Auto-zero/Auto-calibration

From Wikimization

(Difference between revisions)

Revision as of 11:51, 21 August 2010

1 Motivation
2 Mathematical Formulation
3 Examples
- 3.1 Biochemical temperature control
- 3.2 Infrared Gas analysers
4 Various forms of
5 Areas of optimization

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

Nomenclature: It will be presumed, unless otherwise stated, that collected variables compose a (topological) manifold; i.e. a collection designated by single symbol and id. Not necessarily possessing a differential geometry metric. The means that there is no intrinsic two dimensional tensor, $LaTeX: g_{ij} \,$ , allowing a identification of contravarient vectors with covarient ones. These terms are presented to provide a mathematically precise context to distinguish: contravarient and covariant vectors, tangent spaces and underlying coordinate spaces. Typically they can be ignored.

The quintessential example of covariant tensor is the differential of a scaler, although the vector space formed by differentials is more extensive than differentials of scalars.
The quintessental contravariant vector is the derivative of a path $LaTeX: p \,$ with component values $LaTeX: e^i =f_i(s) \,$ on the manifold. With $LaTeX: (p)^j=\frac{\partial e^j}{\partial s}\cdot \frac {\partial}{\partial e^j} \,$ being a component of the contravariant vector along $LaTeX: p \,$ parametrized by "s".
Using (see directly below) as an example
- refers to collection of variables identified by "id"
  - Although a collection does not have the properties of a vector space; in some cases we will assume (restrict) it have those properties. In particular this seems to be needed to state that the Q() functions are convex.
- $LaTeX: e_{id}^i \,$ refers to the $LaTeX: i'th \,$ component of $LaTeX: e_{id} \,$
- $LaTeX: (e_{id})_j^i\,$ refers to the tangent/cotangent bundle with $LaTeX: i \,$ selecting a contravariant component and $LaTeX: j \,$ selecting a covariant component
- $LaTeX: (expr)_{|x\leftarrow c} \,$ refers to an expression, "expr", where $LaTeX: (expr)$ is evaluated with $LaTeX: x=c$
- $LaTeX: (expr)_{|x\rightarrow c} \,$ refers to an expression, "expr", where $LaTeX: (expr)$ is evaluated as the limit of "x" as it approaches value "c"

Definitions:

$LaTeX: x\,$ a collection of some environmental or control variables that need to be estimated
$LaTeX: \bar{x}$ a collection of calibration points
$LaTeX: \hat{x}$ be the estimate of $LaTeX: x\,$
$LaTeX: p \,$ a collection of parameters that are constant during operations but selected at design time. The system "real" values during operation are typically $LaTeX: p+e \,$ ; although other modifications, $LaTeX: E(p,e) \,$ are possible indicating variance of parameters from nominal. "p" are mostly included in symbolic formulas to allow sensitivity calculations or completeness in symbolic expressions.
$LaTeX: e\,$ be errors: assumed to vary, but constant during intervals between calibrations and real measurements
be the results of a measurement processes attempting to measure
- $LaTeX: y=Y(x;p,e)\,$ 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
be estimates of derived from
- $LaTeX: \hat{e}=E(\bar{x},\bar{y})$
be a quality measure of resulting estimation; for example
- 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 so as to minimize
- 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 during design and analysis. The direct inversion of for naturally invokes the error space as a link to .
- Inversion for
  - $LaTeX: R_t=\frac{(V_{off}-V_t)(e_t+R_b+e_b)+e_t(V_{ref}+e_{ref})}{V_t-V_{off}-V_{ref}-e_{ref}}$
- 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 .
- 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

Retrieved from "http://www.convexoptimization.com/wikimization/index.php/Auto-zero/Auto-calibration"

@@ Line 7: / Line 7: @@
 ==  Mathematical Formulation ==
-Let
+Nomenclature:
-*<math> x\, </math> a vector of some environmental or control variables that need to be estimated
+It will be presumed, unless otherwise stated, that collected variables compose a (topological) manifold;  i.e. a collection designated by single symbol and id.  Not necessarily possessing a differential geometry metric.  The means that there is no intrinsic two dimensional tensor, <math>g_{ij} \,</math>, allowing a identification of contravarient vectors with covarient ones.  These terms are presented to provide a mathematically precise context to distinguish: contravarient and covariant vectors, tangent spaces and underlying coordinate spaces.  Typically they can be ignored.
-*<math>\bar{x}</math> a vector of calibration points
+* The quintessential example of covariant tensor is the differential of a scaler, although the vector space formed by differentials is more extensive than differentials of scalars.
+* The quintessental contravariant vector is the derivative of a path <math>p \,</math> with component values <math>e^i =f_i(s) \,</math> on the manifold.  With <math>(p)^j=\frac{\partial e^j}{\partial s}\cdot \frac {\partial}{\partial e^j} \,</math> being a component of the contravariant vector along <math>p \,</math> parametrized by "s".
+*Using <math>e \,</math> (see directly below) as an example
+** <math>e_{id} \,</math> refers to collection of variables identified by "id"
+*** Although a collection does not have the properties of a vector space; in some cases we will assume (restrict) it have those properties.  In particular this seems to be needed to state that the Q() functions are convex.
+** <math>e_{id}^i \,</math> refers to the <math>i'th \,</math> component of <math>e_{id} \,</math>
+** <math>(e_{id})_j^i\,</math> refers to the tangent/cotangent bundle with <math>i \,</math> selecting a contravariant component and  <math>j \,</math> selecting a covariant component
+** <math>(expr)_{|x\leftarrow c} \,</math> refers to an expression, "expr", where <math>(expr)</math> is evaluated with <math>x=c</math>
+** <math>(expr)_{|x\rightarrow c} \,</math> refers to an expression, "expr", where <math>(expr)</math> is evaluated as the limit of "x" as it approaches value "c"
+Definitions:
+*<math> x\, </math> a collection of some environmental or control variables that need to be estimated
+*<math>\bar{x}</math> a collection of calibration points
 *<math>\hat{x}</math> be the estimate of <math>x\,</math>
+*<math>p \,</math> a collection of parameters that are constant during operations but selected at design time.  The system "real" values during operation are typically <math>p+e \,</math>; although other modifications, <math>E(p,e) \,</math> are possible indicating variance of parameters from nominal.  "p" are mostly included in symbolic formulas to allow sensitivity calculations or completeness in symbolic expressions.
 *<math>e\,</math> be errors: assumed to vary, but constant during intervals between calibrations and real measurements
 * <math>y\,</math> be the results of a measurement processes attempting to measure <math>x\,</math>
-** <math>y=Y(x;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};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
 *<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>
@@ Line 20: / Line 33: @@
 **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.
-** <math>\hat{x}</math> is typically decomposed into a chain using <math>\hat{e}</math>
+** <math>\hat{x} \,</math> is typically decomposed into a chain using <math>\hat{e}</math>
 Then the problem can be formulated as:
 *Given   <math>\bar{x},\bar{y}</math>
-*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>
+*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 <math>\hat{X}</math> is that many correction schemes are feedback controlled; <math>\hat{X}</math> is never computed, internally, although it might be necessary in design or analysis.
+** The reason for the process term <math>\hat{X} \,</math> is that many correction schemes are feedback controlled; <math>\hat{X} \,</math> is never computed, internally, although it might be necessary in design or analysis.
 == Examples ==

Auto-zero/Auto-calibration

From Wikimization

Revision as of 11:51, 21 August 2010

Contents

Motivation

Mathematical Formulation

Examples

Biochemical temperature control

Infrared Gas analysers

Various forms of $LaTeX: Q()\,$

Areas of optimization

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