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 7: Generating the STM

2. Announcements • Lecture Quiz – Due Wednesday • Homework 2 – Due September 13 • Bring pen and paper on Friday!

3. Today’s Lecture • Previous Lecture • Linearization – State Transition Matrix (STM) • Alternative Derivation • Solution Methods

4. Last Lecture – A Couple of Clarifications

5. Least Squares: Derivation of Normal Form For now, assume G is a linear function • Differentiate with respect to x. What is the answer?

6. Linearize About Reference

7. Linearization – State Transition Matrix

8. State Transition Matrix • Since x is linear (note lower case!) then there exists a solution to the linear, first order system of differential equations: • The solution is of the form: • Φ(t,ti) is the state transition matrix (STM) that maps x(ti) to the state x(t) at time t.

9. STM Differential Equation Constant! • What is the differential equation?

10. Methods to Generate the STM • There are four methods to generate the STM: • Solve from the direct equation (next lecture) • If A is constant, use the Laplace Transform or eigenvector/value analysis • Analytically integrate the differential equation directly • Numerically integrate the equations (ode45)

11. State Transition Matrix – Alternative Derivation

12. STM – Alternative Derivation • Expand X(t) in a Taylor series about X*(t):

13. STM – Alternative Derivation

14. Flat Earth Problem (FEP)STM – Alternative Derivation

15. State Transition Matrix – Laplace Transform

16. Laplace Transforms • Laplace Transforms are useful for analysis of linear time-invariant systems: • electrical circuits, • harmonic oscillators, • optical devices, • mechanical systems, • even orbit problems. • Transformation from the time domain into the frequency domain. • Inverse Laplace Transform converts the system back.

17. Laplace Transform Tables

18. Example • Solve the ODE • We can solve this using “traditional” calculus:

19. Example • Solve the ODE • Or, we can solve this using Laplace Transforms:

20. Applied to Stat OD • Solve the ODE

21. FEP STM – Laplace Transform

22. FEP STM – Laplace Transform

23. State Transition Matrix – Direct Approach

24. ‘Direct Approach’ to Solving STM • Leverage the differential equation and combine it with analytic diff. eq. methods • Compatible with simple equations, but not with larger estimated state vectors or complicated dynamics

25. Example Direct Approach

26. Solution

27. State Transition Matrix – Numeric Integration

28. Numeric Integration • For more complicated dynamics, must integrate X*(t) and Φ(t,t0) simultaneously in propagator • Up to n+n2 propagated states • Derivative function must include the evaluation of the [A(t)]* matrix in addition to F(X,t)

29. Example Implementation • Use the MATLAB reshape() command to turn matrix into a vector • v = reshape( V, nrows*ncols, 1 ); • MATLAB Demo…