Auto-zero/Auto-calibration

Revision as of 09:16, 14 August 2010 by Rrogers314 (Talk | contribs)

Mathematical Formulation

Let

$LaTeX: x\,$ a vector of some environmental or control variables that need to be estimated
$LaTeX: \bar{x}$ a vector of calibration points
$LaTeX: \hat{x}$ be the estimate of $LaTeX: x\,$
a vector of nominal values of uncertain parameters affecting the measurement
- Assumed constant or designed in
be the errors in
- Assumed to vary but constant in the intervals between calibrations and real measurements
be the results of a measurement processes attempting to measure
- $LaTeX: y=Y(x;p,e)\,$ where $LaTeX: e\,$ might be additive, multiplicative, or some other form.
- $LaTeX: \bar{y}=Y(\bar{x};p,e)$ the reading values at the calibration points

Subsequently $LaTeX: p\,$ will be assumed fixed for the problem realm; and dropped from notation

$LaTeX: \hat{e}_k$ be estimates of $LaTeX: e_k\,$ derived from $LaTeX: \bar{y}, \bar{y}$
$LaTeX: Q(x,\hat{x})$ be a quality measure of resulting estimation; for example $LaTeX: \sum{(x_i-\hat{x_i})^2}$

The example is oversimplified as will be demonstrated below.

Then the problem can be formulated as: