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ASEN 5070: Statistical Orbit Determination I Fall 2013 Marco L. Balducci

ASEN 5070: Statistical Orbit Determination I Fall 2013 Marco L. Balducci Professor Brandon A. Jones Professor George H. Born Lecture 10: Minimum Norm and Weighted Least Squares With a priori. Overview. Minimum Norm Why it may be necessary Weighted a priori A derivation

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ASEN 5070: Statistical Orbit Determination I Fall 2013 Marco L. Balducci

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  1. ASEN 5070: Statistical Orbit Determination I Fall 2013 Marco L. Balducci Professor Brandon A. Jones Professor George H. Born Lecture 10: Minimum Norm and Weighted Least Squares With a priori

  2. Overview • Minimum Norm • Why it may be necessary • Weighted a priori • A derivation • Selected Topic

  3. Review • What is n ? • Dimensions in the state vector • What is l ? • Number of observations • What is p ? • Dimensions in the observation vector • What is m ? • Number of equations m = p x l

  4. Review • What is n ? • Dimensions in the state vector • What is l ? • Number of observations • What is p ? • Dimensions in the observation vector • What is m ? • Number of equations m = p x l

  5. Review • What is n ? • Dimensions in the state vector • What is l ? • Number of observations • What is p ? • Dimensions in the observation vector • What is m ? • Number of equations m = p x l

  6. Review • What is n ? • Dimensions in the state vector • What is l ? • Number of observations • What is p ? • Dimensions in the observation vector • What is m ? • Number of equations m = p x l

  7. Review • What is n ? • Dimensions in the state vector • What is l ? • Number of observations • What is p ? • Dimensions in the observation vector • What is m ? • Number of equations m = p x l

  8. Information • 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 = mtotal equations. • Three situations: • n < m: Least Squares • n = m: Deterministic • n > m: Minimum Norm

  9. Least Squares • 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!

  10. The Minimum Norm Solution

  11. Minimum Norm 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

  12. Minimum Norm 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

  13. Minimum Norm Recall: Want to minimize the sum of the squares of the difference given = 0 That is

  14. Minimum Norm Therefore the performance index becomes:

  15. Minimum Norm Therefore the performance index becomes:

  16. Minimum Norm Therefore the performance index becomes: Hint:

  17. Minimum Norm Hence the performance index becomes:

  18. Pseudo-Inverses Apply when there are more unknowns than equations or more equations than unknowns

  19. Least Squares • Least Squares • Weighted Least Squares

  20. Weighted Least Squares with a priori information

  21. Derivation

  22. Derivation

  23. Summary So Far • Least Squares • Weighted Least Squares • Least Squares with a priori • Minimum Norm

  24. Solving for State Vectors

  25. Solving for State Vectors

  26. Solving for State Vectors

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