can use the Minimum Variance Estimator to generate the best estimate of the state, given any statistical information about the observations. ◦ Varying standard deviations ◦ Correlations of observations over time ◦ Observation drift ◦ Etc. 46

47. University of Colorado Boulder  Given: System of state-propagation equations and observation state equations:  Find: The linear, unbiased, minimum variance estimate,, of the state. 47

48. University of Colorado Boulder  Given: System of state-propagation equations and observation state equations:  Find: The linear, unbiased, minimum variance estimate,, of the state.  Linear: 48

49. University of Colorado Boulder  Given: System of state-propagation equations and observation state equations:  Find: The linear, unbiased, minimum variance estimate,, of the state.  Unbiased: 49

50. University of Colorado Boulder  Given: System of state-propagation equations and observation state equations:  Find: The linear, unbiased, minimum variance estimate,, of the state.  Min Variance Estimate:  (Jump from the derivation to 4.4.17 – see book for details) 50

51. University of Colorado Boulder  Given: System of state-propagation equations and observation state equations:  Find: The linear, unbiased, minimum variance estimate,, of the state.  Best Estimate: 51

52. University of Colorado Boulder  Least Squares  Weighted Least Squares  Least Squares with a priori  Min Variance 52

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54. University of Colorado Boulder  Least Squares  Weighted Least Squares  Least Squares with a priori  Min Variance  Min Variance with a priori 54

55. University of Colorado Boulder  Homework 3 Graded ◦ Comments included on D2L ◦ Any questions, talk with us this week  Homework 4 due Today  Homework 5 due next week  Bobby Braun will be a guest speaker next Thursday. ◦ NASA Chief Technologist ◦ Now at Georgia Tech.  Exam on 10/11.  I’ll be out of the state 10/9 – 10/16; we’ll have guest lecturers then. 55