Auto-zero/Auto-calibration

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** <math>V_t=V_{off}+\frac{(V_{ref} +e_{ref}) \cdot (R_t +e_x)}{(R_t+e_x + R_b+e_b)}</math>
** <math>V_t=V_{off}+\frac{(V_{ref} +e_{ref}) \cdot (R_t +e_x)}{(R_t+e_x + R_b+e_b)}</math>
* The problem is to optimally, an ambiguous term during design, estimate <math>R_t \,</math> based upon <math>V_t \,</math> and <math>V_c \,</math>
* The problem is to optimally, an ambiguous term during design, estimate <math>R_t \,</math> based upon <math>V_t \,</math> and <math>V_c \,</math>
-
* The direct inversion formula illustrates the utility of mathematically using the error space, <math>[V_{off},e_x,e_b] \,</math>. during design and analysis. The direct inversion of <math>V_t \,</math> for <math>R_t \,</math> naturally invokes the error space as a link to <math>V_c \,</math>.
+
* The direct inversion formula illustrates the utility of mathematically using the error space, <math>[V_{off},e_x,e_b,e_{ref}] \,</math>. during design and analysis. The direct inversion of <math>V_t \,</math> for <math>R_t \,</math> naturally invokes the error space as a link to <math>V_c \,</math>.
** Inversion for <math>R_t \,</math>
** Inversion for <math>R_t \,</math>
**
**

Revision as of 10:10, 17 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: 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 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 equation only evaluating one calibration channel and one reading channel
    • LaTeX: V_x=V_{off}+\frac{V_{ref} \cdot 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, 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}) \cdot (R_x +e_x)}{(R_x+e_x + R_b+e_b)}
  • Calibration reading
    • LaTeX: V_c=V_{off}+\frac{(V_{ref} +e_{ref}) \cdot (R_c +e_x)}{(R_c+e_x + R_b+e_b)}
  • Thermistor (real) reading.
    • LaTeX: V_t=V_{off}+\frac{(V_{ref} +e_{ref}) \cdot (R_t +e_x)}{(R_t+e_x + R_b+e_b)}
  • The problem is to optimally, an ambiguous term during design, 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_x,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 \,


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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