Autozero/Autocalibration
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(→Algebra of Distributions of Random Variables) 
(→Biochemical temperature control) 

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** <math>V_x=v_{off}+\frac{(V_{ref} +e_{ref})(R_x +e_{com})}{(R_x+e_{com} + R_b+e_b)}</math>  ** <math>V_x=v_{off}+\frac{(V_{ref} +e_{ref})(R_x +e_{com})}{(R_x+e_{com} + R_b+e_b)}</math>  
* Implicit formula for PDF distributions. Generally this form two unknown distributions <math>\overline{V_{t}},\overline{R_{t}}\,</math> but in any particular application one or the other is known. In addition the constants, <math>1, V_{ref}, V_b \,</math> in the formula stand for the Dirac delta distribution positioned on the constant.  * Implicit formula for PDF distributions. Generally this form two unknown distributions <math>\overline{V_{t}},\overline{R_{t}}\,</math> but in any particular application one or the other is known. In addition the constants, <math>1, V_{ref}, V_b \,</math> in the formula stand for the Dirac delta distribution positioned on the constant.  
  **<math>\overline{V_{t}}\ominus\overline{v_{off}}\ominus\left(\left(V_{ref}\oplus\overline{e_{ref}}\right)\oslash\left(1\oplus\left(\left(R_{b}\oplus\overline{e_{b}}\right)\oslash\left(\overline{R_{t}}\oplus\overline{e_{com}}\right)\right)\right)\right)=0\,  +  **<math>\overline{V_{t}}\ominus\overline{v_{off}}\ominus\left(\left(V_{ref}\oplus\overline{e_{ref}}\right)\oslash\left(1\oplus\left(\left(R_{b}\oplus\overline{e_{b}}\right)\oslash\left(\overline{R_{t}}\oplus\overline{e_{com}}\right)\right)\right)\right)=0\,</math> 
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* Calibration reading  * Calibration reading  
** <math>V_c=\left.V_x\right_{x\leftarrow c}</math>  ** <math>V_c=\left.V_x\right_{x\leftarrow c}</math>  
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*** <math>R_t=\frac{(V_tv_{off})(e_{com}+R_b+e_b)e_{com}(V_{ref}+e_{ref})}{V_{ref}v_{off}V_te_{ref}}</math>  *** <math>R_t=\frac{(V_tv_{off})(e_{com}+R_b+e_b)e_{com}(V_{ref}+e_{ref})}{V_{ref}v_{off}V_te_{ref}}</math>  
** Inversion in terms of estimates  ** Inversion in terms of estimates  
  ***<math>\widehat{R_t}=\frac{(V_t\widehat{v_{off}}) (\widehat{e_{com}}+R_b+\widehat{e_b})\widehat{e_{com}}(V_{ref}+\widehat{e_{ref}})}{V_{ref}\widehat{v_{off}}V_{t}\widehat{e_{ref}}}  +  ***<math>\widehat{R_t}=\frac{(V_t\widehat{v_{off}}) (\widehat{e_{com}}+R_b+\widehat{e_b})\widehat{e_{com}}(V_{ref}+\widehat{e_{ref}})}{V_{ref}\widehat{v_{off}}V_{t}\widehat{e_{ref}}}</math> 
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** Mode estimate: Find the maximum value of <math>\overline{R_{t}'} \,</math>. A necessary condition for differentiable functions is <math>\frac{d\overline{R_{t}'}_{{p'}}}{dp'}=0 \,</math> In polynomial cases this can theoretically be solved via Groebner basis; but even given the "exact" solutions, one is forced into suboptimal/approximate estimates.  ** Mode estimate: Find the maximum value of <math>\overline{R_{t}'} \,</math>. A necessary condition for differentiable functions is <math>\frac{d\overline{R_{t}'}_{{p'}}}{dp'}=0 \,</math> In polynomial cases this can theoretically be solved via Groebner basis; but even given the "exact" solutions, one is forced into suboptimal/approximate estimates.  
** Mean estimate: Find the expectation, that is <math>\widehat{R_t}=\int R'\cdot \overline{R_{t}'}_{{R'}} dR' </math>  ** Mean estimate: Find the expectation, that is <math>\widehat{R_t}=\int R'\cdot \overline{R_{t}'}_{{R'}} dR' </math> 
Revision as of 11:48, 24 November 2011
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 Autocalibrations 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, , 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 with component values on the manifold. With being a component of the contravariant vector along 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.
 refers to the component of
 refers to the tangent/cotangent bundle with selecting a contravariant component and selecting a covariant component
 refers to an expression, "expr", where is evaluated with
 refers to an expression, "expr", where is evaluated as the limit of "x" as it approaches value "c"
 refers to collection of variables identified by "id"
Definitions:
 a collection of some environmental or control variables that need to be estimated
 a collection of calibration points
 the PDF of . This is a representation of a function.
 the evaluation of the PDF at .
 be the estimate of
 a collection of parameters that are constant during operations but selected at design time. The system "real" values during operation are typically ; although other modifications are possible.
 be errors: assumed to vary, but constant during intervals between calibrations and real measurements
 the PDF of . This is a representation of a function.
 the evaluation of the PDF at x.
 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 for an instantiation of
 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.
 is typically decomposed into a chain using
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; is never computed, internally, although it might be necessary in design or analysis.
Algebra of Distributions of Random Variables
This section discusses algebraic combinations of variables realized from distributions.
 is the Fourier transform of .
 is the Mellin transform of .
 :Let
 ,
 the PDF's of .
 :Let
 ,
 the PDF's of .
 :Let

 Springer as an extension to
 the PDF's of .

 :Let

 Springer as an extension to
 the PDF's of .

Reference: The algebra of random variables, Springer, M.D., 1979, Wiley, New York
Examples
Biochemical temperature control
where multiple temperature sensors are multiplexed into a data stream and one or more channels are set aside for Autocalibration. 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.
 can be either the calibration resistor or the unknown resistance of the thermistor
 is the corresponding voltage read: or
 is the reading offset value, an error
 the PDF of
 the nominal bias voltage and bias voltage error
 the PDF of
 the nominal bias resistor and bias resistor error
 the PDF of
 an unknown constant resistance in series with for all readings
 the PDF of
 the restricted form of
 Hereafter will signify a restricted form of
 With errors
 Implicit formula for PDF distributions. Generally this form two unknown distributions but in any particular application one or the other is known. In addition the constants, in the formula stand for the Dirac delta distribution positioned on the constant.
 Calibration reading
 Thermistor (real) reading
 The problem is to optimally estimate based upon and
 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
 Inversion in terms of estimates
 Mode estimate: Find the maximum value of . A necessary condition for differentiable functions is In polynomial cases this can theoretically be solved via Groebner basis; but even given the "exact" solutions, one is forced into suboptimal/approximate estimates.
 Mean estimate: Find the expectation, that is
 Worst case: where points considered where the constraint meets some boundary; say + .01%
 Any of the above extended to cover a range of as well as the range of errors.
 Inversion for
 It should be mentioned that, in this case is not a good (or natural) function. A better function for both results and calculations is . 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.
Optimization Format
 The setpoint case with a full optimzer available at runtime. The purpose is to choose error space values that minimize some likelihood function. The selected error values can then be reused to calculate for different values of . It is presumed that we choose likelihoods such that surfaces of constant likelihood form a series of nested n1 dimensional convex sets/shells around a center.
 Absolute value case choosing error values for subsequent calculations.
minimize w.r.t. subject to
Optimization Format 2
 Reformulation in extended matrix format. A derivation is in the discussion under /* Optimization Format */.
 We introduce a new variable to produce coupled equations: . This is meant to simplify the nonlinearities.
minimize w.r.t. subject to
Optimization Format 3
 Working directly on choosing the most likely value of .
minimize r w.r.t subject to
Optimization Format 4
 An alternate selection criteria; probably not usable.
minimize r w.r.t subject to
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
 Weighted least squares of over the range of
 Minimize mode of with respect to the range of and the measurements
 Minimize the mean of with respect to the range of and the measurements
 Minimize the worst case of over the range of
 Some weighting of the error interval with respect to
Areas of optimization
Design
Runtime
Calibration usage