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

1. ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 5: Linear Algebra and Linearization

2. Announcements • Homework 0 & 1 were due today • Homework 2 Posted – Due September 13 • Quiz for Lectures 1-5 active at noon • Due by Monday’s Lecture at 11am • Five Questions (10-15 minutes) • Expect to have one every couple of lectures

3. Today’s Lecture • Linear Algebra (Appendix B) • Linearization

4. Select Topics in Linear Algebra

5. Matrix Basics • Matrix A is comprised of elements ai,j • The matrix transpose swaps the indices

6. Matrix Basics • Matrix inverse A-1 is the matrix such that • For the inverse to exist, A must be square • We will treat vectors as n×1 matrices

7. Matrix Basics

8. Transpose/Inverse Identities

9. Matrix Determinant • The square matrix determinant, |A|, describes if a solution to a linear system exists: • It also describes the change in area/volume/etc. due to a linear operation:

10. Matrix Determinant Identities

11. Linear Independence • A set of vectors are linearly independent if none of them can be expressed as a linear combination of other vectors in the set • In other words, no scalars αi exist such that for some vector vj in the set {vi}, i=1,…,n,

12. Matrix Rank • The matrix column rank is the number of linearly independent columns of a matrix • The matrix row rank is the number of linearly independent rows of a matrix • rank(A) = min( col. rank of A, row rank of A)

13. Exercise • What is the rank of the following matrices:

14. Rank Identities

15. Vector Differentiation • When differentiating a scalar function w.r.t. a vector:

16. Vector Differentiation • When differentiating a function with vector output w.r.t. a vector:

17. Matrix Derivative Identities • If A and B are n×1 vectors that are functions of X:

18. Positive Definite Matrices • The n×n matrix A is positive definite if and only if: • The n×n matrix A is positive semi-definite if and only if:

19. Minimum of a function • The point x is a minimum if and is positive definite.

20. Eigenvalues/vectors • Given the n×n matrix A, there are n eigenvalues λ and vectors X≠0 where

21. Book Appendix B • Other identities/definitions in Appendix B of the book • Matrix Trace • Maximum/Minimum Properties • Matrix Inversion Theorems • Review the appendix and make sure you understand the material

22. Linearization – Why do we need it?

23. Estimated State Vector • We want to get the best estimate of X possible • Ex. force model parameters: CD, CR, J2, etc. • Ex. measurement params: station coordinates, observation biases, etc.

24. Orbit State Vector Dynamics • “Solve-for” parameters are usually constant (but not always…) • More generally:

25. Observation Vector • Example measurement types: • Range, Range-Rate • Right Ascension/Declination • GPS pseudorange and carrier phase • Star tracker and angular rate gyro • At each epoch ti we have a measurement model G(Xi, ti) • εirepresents the model error in G(Xi, ti) • May result from statistical uncertainty • Could be a result of modeling error • What are some examples of modeling error?

26. General Estimation Problem • How do we estimate X ? • How do we estimate the errors εi? • How do we account for force and observation model errors?

27. Why not use the NR estimator? • It works for HW 1, why don’t we do it in practice? • Assumed the same number of observations as unknowns • What about when we have more observations than unknowns? • Did not rigorously account for observation errors • How do we account for statistical uncertainties?

28. # Unknowns vs. # Measurements • Known: p×lobservations • Unknowns: • n×lunknown state variables • p×lunknown observation errors • (n+p)×l total unknown values • We have more unknowns that observations, what do we do now? • X(t) is a function of X(t0) • Well, now we are down to n+(p×l) unknowns…

29. Cost Function J(x) • We introduce a “cost function” that we seek to minimize • We now select X to minimize J(X) • No longer estimating εi! • This gives us n+p×lequations and only n unknowns • This is known as: Least Squares Estimation