Auto-zero/Auto-calibration
From Wikimization
(Difference between revisions)
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+ | == Motivation == | ||
+ | In instrumentation, both in a supporting role and as a prime objective, measurements are taken that are subject to systematic errors. Routes to minimizing the effects of these errors are | ||
+ | * Spend more money on the hardware. This is valid but hits areas of diminishing returns. That is the price rises disproportionately with respect to increased accuracy. | ||
+ | * Apparently in the industrial processing industry various measurement points are implemented and regressed to find "subspaces" that the process has to be operating on. Due to lack of experience I (RR) will not be covering that here; although others are welcome too (and replace this statement). This is apparently called "data reconciliation". | ||
+ | * Calibrations are done and incorporated into the instrument. This might be by analog adjustments or customization via. writeable stores for software to use. | ||
+ | * Runtime Auto-calibrations are done at regular intervals. These are done at a variety of time intervals: every .01 seconds to 30 minutes. I can speak to these most directly; but I consider the "Calibrations" to be a special case. | ||
+ | |||
== Mathematical Formulation == | == Mathematical Formulation == | ||
Let | Let | ||
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Then the problem can be formulated as: | Then the problem can be formulated as: | ||
*Given <math>\bar{x},\bar{y}</math> | *Given <math>\bar{x},\bar{y}</math> | ||
- | *Find a formula | + | *Find a formula or process to select <math>(\bar{x},\bar{y})\xrightarrow{\hat{X}} \hat{x}</math> so as to minimize <math>Q(x,\hat{x})</math> |
+ | ** The reason for the process term is that many correction schemes are feedback controlled and internally never compute <math>\hat{X}</math> ; although it might be necessary in design or analysis. |
Revision as of 09:52, 14 August 2010
Motivation
In instrumentation, both in a supporting role and as a prime objective, measurements are taken that are subject to systematic errors. Routes to minimizing the effects of these errors are
- Spend more money on the hardware. This is valid but hits areas of diminishing returns. That is the price rises disproportionately with respect to increased accuracy.
- Apparently in the industrial processing industry various measurement points are implemented and regressed to find "subspaces" that the process has to be operating on. Due to lack of experience I (RR) will not be covering that here; although others are welcome too (and replace this statement). This is apparently called "data reconciliation".
- Calibrations are done and incorporated into the instrument. This might be by analog adjustments or customization via. writeable stores for software to use.
- Runtime Auto-calibrations are done at regular intervals. These are done at a variety of time intervals: every .01 seconds to 30 minutes. I can speak to these most directly; but I consider the "Calibrations" to be a special case.
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 or process to select so as to minimize
- The reason for the process term is that many correction schemes are feedback controlled and internally never compute ; although it might be necessary in design or analysis.