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 31: State Noise Compensation and GMPs

2. Announcements • Homework 9 due Friday

3. Project Grading

4. Project Grading • Grading rubric generated for the projects • Will be posted to the Project Report Suggestions page • Reserve the right to edit/clarify, but the core content will not change

5. Clarifications and Corrections

6. Observation Editing • With the prediction residual information, we have a means to identify outliers based on criteria τ : True for EKF True for CKF

7. Condition Number of Inverted Matrix • Let P be a square matrix and non-singular • If the condition number of P is C, what is the condition number of P-1 ?

8. Process Noise – State Noise Compensation

9. General Estimation Problem (Revisited) • What happened to u(modeling error) ? • This is true process noise… • Can we ignore it? • How do we account for it?

10. Noise Process Description • For the sake of our discussion, assume: • In other words, Gaussian with zero mean and uncorrelated in time

11. Process Noise in Continuous Linear System • State update equation: • Covariance Matrix update equation:

12. Model for Discrete Process Noise • If we instead use the discrete form of the previous random process:

13. Sequential vs. Batch • The addition of a noise process is better suited for a sequential filter • Must include the process noise transition matrix in the Batch formulation • Tapley, Schutz, and Born (p. 229) argue that this is cumbersome and impractical for real-world application • An intern we had at CCAR a couple of years ago agrees • Advantage to Kalman, EKF, Potter, and others

14. Process Noise in Linear System • Random process u maps to the state through the matrix B • Consider it a random process for our purposes • Usually (for OD), we consider random accelerations:

15. Special Case • Let’s derive the process noise model for a simple case • Noise process defined in the acceleration • Time between measurement small enough to treat velocity as constant • What is the process noise transition matrix (PNTM)?

16. Block Form of the STM • Recall that:

17. Simplification of PNTM

18. Simplification of PNTM • Recall that we are assuming a small time between observations, i.e., velocity is constant

19. Simplification of PNTM • If velocity is constant, change in position is linear in time:

20. Simplification of PNTM

21. State Noise Compensation (SNC) PNTM • Derived under the assumptions that: • Noise process defined only in the acceleration • Time between measurement small enough to treat velocity as constant

22. Full SNC Covariance Correction Term

23. Selection of Q • This definition of Q is in the inertial Cartesian frame • Is that always a good idea? • What are some of the things you should consider?

24. Definition of Q in Different Frame • We can define Q in any frame, and rotate the matrix:

25. Selection of Q • The previous derivation assumed Q was known • How do we select Q ? • Ideally, Q describes the magnitude of the uncertainty of the acceleration acting on the satellite • What if we don’t know the magnitude? (after all, we are trying to account for an unknown acceleration) • Often determined by trial and error • You will do this in Homework 11

26. Better Ways to Empirically Determine Process Noise? • Can we estimate the Q matrix or other parameters of the process noise? • Gauss Markov Process (Dynamic Model Compensation) • Multiple Model Adaptive Estimation (MMAE) and Heirarchical Mixture of Experts (HME) • Others in the literature

27. Introduction to Gauss-Markov Processes

28. What is the Markov property? • The Markov property describes a random (stochastic) process where knowledge of the future is only dependent on the present:

29. Example Markov Processes • Basic random walk process • Number of popcorn kernels popped over time

30. Is this a Markov Process? • On a given day, the printer in our office is either working or broken. If it is working one day, the probability of it breaking the next day is b. If it was broken on one day, the probability of it being repaired the next day is r. • If r and b are independent, is this a Markov process? • If r and b are dependent, is this a Markov process?

31. Is this a Markov Process? • I have a deck of cards in my pocket. I pull out five cards: • 5 of hearts • Queen of diamonds • 2 of clubs • I then pull out: • Ace of spades • 4 of clubs • What is the probability that the next card is a ten of any suit?

32. Is this a Markov Process? • An object under linear motion? • A satellite in a chaotic orbit? • An object under stochastic linear or nonlinear motion?

33. Gauss-Markov Processes • Introduction of the random, uncorrelated (in time), Gaussian process noise u(t) makes η a Gauss-Markov process • We will use the GMP to develop another form of process noise