I have seen several forums related to this question about the difference between observers and estimators in control theory and could not understand it clearly. I wanted to know how the state estimators and observers originated in first place. What was the motivation behind this concept? Does it have any relation with statistics?

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    $\begingroup$ Not exactly my area, but IIRC it's pretty well centered in mid 20th century state space control theory. motivation was aerospace and process chemistry. IMO not really stats at all (which is about probability IMO), but there is some overlap with stats methods used in signal processing, especially once you get into optimal control / observer. Maybe ask in the DSP stackexchange? $\endgroup$
    – Pete W
    Apr 26 at 12:26
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    $\begingroup$ I believe estimators are for systems that are perturbed by stochastic terms, while observers consider completely deterministic systems (and therefore would allow for asymptotic convergence of the state error to zero). $\endgroup$
    – fibonatic
    Apr 26 at 13:07
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    $\begingroup$ The motivation behind the concept is rather simple: Most state-space controllers use the state as the control variable ($u = -Kx$). Using the state is beneficial as it provides guarantees for stability and optimality. However, the states are usually not directly measurable. Therefore, one needs a method that computes a state-estimate online. Observers provide this estimate with guarantees of convergence and robustness against gaussian noise. $\endgroup$
    – Petrus1904
    Apr 26 at 13:14
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    $\begingroup$ @Petrus1904 With robustness against gaussian noise I assume you might be referring to Kalman filters, but those are also called Linear Quadratic Estimators, so estimator instead of observer. $\endgroup$
    – fibonatic
    Apr 26 at 15:36

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