Auto-zero/Auto-calibration

From Wikimization

(Difference between revisions)
Jump to: navigation, search
Line 1: Line 1:
== Mathematical Formulation ==
== Mathematical Formulation ==
Let
Let
-
*<math> x_i\, </math> be some environmental or control variables that need to be estimated
+
*<math> x\, </math> a vector of some environmental or control variables that need to be estimated
-
*<math>\hat{x}_i</math> be the estimates of <math>x_i\,</math>
+
*<math>\bar{x}</math> a vector of calibration points
-
*<math>Q(x,\hat{x})</math> be a quality measure of resulting estimation; for example <math>\sum{(x_i-\hat{x_i})^2}</math>.
+
*<math>\hat{x}</math> be the estimate of <math>x\,</math>
-
* <math>y_j\,</math> be the results of a measurement processes attempting to measure <math>x_i\,</math>
+
*<math>p\,</math> a vector of nominal values of uncertain parameters affecting the measurement
-
*<math>p_k\,</math> be uncertain parameters affecting the measurement
+
** Assumed constant or designed in
-
*<math>e_k\,</math> be the errors in <math>p_k\,</math> referred to nominal.
+
*<math>e\,</math> be the errors in <math>p\,</math>
-
*<math>\hat{e}_k</math> be the estimates of <math>e_k\,</math>
+
** Assumed to vary but constant in the intervals between calibrations and real measurements
 +
* <math>y\,</math> be the results of a measurement processes attempting to measure <math>x\,</math>
 +
** <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}_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:
Then the problem can be formulated as:
*Given <math>y_j\,</math>
*Given <math>y_j\,</math>
-
*Find a formula/process
+
*Find a formula/process to minimize <math>Q(x,\hat{x})</math>

Revision as of 09:16, 14 August 2010

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\,
  • LaTeX: p\, a vector of nominal values of uncertain parameters affecting the measurement
    • Assumed constant or designed in
  • LaTeX: e\, be the errors in LaTeX: p\,
    • Assumed to vary but constant in the intervals between calibrations and real measurements
  • LaTeX: y\, be the results of a measurement processes attempting to measure LaTeX: x\,
    • 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:

  • Given LaTeX: y_j\,
  • Find a formula/process to minimize LaTeX: Q(x,\hat{x})
Retrieved from "http://www.convexoptimization.com/wikimization/index.php/Auto-zero/Auto-calibration"
Personal tools