Auto-zero/Auto-calibration

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Formulation

Let

$LaTeX: x_i$ be some environmental or control variables that need to be estimated
$LaTeX: \hat{x}_i$ be the estimates of $LaTeX: x_i$
$LaTeX: y_j$ be the results of a measurement processes attempting to measure $LaTeX: x_i$
$LaTeX: p_k$ be uncertain parameters affecting the measurement
$LaTeX: e_k$ be the errors in $LaTeX: p_k$ referred to nominal.
$LaTeX: \hat{e}_k$ be the estimates of $LaTeX: e_k$

Then the problem can be formulated as:

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