Auto-zero/Auto-calibration
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(Difference between revisions)
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** <math>y=Y(x;p,e)\,</math> where <math>e\,</math> might be additive, multiplicative, or some other form. | ** <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 | ** <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: | Then the problem can be formulated as: | ||
- | *Given | + | *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> |
Revision as of 09:35, 14 August 2010
Mathematical Formulation
Let
- a vector of some environmental or control variables that need to be estimated
- a vector of calibration points
- be the estimate of
- 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
- where might be additive, multiplicative, or some other form.
- the reading values at the calibration points
- Subsequently will be assumed fixed for the problem realm; and dropped from notation
- be estimates of derived from
- be a quality measure of resulting estimation; for example
- Where is allowed to vary over a domain for fixed
- The example is oversimplified as will be demonstrated below.
Then the problem can be formulated as:
- Given
- Find a formula/process to select so as to minimize