University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 32: Gauss-Markov Processes and Dynamic Model Compensation

University of Colorado Boulder  Homework 9 due Friday  Lecture Quiz due Friday at 5pm  Exam 2 ◦ Returned to students and discussion on Friday 11/13 ◦ Lecture 33 will not be posted to D2L until Monday 11/16 2

University of Colorado Boulder 3 Project Grading / Discussion

University of Colorado Boulder  Grading rubric generated for the projects ◦ Posted to the Project Report Suggestions page ◦ Reserve the right to edit/clarify, but the core content will not change 4

University of Colorado Boulder 5 Introduction to Gauss-Markov Processes

University of Colorado Boulder  The Markov property describes a random (stochastic) process where knowledge of the future is only dependent on the present: 6

University of Colorado Boulder  Basic random walk process 7 Image credit: Google Maps

University of Colorado Boulder  Number of popcorn kernels popped over time 8

University of Colorado Boulder  On a given day, the CCAR photocopier 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? 9

University of Colorado Boulder  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? 10

University of Colorado Boulder  An object under linear motion?  A satellite in a chaotic orbit?  An object under stochastic linear or nonlinear motion?  The estimated state in a Kalman filter? 11

University of Colorado Boulder 12 Dynamic Model Compensation (DMC)

University of Colorado Boulder  For the sake of our discussion, assume: 13  In other words, Gaussian with zero mean and uncorrelated in time ◦ Dubbed State Noise Compensation (SNC)

University of Colorado Boulder  If the dynamics noise is systematic, then the correlations in acceleration error are likely correlated in time ◦ For example, the error due to truncated gravity field is a smooth function of position  What other options exist to account for correlations in time? 14

University of Colorado Boulder  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 15

University of Colorado Boulder  Stochastic integral cannot be solved analytically, but has a statistical description: 16 Deterministic Stochastic

University of Colorado Boulder  Stochastic integral cannot be solved analytically, but has a statistical description: 17 Deterministic Stochastic

University of Colorado Boulder  It may be shown that: 18  In other words: ◦ The process is exponentially correlated in time ◦ Rate of the correlation fade is determined by β ◦ For large β, the faster the decay

University of Colorado Boulder  Instead, let’s use an equivalent process 19  L k has the same statistical description as the stochastic integral  Hence, it is an equivalent process

University of Colorado Boulder  Process behavior varies with the equation parameters 20 Add dependence on time to emphasize different realizations for different times

University of Colorado Boulder  What happens if β  0? 21

University of Colorado Boulder  What happens if σ  0? 22

University of Colorado Boulder 23

University of Colorado Boulder  Augment the state vector to include the accelerations  Requires new F(t) and A(t)  The random portion determines the process noise matrix Q(t) (see Appendix F, p ) 24

University of Colorado Boulder 25

University of Colorado Boulder 26 Image: Leonard, Nievinski, and Born, 2013