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==The need for Estimation==
[[File:LeastSquares1.jpg|thumb|frame| An example robot]]
A system can be characterized by quantifying some of its physical properties which are relevant to the process it is undergoing and all the quantified values stacked together in a vector form is termed as state. Now to ‘’’uniquely’’’ '''uniquely''' determine the state of the system at a particular instant of time, ‘’’sufficient’’’ '''sufficient''' amount of ‘’’certain’’’ '''certain''' information (the information could be an indirect function of state variables) about the state is required, but often the source of the information are measurements provided by sensors which are almost always ‘’’uncertain’’’ '''uncertain''' in which case the state cannot be ‘’’deterministically calculated’’’'''deterministically calculated'''. Still, controlling a system requires a good approximation or an estimate of the state if not exactly the true state, hence the need of ‘’’estimating’’’ '''estimating''' it. <br \>
The point to be noted here is that the need of ‘’’sufficiency’’’ of the information is still there for state estimation although the exact definition of sufficiency can only be understood after delving deeper into the subject of estimation.
The following example <ref>http://www.bzarg.com/p/how-a-kalman-filter-works-in-pictures/</ref> will provide more clarity to the above discussion.
== Formulation of any state estimation problem ==
Let <math>\vec{X} \in \mathbb{R}^n</math> represent the state of a system and <math>\vec{Y} \in \mathbb{R}^r</math> represent the vector containing all the measured information about the state. Physical properties of the system being measured will be known beforehand and can be written as some function of state vector <math>\vec{X}</math>, say <math>\vec{f}(\vec{X}): \mathbb{R}^n \rightarrow \mathbb{R}^r</math>. Ideally, without any measurement errors, <math>\vec{f}(\vec{X}): </math> should be equal to <math>\vec{Y}</math> but the presence of uncertainty makes them unequal, thus
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