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ASEN 5070
Statistical Orbit Determination I
Fall 2012
Professor Jeffrey S. Parker
Professor George H. Born
Lecture 19: Numerical Compensations
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Homework 8 due next week.◦ Make sure you spend time studying for the exam
Exam 2 in one week (Thursday). Review on Tuesday.
Exam 2 will cover:◦ Batch vs. CKF vs. EKF◦ Probability and statistics (good to keep this up!)
Haven’t settled on a question yet, but it will probably be a conditional probability question. I.e., what’s the probability of X given that Y occurs?
◦ Observability◦ Numerical compensation techniques, such as the Joseph and Potter
formulation.◦ No calculators should be necessary◦ Open Book, Open Notes
Announcements
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Quiz 15 Review
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Quiz 15 Review
Use Matlab and try it out
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Quiz 15 Review
This is TRUE for the Batch, but you may run into a problem with the Kalman filters.
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Quiz 15 Review
The best scan of the book ever:
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University of ColoradoBoulder 7
Quiz 15 Review
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Quiz 15 Review
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Quiz 15 Review
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Quiz 15 Review
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University of ColoradoBoulder 11
Quiz 15 Review
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Quiz 15 Review
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Due in 7 days
HW#8
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HW#8
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HW#8 Solutions to (2)
Note that these plots aren’t 100% well-labeled!
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HW#8
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The biggest pitfall
When processing the 2nd observation, set
that is, use the most current covariance you have as the a priori! Not the original one.
HW#8
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Processing an observation vector one element at a time.
Whitening Cholesky Joseph
Today◦ Positive Definiteness◦ Conditioning Number◦ Potter◦ Householder
Previous Lecture
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Definition
Positive Definite Matrices
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Properties of PD matrices
Positive Definite Matrices
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CCARColorado Center for
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Properties of PD matrices
Positive Definite Matrices
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Quick Break
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CCARColorado Center for
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University of ColoradoBoulder
Summary of Results2P
Exact to order 1 2 (1 3 )
(1 3 ) 2 4
Conventional Kalman
1 11
1 11 2
Joseph
1 2 (1 3 )
(1 3 ) 2
Batch
1 2 (1 3 )
(1 3 ) 2 4
Example Illustrating Numerical Instability of Sequential (Kalman) Filter (see 4.7.1)
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Conditioning Number of matrix A: C(A)
where the gammas are the min/max eigenvalue of the matrix.
Inverting A with p digits of precision becomes error-prone as C(A) 10p.
If we invert the square root of A:
then C(W)=sqrt(C(A)). Numerical difficulties will arise as C(W) 102p.
Conditioning Number
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Square Root Filter Algorithms
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Motivation:◦ Loss of significant digits that occurs in computing
the measurement update of the state error covariance matrix (P) at the observation epoch (Kaminski et al., 1971)
◦ If eigenvalues spread a wide range, then the numerical errors can destroy the symmetry and PD-ness of the P matrix. Filter divergence can occur.
Square Root Filter Algorithms
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Define W, the square root of P:
Observe that if we have W, then computing P in this manner will always result in a symmetric PD matrix.
Note: Square root filters are typically derived to process one observation at a time. Hence,
Square Root Filter Algorithms
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(Sorry for just scanning the text, but it’s a pretty concise description!)
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CCARColorado Center for
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University of ColoradoBoulder
(Sorry for just scanning the text, but it’s a pretty concise description!)
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CCARColorado Center for
Astrodynamics Research
University of ColoradoBoulder
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CCARColorado Center for
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This is a key
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CCARColorado Center for
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University of ColoradoBoulder
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CCARColorado Center for
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Potter Square Root Filter
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CCARColorado Center for
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Potter Square Root Filter
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CCARColorado Center for
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Potter Square Root Filter
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CCARColorado Center for
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Potter Square Root Filter
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CCARColorado Center for
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Potter Square Root Filter
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CCARColorado Center for
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University of ColoradoBoulder
Potter Square Root Filter
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CCARColorado Center for
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Potter Square Root Filter
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CCARColorado Center for
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Potter Square Root Filter
(say, using Cholesky)
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CCARColorado Center for
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Potter Square Root Filter
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Potter Square Root Filter
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CCARColorado Center for
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Potter Square Root Filter
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CCARColorado Center for
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Potter Square Root Filter
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CCARColorado Center for
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Potter Square Root Filter
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CCARColorado Center for
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Potter Square Root Filter
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CCARColorado Center for
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Potter Square Root Filter
Note: if you are given an a priori P matrix, convert it to W using Cholesky or equivalent at start.
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CCARColorado Center for
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Potter Square Root Filter
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CCARColorado Center for
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Potter Square Root Filter
And page 339 of Stat OD text.
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CCARColorado Center for
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Numerical Instability of Kalman Filter
1 2 (1 )
(1 ) 2 4
Summary of Results
2PExact to order
1 2 (1 3 )
(1 3 ) 2 4
Conventional Kalman
1 11
1 11 2
Joseph
1 2 (1 3 )
(1 3 ) 2
Batch
1 2 (1 3 )
(1 3 ) 2 4
Potter Algorithm
![Page 51: ASEN 5070 Statistical Orbit Determination I Fall 2012 Professor Jeffrey S. Parker](https://reader030.vdocuments.mx/reader030/viewer/2022032414/568132cb550346895d998f50/html5/thumbnails/51.jpg)
CCARColorado Center for
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University of ColoradoBoulder 51
Homework 8 due next week.◦ Make sure you spend time studying for the exam
Exam 2 in one week (Thursday). Review on Tuesday.
Exam 2 will cover:◦ Batch vs. CKF vs. EKF◦ Probability and statistics (good to keep this up!)
Haven’t settled on a question yet, but it will probably be a conditional probability question. I.e., what’s the probability of X given that Y occurs?
◦ Observability◦ Numerical compensation techniques, such as the Joseph and Potter
formulation.◦ No calculators should be necessary◦ Open Book, Open Notes
The End (unless there’s a lot more time)