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ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones

ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 24: Numeric Considerations and Introduction to Square-Root Algorithms. Bierman Example of Poorly Conditioned System. Problem Statement. Derivation of Exact Solution.

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ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones

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  1. ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 24: Numeric Considerations and Introduction to Square-Root Algorithms

  2. Bierman Example of Poorly Conditioned System

  3. Problem Statement

  4. Derivation of Exact Solution • Process the first observation:

  5. Derivation of Exact Solution • Process the second observation:

  6. Fixed-Point Arithmetic Error • Consider the implementation on a computer with a limited precision:

  7. Kalman Filter Result with Limited Precision

  8. Joseph Formulation • Exact to order ε

  9. Covariance Matrix Solutions

  10. Potter Algorithm – Motivation and Derivation

  11. Covariance Condition Number • The condition number of P may be defined 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?

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

  13. Time Update for W (one method)

  14. Square-Root Measurement Update

  15. Square-Root Measurement Update

  16. 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?

  17. Potter Square-Root Filter Derivation

  18. Potter Square-Root Filter Derivation

  19. Potter Square-Root Filter Derivation

  20. Potter Square-Root Filter Derivation

  21. 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. 335-340)

  22. 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 (next week)

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