least_squares_2009_10_14_01_2up

# least_squares_2009_10_14_01_2up - 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 k y meas − Ax k 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 don’t 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 Σ V T then range( A ) = span { u 1 , . . . , u r } also the SVD of A T is A T = V Σ 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 Σ T U T U Σ V T = V Σ T Σ V T which gives an SVD of A T A . Σ T Σ has the same number of non-zero elements as Σ , 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 = ˆ U ˆ Σ V T = A U ê Î ê V T here • ˆ Σ is square, diagonal, positive definite • ˆ U and ˆ V are skinny, orthonormal columns the pseudo-inverse of A is A † = ˆ V ˆ Σ − 1 ˆ 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 . 33 0 . 31 . 79 . 39 − . 15 . 38 0 . 20 − . 11 − . 53 − . 73 pseudo-inverse: A † = . 33 . 38 . 31 . 20 . 79 − . 11 . 39 − . 53 − . 15 − . 73 1 35 . 69 1 7 . 02 − . 49 . 12 0 . 86 . 30 − . 91 0 . 30 8 - 7 Least Squares S. Lall, Stanford 2009.10.14.01 key point: pseudo-inverse solves • least-squares estimation problems • minimum-norm control problems properties of the pseudo-inverse...
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