Auto-zero/Auto-calibration

(Difference between revisions)

Revision as of 09:35, 14 August 2010

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
be estimates of derived from
- $LaTeX: \hat{e}=E(\bar{x},\bar{y})$
be a quality measure of resulting estimation; for example
- Where $LaTeX: x\,$ is allowed to vary over a domain for fixed $LaTeX: \hat{x}$
- The example is oversimplified as will be demonstrated below.

Then the problem can be formulated as:

Given $LaTeX: \bar{x},\bar{y}$
Find a formula/process to select $LaTeX: (\bar{x},\bar{y})\larrow \hat{x}$ so as to minimize $LaTeX: Q(x,\hat{x})$

@@ Line 11: / Line 11: @@
 ** <math>y=Y(x;p,e)\,</math> where <math>e\,</math> might be additive, multiplicative, or some other form.
 ** <math>\bar{y}=Y(\bar{x};p,e)</math> the reading values at the calibration points
+**Subsequently <math>p\,</math> will be assumed fixed for the problem realm; and dropped from notation
+*<math>\hat{e}</math> be estimates of <math>e\,</math> derived from <math>\bar{x}, \bar{y}</math>
+**<math>\hat{e}=E(\bar{x},\bar{y})</math>
+*<math>Q(x,\hat{x})</math> be a quality measure of resulting estimation; for example <math>\sum{(x_i-\hat{x_i})^2}</math>
+**Where <math>x\,</math> is allowed to vary over a domain for fixed <math>\hat{x}</math>
+**The example is oversimplified as will be demonstrated below.
-Subsequently <math>p\,</math> will be assumed fixed for the problem realm; and dropped from notation
-*<math>\hat{e}_k</math> be estimates of <math>e_k\,</math> derived from <math>\bar{y}, \bar{y}</math>
-*<math>Q(x,\hat{x})</math> be a quality measure of resulting estimation; for example <math>\sum{(x_i-\hat{x_i})^2}</math>
-The example is oversimplified as will be demonstrated below.
 Then the problem can be formulated as:
-*Given  <math>y_j\,</math>
+*Given   <math>\bar{x},\bar{y}</math>
-*Find a formula/process to minimize <math>Q(x,\hat{x})</math>
+*Find a formula/process to select <math>(\bar{x},\bar{y})\larrow \hat{x}</math> so as to minimize <math>Q(x,\hat{x})</math>