Auto-zero/Auto-calibration

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Contents

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; 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

  • 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} be estimates of LaTeX: e\, derived from LaTeX: \bar{x}, \bar{y}
    • LaTeX: \hat{e}=E(\bar{x},\bar{y})
  • LaTeX: Q(x,\hat{x}) be a quality measure of resulting estimation; for example LaTeX: \sum{(x_i-\hat{x_i})^2}
    • Where LaTeX: x\, is allowed to vary over a domain for fixed LaTeX: \hat{x}
    • The example is oversimplified as will be demonstrated below.
    • LaTeX: \hat{x} is typically decomposed into a chain using LaTeX: \hat{e}


Then the problem can be formulated as:

  • Given LaTeX: \bar{x},\bar{y}
  • Find a formula or process to select LaTeX: (\bar{x},\bar{y})\xrightarrow{\hat{X}} \hat{x} so as to minimize LaTeX: Q(x,\hat{x})
    • The reason for the process term is that many correction schemes are feedback controlled and internally never compute LaTeX: \hat{X} ; although it might be necessary in design or analysis.


Examples

Biochemical temperature control where multiple temperature sensors are multiplexed into a data stream and one or more channels are set aside for Auto-calibration

Infrared Gas analyzers with either multiple stationary filters or a rotating filter wheel. In either case the components, sensors, and physical structure are subject to significant variation.


Various forms of LaTeX: Q()\,

  • Weighted least squares of LaTeX: Q()\, over the range of LaTeX: x\,
  • Minimize mode of LaTeX: \hat{e} at calibration
  • Minimize the mean of LaTeX: \hat{e} with respect to the range of LaTeX: e\, and the measurements LaTeX: \bar{y}=Y(\bar{x};p,e)
  • Minimize the worst case of LaTeX: Q()\, over the range of LaTeX: x\,
  • Some weighting of the error interval with respect to LaTeX: Q()\,
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