1. University of Colorado Boulder ASEN 5070 Statistical Orbit Determination I Fall 2012 Professor Jeffrey S. Parker Professor George H. Born Lecture 23: Process Noise 1

2. University of Colorado Boulder  How was break?  HW 10 due Thursday.  HW 11 due next week.  Grading  You have the tools needed to finish the project. Only things missing are bonus pieces.  I may be out tomorrow; if you need office-hour help, check the TAs or visit me Thursday. 2 Last Day of Classes Final Project Due All HW Due Take-Home Exam Due

3. University of Colorado Boulder 3

4. University of Colorado Boulder 4

5. University of Colorado Boulder 5

6. University of Colorado Boulder 6

7. University of Colorado Boulder 7

8. University of Colorado Boulder 8

9. University of Colorado Boulder 9

10. University of Colorado Boulder 10

11. University of Colorado Boulder 11

12. University of Colorado Boulder 12

13. University of Colorado Boulder  Due the Thursday after HW10 13

14. University of Colorado Boulder  Process Noise 14

15. University of Colorado Boulder  Applying the sequential algorithm to a large amount of data will cause the covariance to shrink down too far. 15

16. University of Colorado Boulder  Sequential filter measurement update:  Large number of observations will drive the covariance to zero. ◦ (which is great if everything is perfectly modeled! But that’s never quite so.)

17. University of Colorado Boulder  Trace of covariance over time.

18. University of Colorado Boulder  As 18

19. University of Colorado Boulder  As The filter will begin ignoring observations and the best estimate will remain constant over time. 19

20. University of Colorado Boulder  This is the example given in Appendix F  I coded it up, and you can too. It’s pretty straightforward. Plus this way I could investigate different aspects of the scenario.  Given: ◦ A particle moving along the x-axis in a positive direction. ◦ It is nominally moving at a constant 10 m/s velocity. ◦ It is actually being perturbed by an unknown acceleration – unmodeled! ◦ The acceleration is a small-amplitude oscillation. ◦ The particle is being tracked by an observer, also on the x-axis at 10 Hz observation frequency.  Range with white noise N(0,1) m  Range-rate with white noise N(0,0.1) m/s  Can we estimate the state and even the unmodeled force? 20

21. University of Colorado Boulder  Our system  Notice that the acceleration is not in our model. 21

22. University of Colorado Boulder 22

23. University of Colorado Boulder  Observations  Range N(0,1) meter  Range Rate N(0,0.1) m/s 23

24. University of Colorado Boulder  If you KNEW the acceleration, then this is what you would see using an EKF 24

25. University of Colorado Boulder  You don’t know the acceleration, so this is what you DO see with an EKF 25

26. University of Colorado Boulder  We need a way to add expected process noise to our filter.  Many ways to do that: ◦ State Noise Compensation  Constant noise  Piecewise constant noise  Correlated noise  White noise ◦ Dynamic Model Compensation  1 st order linear stochastic noise  More complex model compensations 26 Example

27. University of Colorado Boulder  Let’s say that we know that our particle is being perturbed by some acceleration.  We characterize it by simple white noise (note: we know that it is far more structured than that!) stationary Gaussian process with a mean of zero and a variance of (Dirac delta function) 27

28. University of Colorado Boulder  Adjust our system accordingly 28 A B (No perturbation)

29. University of Colorado Boulder  We can show that the time update for the sequential algorithm does not change: (we’ll show this later)  The time update for the covariance does change (we’ll derive it later) 29

30. University of Colorado Boulder  Introduce the Process Noise Covariance Matrix Q (this equation is specific to this example and we’ll derive the general equation later)  For our problem, we have Phi and B. Perform the integration and we find 30

31. University of Colorado Boulder  Further, since the time update interval is always 0.1 sec (10 Hz observation frequency) 31 has units of m 2 /s 3

32. University of Colorado Boulder  Implication: original deterministic constant velocity model of the motion is modified to include ◦ Random component ◦ Constant-diffusion Brownian motion process ◦ is the diffusion coefficient and is a tuning parameter  Remember: this is a very broad, simple process noise compensation. This assumes white noise acceleration. 32

33. University of Colorado Boulder  New time update:  Old:  New for Example:  General: 33

34. University of Colorado Boulder 34  Old Result

35. University of Colorado Boulder 35  New Result

36. University of Colorado Boulder 36  Comparison of estimated velocity

37. University of Colorado Boulder 37  This does depend on the value of Too low: not enough help Too high: no carry-over information

38. University of Colorado Boulder  We need a way to add expected process noise to our filter.  Many ways to do that: ◦ State Noise Compensation  Constant noise  Piecewise constant noise  Correlated noise  White noise ◦ Dynamic Model Compensation  1 st order linear stochastic noise  More complex model compensations 38 Example 1 Example 2

39. University of Colorado Boulder  Let’s reformulate our compensation assumption.  Rather than assuming that our particle is under the influence of some white noise acceleration, let’s assume there’s some structure.  DMC assumes that the unknown acceleration can be characterized as: ◦ 1 st -order linear stochastic differential equation ◦ Known as the Langevin equation 39

40. University of Colorado Boulder  What does this expression mean?  is our estimated acceleration  is the inverse of a correlation time (another tuning parameter!)  is a white zero-mean Gaussian process  Solution: 40 Deterministic Stochastic

41. University of Colorado Boulder  We can add the deterministic component of this acceleration to our state. 41 Deterministic Stochastic

42. University of Colorado Boulder  Integrate this to observe its effects on the dynamical model 42 Deterministic Stochastic

43. University of Colorado Boulder  Note: tau (the correlation constant) may also be added to the state to produce a 4-element state.  This example proceeds by setting tau to be a constant. ◦ Users could fiddle with tau to tune the filter. 43

44. University of Colorado Boulder  Re-derive the H-tilde, A, Phi, etc matrices to accommodate the new state parameters.  Results: ◦ A great estimate of the state ◦ A much wider range of acceptable values for the tuning parameters – which is very useful when we don’t know the answer! ◦ A good estimate of the unmodeled acceleration profile – which may be used as information to track down missing pieces of one’s dynamical model. 44

45. University of Colorado Boulder  Estimated State 45

46. University of Colorado Boulder  Estimated State 46

47. University of Colorado Boulder  Estimated State 47

48. University of Colorado Boulder  DMC is less sensitive to tuning than SNC 48

49. University of Colorado Boulder  What if we changed the perturbing acceleration? ◦ Triangle wave ◦ Square wave ◦ Constant acceleration ◦ White noise 49

50. University of Colorado Boulder  Square Wave 50

51. University of Colorado Boulder  Constant Acceleration 51

52. University of Colorado Boulder  ~White Noise 52

53. University of Colorado Boulder  ~Impulses 53

54. University of Colorado Boulder  Shown an example  Now we’ll derive the SNC and DMC formulations  Then we’ll apply them to our satellite state estimation problem.  But first! A quick break. 54

55. University of Colorado Boulder  Our setting: ◦ We are not modeling everything perfectly – and we either suspect it or know it.  State dynamics of a linear system under the influence of process noise: ◦ A(t) and B(t) are known. ◦ u(t) is mx1 ◦ B(t) is nxm 55

56. University of Colorado Boulder  State dynamics of a linear system under the influence of process noise: ◦ u(t) can include all kinds of processes.  White noise, constant noise, piecewise constant, correlated, etc. ◦ The standard State Noise Compensation (SNC) algorithm assumes that u(t) is white noise ◦ Dynamic Model Compensation assumes that u(t) is more complex, as we saw earlier. ◦ SNC: 56

57. University of Colorado Boulder  Solve the linear system via a method of variation of parameters.  Homogeneous equation:  Solution of the form:  Select C 0 as a function of time so that 57

58. University of Colorado Boulder  Then we can take the derivative to find 58

59. University of Colorado Boulder  Then we can take the derivative to find  Plugging this into 59

60. University of Colorado Boulder  Recall our old friend: 60

61. University of Colorado Boulder  Substitute the solution from above: 61

62. University of Colorado Boulder  Hence  Integrating yields 62

63. University of Colorado Boulder  Substituting into yields 63

64. University of Colorado Boulder  Initial conditions We can show that C 0 = x 0 64

65. University of Colorado Boulder  This equation is the general solution for the inhomogeneous equation  It indicates how the true state propagates under the influence of process noise. 65

66. University of Colorado Boulder Next, we wish to understand how the estimate of the state propagates in the presence of process noise. is a stochastic integral and cannot be evaluated in a deterministic sense. for but

67. University of Colorado Boulder so We have our familiar solution This is good news! It means that we don’t have to change our methods to update x-bar in our sequential algorithm.

68. University of Colorado Boulder  Since we have shown that the time update of the state is unchanged  If the mean of the process noise is nonzero 68 (defined later)

69. University of Colorado Boulder  How was break?  HW 10 due Thursday.  HW 11 due next week.  Grading  You have the tools needed to finish the project. Only things missing are bonus pieces.  I may be out tomorrow; if you need office-hour help, check the TAs or visit me Thursday. 69 Last Day of Classes Final Project Due All HW Due Take-Home Exam Due