Least Squares Problems and Optimal State Estimates

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We now return once more to the equation LaTeX: y=Ax $y=Ax$ , but instead study the situation when this equation has no solutions. This time we investigate how to pick LaTeX: x $x$ to make LaTeX: Ax $Ax$ as close as possible to $y$ .

This type of least squares problem also has strong connections to optimal control. This time problems related to observer design are more natural (estimating the state that best fits the observed output). We study an example, again showing that the same ideas hold for more general linear mappings than just the matrix equation LaTeX: y=Ax $y=Ax$ .