Auto-zero/Auto-calibration
From Wikimization
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
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
- 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 indicating variance of parameters from nominal. "p" are mostly included in symbolic formulas to allow sensitivity calculations or completeness in symbolic expressions.
- be errors: assumed to vary, but constant during intervals between calibrations and real measurements
- 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
- 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.
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.
- 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 nominal bias voltage and bias voltage error
- the nominal bias resistor and bias resistor error
- an unknown constant resistance in series with for all readings
- With errors
- 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
- Setpoint: when then minimize . This might seem a little strange, but applies when one is trying to set to a setpoint but errors occur during measurement. The division is induced when and is the real variable of interest.
- Mode estimate: Maximize the probability of an estimate treating the calibration measurement as a constraint hypersurface in the error space; requiring 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 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 . 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
- 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