Transcript Lecture 32 - 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
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Homework 9 due Friday
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Lecture Quiz due Friday at 5pm
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Exam 2
◦ Returned to students and discussion on Friday
11/13
◦ Lecture 33 will not be posted to D2L until Monday
11/16
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Project Grading / Discussion
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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
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Introduction to Gauss-Markov Processes
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The Markov property describes a random
(stochastic) process where knowledge of the
future is only dependent on the present:
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Basic random walk process
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Image credit: Google Maps
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Number of popcorn kernels popped over time
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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?
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I have a deck of cards in my pocket. I pull out
five cards:
◦ 5 of hearts
◦ Queen of diamonds
◦ 2 of clubs
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I then pull out:
◦ Ace of spades
◦ 4 of clubs
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What is the probability that the next card is a ten
of any suit?
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An object under linear motion?
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A satellite in a chaotic orbit?
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An object under stochastic linear or nonlinear
motion?
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The estimated state in a Kalman filter?
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Dynamic Model Compensation (DMC)
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For the sake of our discussion, assume:
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In other words, Gaussian with zero mean and
uncorrelated in time
◦ Dubbed State Noise Compensation (SNC)
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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
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What other options exist to account for
correlations in time?
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Introduction of the random, uncorrelated (in time), Gaussian
process noise u(t) makes η a Gauss-Markov process
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We will use the GMP to develop another form of process noise
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Deterministic
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Stochastic
Stochastic integral cannot be solved analytically,
but has a statistical description:
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Deterministic
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Stochastic
Stochastic integral cannot be solved analytically,
but has a statistical description:
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It may be shown that:
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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
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Instead, let’s use an equivalent process
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Lk has the same statistical description as the
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Hence, it is an equivalent process
stochastic integral
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Add dependence on time to emphasize
different realizations for different times
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Process behavior varies with the equation
parameters
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What happens if β  0?
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What happens if σ  0?
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Augment the state vector to include the accelerations
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Requires new F(t) and A(t)
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The random portion determines the process noise
matrix Q(t) (see Appendix F, p. 507-508)
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Image: Leonard, Nievinski, and Born, 2013
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