MIT6_241JS11_lec02

MIT6_241JS11_lec02 - 6.241 Dynamic Systems and Control...

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Unformatted text preview: 6.241 Dynamic Systems and Control Lecture 2: Least Square Estimation Readings: DDV, Chapter 2 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology February 7, 2011 E. Frazzoli (MIT) Lecture 2: Least Squares Estimation Feb 7, 2011 1 / 9 Outline 1 Least Squares Estimation E. Frazzoli (MIT) Lecture 2: Least Squares Estimation Feb 7, 2011 2 / 9 Least Squares Estimation Consider an system of m equations in n unknown, with m > n , of the form y = Ax . Assume that the system is inconsistent : there are more equations than unknowns, and these equations are non linear combinations of one another. In these conditions, there is no x such that y Ax = 0. However, one can write e = y Ax , and find x that minimizes e . In particular, the problem min e 2 = min y Ax 2 x x is a least squares problem. The optimal x is the least squares estimate . E. Frazzoli (MIT) Lecture 2: Least Squares Estimation Feb 7, 2011 3 / 9 Computing the Least-Square Estimate The set M := { z R m : z = Ax , x R n } is a subspace of R m , called the range of A , R ( A ), i.e., the set of all vectors that can be obtained by linear combinations...
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MIT6_241JS11_lec02 - 6.241 Dynamic Systems and Control...

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