1 / 74

ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born

ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 5: Stat OD. Announcements. Homework 2 due Thursday Homework 3 out today Basic dynamical systems relationships Studies of the state transition matrix Linear algebra

calum
Download Presentation

ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 5: Stat OD

  2. Announcements • Homework 2 due Thursday • Homework 3 out today • Basic dynamical systems relationships • Studies of the state transition matrix • Linear algebra • I’m unavailable this Wednesday. Use those TAs  and email is great of course.

  3. Quiz Results

  4. Quiz Results

  5. Quiz Results

  6. Quiz Results ~N( 0.0, 1.41 )

  7. Quiz Results

  8. Quiz Results ~N( 1.0, 0.41 )

  9. Quiz Results

  10. Quiz Results

  11. Homework 2 • Some popular questions and answers • Energy with Drag

  12. Homework 2 • Some popular questions and answers • Computation of Time of Perigee

  13. Homework 2: Tp Calculation

  14. Homework 2: Tp Calculation

  15. Homework 2: Tp Calculation

  16. Homework 3 • Chapter 4, Problems 1-6 • Solving ODEs • Linear Algebra • Studying the state transition matrix

  17. Today’s Lecture • Review of Differential Equations • Laplace Transforms • Review of Statistics

  18. Review of Diff EQ • Stat OD dynamics: • Solve for given A and

  19. Review of Diff EQ • Stat OD dynamics: • Solve for given A and

  20. Review of Diff EQ • Solve for • w/

  21. Review of Diff EQ • Solve for • w/

  22. Example • Solve the ODE • We can solve this using “traditional” calculus: Check your answer by plugging it back in

  23. 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.

  24. Laplace Transform Tables

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

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

  27. Applied to Stat OD • Solve the ODE

  28. Applied to Stat OD • Solve the ODE

  29. Applied to Stat OD • Solve the ODE

  30. Applied to Stat OD • Solve the ODE

  31. Applied to Stat OD • Solve the ODE

  32. Applied to Stat OD • Solve the ODE

  33. Applied to Stat OD • Solve the ODE

  34. Questions? • Questions on Diff EQ? • Quick Break • Review of Statistics to follow

  35. Review of Statistics • X is a random variable with a prescribed domain. • x is a realization of that variable. • Example: • 0 < X < 1 • x1 = 0.232 • x2 = 0.854 • x3 = 0.055 • etc

  36. Axioms of Probability 2. p(S)=1, S is the certain event

  37. Venn Diagram

  38. Axioms of Probability

  39. Probability Density & Distribution Functions • For the continuous random variable, axioms 1 and 2 become

  40. Probability Density & Distribution Functions • For the continuous random variable, axioms 1 and 2 become • The third axiom becomes • Which for a < b < c

  41. Probability Density & Distribution Functions Using axiom 2 as a guide, solve the following for k:

  42. Probability Density & Distribution Functions Using axiom 2 as a guide, solve the following for k:

  43. Probability Density & Distribution Functions

  44. Probability Density & Distribution Functions Example: • From the definition of the density and distribution functions we have: • From axioms 1 and 2, we find:

  45. Expected Values Note that:

  46. Expected Values

  47. Expected Values

  48. Expected Values

  49. Expected Values

  50. The Gaussian or Normal Density Function

More Related