Transcript Lecture 28
ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 28: Orthogonal Transformations University of Colorado Boulder Lecture quiz due at 5pm Exam 2 – Friday, November 6 Office Hours Next Week: ◦ Prof. Jones – T/Th 12:30-1:30pm ◦ Eduardo – W 2-4pm, Th 2:30-4:30pm University of Colorado Boulder 2 Red indicates topics that could be covered in a numeric problem. Minimum variance as a sequential processor Conventional Kalman Filter (CKF) Extended Kalman Filter (EKF) Pre-fit/predicted/post-fit residuals Handling observation biases Numeric considerations in the CKF/EKF Batch vs. CKF vs. EKF Potter square-root filter Cholesky decomposition w/ forward and backward substitution Singular value decomposition (SVD) methods Orthogonal transforms Analysis of filter results (Monday) University of Colorado Boulder 3 University of Colorado Boulder 4 Least Squares via Orthogonal Transformations University of Colorado Boulder 5 University of Colorado Boulder 6 Recall the least squares cost function: By property 4 on the previous slide and Q an orthogonal matrix: University of Colorado Boulder 7 University of Colorado Boulder 8 University of Colorado Boulder 9 The method for selecting R defines a particular algorithm ◦ Givens Transformations (Section 5.4) ◦ Householder Transformation (Section 5.5) ◦ Gram-Schmidt Orthogonalization Not in the book and we won’t cover it University of Colorado Boulder 10 LS Solution via Givens Transformations University of Colorado Boulder 11 University of Colorado Boulder 12 University of Colorado Boulder 13 Consider the desired result To achieve this, we select the Givens matrix such that We then use this transformation in the top equation University of Colorado Boulder 14 We do not want to add non-zero terms to the previously altered rows, so we use the identity matrix except in the rows of interest: University of Colorado Boulder 15 After applying the transformation, we get: Repeat for all remaining non-zero elements in the third column What if the term is already 0 ? University of Colorado Boulder 16 Need to find the orthogonal matrix Q to yield a matrix of the form of the RHS Q is generated using a series of Givens transformations G University of Colorado Boulder 17 We select G to get a zero for the term in red: To achieve this, we use: University of Colorado Boulder 18 We select G to get a zero for the term in red: To achieve this, we use: University of Colorado Boulder 19 We select G to get a zero for the term in red: To achieve this, we use: University of Colorado Boulder 20 We select G to get a zero for the term in red: To achieve this, we use: University of Colorado Boulder 21 We select G to get a zero for the term in red: To achieve this, we use: University of Colorado Boulder 22 We select G to get a zero for the term in red: To achieve this, we use: University of Colorado Boulder 23 We select G to get a zero for the term in red: To achieve this, we use: University of Colorado Boulder 24 We now have the required Q matrix (for this conceptual example): University of Colorado Boulder 25 University of Colorado Boulder 26 Givens Transformations – An New Example University of Colorado Boulder 27 Consider the case where: University of Colorado Boulder 28 University of Colorado Boulder 29 University of Colorado Boulder 30 University of Colorado Boulder 31 We then have the matrices needed to solve the system: University of Colorado Boulder 32 Batch vs. Givens University of Colorado Boulder 33 Consider the case where: The exact solution is: After truncation: University of Colorado Boulder 34 Well, the Batch can’t handle it. What about Cholesky decomposition? Darn, that’s singular too. Let’s give Givens a shot! University of Colorado Boulder 35 University of Colorado Boulder 36 University of Colorado Boulder 37 University of Colorado Boulder 38 Hence, Givens transformations give us a solution for the state Home Exercise: Why is this true? Note: R is not equal to H ! University of Colorado Boulder Still a problem w/ P ! 39 Givens uses a sequence of rotations to generate the R matrix Instead, Householder transformations use a sequence of reflections to generate R ◦ See the book for details University of Colorado Boulder 40