1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 7: Linearization and the State Transition Matrix

2. University of Colorado Boulder  Homework 2 – Due September 12  Lecture Quiz – Due Friday @ 5pm ◦ Full credit for doing it ◦ We will discuss the answers/results in class on Monday 2

3. University of Colorado Boulder  Linearization – How we do it? (wrap-up)  State Transition Matrix (STM) ◦ Derivation ◦ Solution Methods 3

4. University of Colorado Boulder 4 Linearization – Why do we need it? (review)

5. University of Colorado Boulder 5  How do we estimate X ?  How do we estimate the errors ε i ?  How do we account for force and observation model errors?

6. University of Colorado Boulder  This is the “normal form” of the least squares estimator  We assumed that the state-observation relationship was linear, but the orbit determination problems is nonlinear ◦ We will linearize the formulation of the problem 6

7. University of Colorado Boulder 7

8. University of Colorado Boulder 8 Linearization – How do we do it? (continued)

9. University of Colorado Boulder 9

10. University of Colorado Boulder 10

11. University of Colorado Boulder 11 Computed, not measured values!

12. University of Colorado Boulder 12 Linearization – State Transition Matrix

13. University of Colorado Boulder  Since x is linear (note lower case!) then there exists a solution to the linear, first order system of differential equations: 13  The solution is of the form:  Φ(t,t i ) is the state transition matrix (STM) that maps x(t i ) to the state x(t) at time t.

14. University of Colorado Boulder 14 Constant!

15. University of Colorado Boulder 15

16. University of Colorado Boulder  There are four methods to generate the STM: ◦ Solve from the direct Taylor expansion ◦ If A is constant, use the Laplace Transform or eigenvector/value analysis ◦ Analytically integrate the differential equation directly ◦ Numerically integrate the equations (ode45) 16

17. University of Colorado Boulder 17 State Transition Matrix – Alternative Derivation

18. University of Colorado Boulder  Expand X(t) in a Taylor series about X * (t): 18

19. University of Colorado Boulder 19

20. University of Colorado Boulder 20

21. University of Colorado Boulder 21 State Transition Matrix – Laplace Transform

22. University of Colorado Boulder 22  Laplace Transforms are useful for analysis of linear time-invariant systems: ◦ electrical circuits, ◦ harmonic oscillators, ◦ optical devices, ◦ mechanical systems, ◦ even some orbit problems.  Transformation from the time domain into the Laplace domain.  Inverse Laplace Transform converts the system back.

23. University of Colorado Boulder 23

24. University of Colorado Boulder  Solve the ODE  We can solve this using “traditional” calculus: 24

25. University of Colorado Boulder  Solve the ODE  Or, we can solve this using Laplace Transforms: 25

26. University of Colorado Boulder  Solve the ODE: 26

27. University of Colorado Boulder 27

28. University of Colorado Boulder 28

29. University of Colorado Boulder 29 State Transition Matrix – Analytic Approach

30. University of Colorado Boulder  Leverage the differential equation 30 and combine it with classic methods  Compatible with simple equations, but not with larger estimated state vectors or complicated dynamics

31. University of Colorado Boulder 31

32. University of Colorado Boulder 32

33. University of Colorado Boulder 33

34. University of Colorado Boulder 34

35. University of Colorado Boulder 35

36. University of Colorado Boulder 36

37. University of Colorado Boulder 37

38. University of Colorado Boulder 38

39. University of Colorado Boulder 39

40. University of Colorado Boulder 40 State Transition Matrix – Numeric Integration

41. University of Colorado Boulder  For more complicated dynamics, must integrate X * (t) and Φ(t,t 0 ) simultaneously in propagator ◦ Up to n+n 2 propagated states ◦ Derivative function must include the evaluation of the [A(t)] * matrix in addition to F(X,t) 41

42. University of Colorado Boulder  Use the MATLAB reshape() command to turn matrix into a vector ◦ v = reshape( V, nrows*ncols, 1 );  MATLAB Demo… 42