least_squares_2009_10_14_01

least_squares_2009_10_14_01 - 8 - 1 Least Squares S. Lall,...

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Unformatted text preview: 8 - 1 Least Squares S. Lall, Stanford 2009.10.14.01 8. Least Squares The pseudo-inverse Example: pseudo-inverse Estimation and least-squares Effects of noise on estimation Example: navigation Regression or curve-fitting Example: fitting polynomials Example: rocket Control and minimum-norm problems Example: force on mass Matlab and the pseudo-inverse History of least-squares 8 - 2 Least Squares S. Lall, Stanford 2009.10.14.01 The Key Points of This Section estimation problems: given y meas , find the least-squares solution x , that minimizes bardbl y meas Ax bardbl control problems: given y des , find the minimum-norm x that satisfies y des = Ax the SVD gives a computational approach it also gives useful information even when important assumptions dont hold estimation: usually need A skinny and full rank control: usually need A fat and full rank it gives us quantitative information about the usefulness of the solutions 8 - 3 Least Squares S. Lall, Stanford 2009.10.14.01 important facts null( A T ) = range( A ) easy via the SVD: because if the SVD of A is A = U V T then range( A ) = span { u 1 ,...,u r } also the SVD of A T is A T = V T U T so null( A T ) = span { u r +1 ,...,u n } 8 - 4 Least Squares S. Lall, Stanford 2009.10.14.01 one more null( A T A ) = null( A ) also easy via the SVD: A T A = V T U T U V T = V T V T which gives an SVD of A T A . T has the same number of non-zero elements as , so both A and A T A have null space span { v r +1 ,...,v n } 8 - 5 Least Squares S. Lall, Stanford 2009.10.14.01 The Pseudo-Inverse the thin SVD is A = U V T = A U V T here is square, diagonal, positive definite U and V are skinny, orthonormal columns the pseudo-inverse of A is A = V 1 U T it is computed using the SVD 8 - 6 Least Squares S. Lall, Stanford 2009.10.14.01 example rank 2 matrix: A = 5 5 14 8 1 1 4 5 4 11 10 24 11 6 the full svd: = . 49 . 30 0 . 82 . 12 . 91 0 . 41 . 86 . 30 0 . 41 35 . 69 0 0 0 7 . 02 0 0 0 0 0 0 . 33 . 31 . 79 . 39 . 15 . 38 . 20 . 11 . 53 . 73 . 25 . 86 . 36 . 25 . 01 . 45 . 26 . 47 . 67 . 25 . 69 . 22 . 14 . 25 . 62 the thin svd: = . 49 . 30 . 12 . 91 . 86 . 30 35 . 69 7 . 02 bracketleftbigg . 33 0 . 31 . 79 . 39 . 15 . 38 0 . 20 . 11 . 53 . 73 bracketrightbigg pseudo-inverse: A = . 33 . 38 . 31 . 20 . 79 . 11 . 39 . 53 . 15 . 73 1 35 . 69 1 7 . 02 bracketleftbigg . 49 . 12 0 . 86 . 30 . 91 0 . 30 bracketrightbigg 8 - 7 Least Squares S. Lall, Stanford 2009.10.14.01 key point: pseudo-inverse solves least-squares estimation problems minimum-norm control problems...
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This note was uploaded on 08/23/2010 for the course EE 263 taught by Professor Boyd,s during the Fall '08 term at Stanford.

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least_squares_2009_10_14_01 - 8 - 1 Least Squares S. Lall,...

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