Auto-zero/Auto-calibration

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*<math>\bar{x}</math> a vector of calibration points
*<math>\bar{x}</math> a vector of calibration points
*<math>\hat{x}</math> be the estimate of <math>x\,</math>
*<math>\hat{x}</math> be the estimate of <math>x\,</math>
-
*<math>p\,</math> a vector of nominal values of uncertain parameters affecting the measurement
+
*<math>e\,</math> be errors: assumed to vary, but constant during intervals between calibrations and real measurements
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** Assumed constant or designed in
+
-
*<math>e\,</math> be the errors in <math>p\,</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\,</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.
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** <math>y=Y(x;e)\,</math> where <math>e\,</math> might be additive, multiplicative, or some other form.
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** <math>\bar{y}=Y(\bar{x};p,e)</math> the reading values at the calibration points
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** <math>\bar{y}=Y(\bar{x};e)</math> the reading values at the calibration points
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**Subsequently <math>p\,</math> will be assumed fixed for the problem realm; and dropped from notation
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*<math>\hat{e}</math> be estimates of <math>e\,</math> derived from <math>\bar{x}, \bar{y}</math>
*<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>\hat{e}=E(\bar{x},\bar{y})</math>

Revision as of 01:37, 20 August 2010

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 to (and replace this statement). This is apparently called "data reconciliation".
  • Calibrations are done and incorporated into the instrument. This can be done by analog adjustments or written into storage mediums for subsequent use by operators or software.
  • Runtime Auto-calibrations 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: e\, be errors: assumed to vary, but constant during intervals between calibrations and real measurements
  • LaTeX: y\, be the results of a measurement processes attempting to measure LaTeX: x\,
    • LaTeX: y=Y(x;e)\, where LaTeX: e\, might be additive, multiplicative, or some other form.
    • LaTeX: \bar{y}=Y(\bar{x};e) the reading values at the calibration points
  • 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 LaTeX: \hat{X} is that many correction schemes are feedback controlled; LaTeX: \hat{X} is never computed, internally, 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. Expected final systems accuracies of .05 degC are needed because mammalian temperature regulation has resulting in processes and diseases that are "tuned" to particular temperatures.

  • A simplified example, evaluating one calibration channel and one reading channel. In order to be more obvious the unknown and calibration readings are designated separately; instead of the convention given above. This is more obvious in a simple case, but in more complicated cases is unsystematic.
    • LaTeX: V_x=V_{off}+\frac{V_{ref}R_x}{R_x + R_b}
    • LaTeX: R_x \, can be either the calibration resistor LaTeX: R_c \, or the unknown resistance LaTeX: R_t \, of the thermistor
    • LaTeX: V_x \, is the corresponding voltage read: LaTeX: V_c\, or LaTeX: V_t\,
    • LaTeX: V_{off} \, is the reading offset value, an error
    • LaTeX: V_{ref} \, the bias voltage
    • LaTeX: R_b \, the bias resistor
  • With errors
    • LaTeX: V_x=V_{off}+\frac{(V_{ref} +e_{ref})(R_x +e_x)}{(R_x+e_x + R_b+e_b)}
  • Calibration reading
    • LaTeX: V_c=\left.V_x\right|_{x\leftarrow c}
  • Thermistor (real) reading
    • LaTeX: V_t=\left.V_x\right|_{x\leftarrow t}
  • The problem is to optimally estimate LaTeX: R_t\, based upon LaTeX: V_t \, and LaTeX: V_c \,
  • The direct inversion formula illustrates the utility of mathematically using the error space LaTeX: [V_{off}\,,e_c\,,e_t\,,e_b\,,e_{ref}] \, during design and analysis. The direct inversion of LaTeX: V_t \, for LaTeX: R_t \, naturally invokes the error space as a link to LaTeX: V_c \,.
    • Inversion for LaTeX: R_t \,
      • LaTeX: R_t=\frac{(V_{off}-V_t)(e_t+R_b+e_b)+e_t(V_{ref}+e_{ref})}{V_t-V_{off}-V_{ref}-e_{ref}}
    • Inversion in terms of estimates
      • LaTeX: \widehat{R_t}=\frac{(\widehat{V_{off}}-V_t)(\widehat{e_x}+R_b+\widehat{e_b})+\widehat{e_x}(V_{ref}+\widehat{e_{ref}})}{V_t-\widehat{V_{off}}-V_{ref}-\widehat{e_{ref}}}
  • The problem is to minimize some LaTeX: Q(R_t,\widehat{R_t}).
    • One point, setpoint: when LaTeX: R_t=R_c \, then minimize LaTeX: (1-\frac{\widehat{R_t}}{R_c}) \,. This might seem a little strange, but applies when one is trying to set LaTeX: R_t \, to a setpoint LaTeX: R_c \, but errors occur during measurement. The division is induced when LaTeX: R_t=ke^{b/t} and LaTeX: t \, is the real variable of interest.
    • Mode estimate: Maximize the probability of an estimate treating the calibration measurement LaTeX: V_c \, as a constraint hypersurface ((n-1)-dimensional foliate in a n-dimensional space) in the error space with a defined PDF function. This can done via KKT; and also extended to more calibration readings. In polynomial cases this can theoretically be solved via Groebner basis; but even given the "exact" solutions, one is forced into sub-optimal/approximate estimates.
    • Mean estimate: Using the same model the expected error of a estimate given all possible values on the constraint surface weighted by a PDF distribution on the constraint surface; is minimized. The projection of the original PDF on n-space, to the constraint surface can be done via differential geometry. There are probably statistical methods, but the statistics descriptions seem to take a cavalier attitude towards some transformations involving integrals.
    • Worst case: where points considered where the constraint meets some boundary; say +- .01%
    • Any of the above extended to cover a range of LaTeX: R_t \, as well as the range of errors.
  • It should be mentioned that, in this case LaTeX: (R_t - \widehat{R_t}) is not a good (or natural) function. A better function for both results and calculations is LaTeX: (1-\frac{R_t}{\widehat{R_t}}). I consider the form of errors to be a natural variation from problem to problem and should be accommodated in any organized procedure.
  • Sensitivities are needed during design in order to determine which errors are tight and find out how much improvement can be had by spending more money on individual parts; and during analysis to determine the most likely cause of failures.

Infrared Gas analysers

with either multiple stationary filters or a rotating filter wheel. In either case the components, sensors, and physical structures 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} with respect to the range of LaTeX: e\, and the measurements LaTeX: \bar{y}=Y(\bar{x};p,e)
  • 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()\,


Areas of optimization

Design

Runtime

Calibration usage

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