Transcript Lecture 32
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 University of Colorado Boulder 2 Project Grading / Discussion University of Colorado Boulder 3 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 University of Colorado Boulder 4 Introduction to Gauss-Markov Processes University of Colorado Boulder 5 The Markov property describes a random (stochastic) process where knowledge of the future is only dependent on the present: University of Colorado Boulder 6 Basic random walk process University of Colorado Boulder Image credit: Google Maps 7 Number of popcorn kernels popped over time University of Colorado Boulder 8 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? University of Colorado Boulder 9 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? University of Colorado Boulder 10 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? University of Colorado Boulder 11 Dynamic Model Compensation (DMC) University of Colorado Boulder 12 For the sake of our discussion, assume: In other words, Gaussian with zero mean and uncorrelated in time ◦ Dubbed State Noise Compensation (SNC) University of Colorado Boulder 13 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? University of Colorado Boulder 14 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 University of Colorado Boulder 15 Deterministic Stochastic Stochastic integral cannot be solved analytically, but has a statistical description: University of Colorado Boulder 16 Deterministic Stochastic Stochastic integral cannot be solved analytically, but has a statistical description: University of Colorado Boulder 17 It may be shown that: 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 18 Instead, let’s use an equivalent process Lk has the same statistical description as the Hence, it is an equivalent process stochastic integral University of Colorado Boulder 19 Add dependence on time to emphasize different realizations for different times Process behavior varies with the equation parameters University of Colorado Boulder 20 What happens if β 0? University of Colorado Boulder 21 What happens if σ 0? University of Colorado Boulder 22 University of Colorado Boulder 23 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. 507-508) University of Colorado Boulder 24 University of Colorado Boulder 25 Image: Leonard, Nievinski, and Born, 2013 University of Colorado Boulder 26