Nov 23, 2012 - 10. 20. 30. 40. 50. 60. 70. 80. 90. 100 temeprature / (°C). Tre f - T re a d. / (°C. ) 5558. 5565 ... Experimental setup. Thermostatic bath ... era tu re / (°C. ) 5548. Unsteady. Steady. Steady. Uncertainties defined. Uncertainties.
A nonparametric model for sensors used in a dynamic context Nicolas Fischer, Eric Georgin, Amine Ismail, Emmanuel Vazquez, Laurent Le Brusquet
To cite this version: Nicolas Fischer, Eric Georgin, Amine Ismail, Emmanuel Vazquez, Laurent Le Brusquet. A nonparametric model for sensors used in a dynamic context. 7th International Workshop on Analysis of Dynamic Measurements, Oct 2012, Paris, France.
HAL Id: hal-00756461 https://hal-supelec.archives-ouvertes.fr/hal-00756461 Submitted on 23 Nov 2012
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A nonparametric model for sensors used in a dynamic context Nicolas Fischer, Eric Georgin, Amine Ismail, Emmanuel Vazquez, Laurent Le Brusquet
Laboratoire national de métrologie et d’essais
Steady state or static conditions Steady state
Calibration procedure and uncertainty budget: well defined
Example: calibration of 2 temperature sensors (Pt100 type) 0
Tref - Tread / (°C)
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5565
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temeprature / (°C)
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Dispersion of sensors in static conditions Steady state
2 sets of Pt100 probes:
Ø= 6 mm and Ø= 3 mm
Calibrated between +10°C and 90°C 0,1 5548
Ø=6 mm
5549
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Ø=3 mm
Tref - Tread / (°C)
5552 5553
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temperature / (°C)
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Experimental setup
Temperature probe
immersion
Probe carrier Thermostatic bath Guiding system
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Sensor response to a step excitation Unsteady state
Real measurement are often unsteady
Example: step excitation in temperature, 1 probe
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Steady
Steady
Unsteady
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temperature / (°C)
30 29 28
Uncertainties not defined
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5548
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Uncertainties defined
24 23 00:00
Uncertainties defined 00:17
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Time / (mm:ss)
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Sensor response to a step excitation Unsteady state
Real measurement are often unsteady
Example: step excitation in temperature, 1 probe Steady
Steady
Unsteady
32 31
temperature / (°C)
30 29 28 27
5548
What is the uncertainty of the measurement in the unsteady region ?
26 25 24 23 00:00
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Time / (mm:ss)
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Sensor response to a step excitation
Unsteady state
Real measurement are often unsteady
Example: step excitation in temperature, several probes 32 31 30 temperature / (°C)
5548
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5549 5550
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Is it possible to determine a coverage region associated to the whole set of trajectories?
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Time / (mm:ss)
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Model based approach for uncertainty propagation Relies on a model M of the sensor and an imput signal u well known
Sensor response
Sensor excitation
Uncertainty on Y(t) ?
Particularly strong issues in industrial cases : • It may be hard to find an accurate model of a sensor with a small number of parameters • the excitation is often unknown or not well characterized
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Measured signals for a set of 20 sensors Simulation of observed data obtained by : • arbitrary choice of the excitation signal u(t) • considering sensors with different time constants
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Modeling the sensor response
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The model of the response is unknown
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The observed signals suggest that the output trajectories are variations around a mean behaviour
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The number of tested sensors is small
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In the case of fluctuations varying slowly in time, it’s no longer valid to consider that the errors at two different times are independent
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Statistical model and inference The response Yobs is modeled as a sum of : • a Gaussian process M invariant through the sensors representing the nominal behaviour • Gaussian processes specific to each sensor j, representing the departure from the nominal behaviour due to specific values of the conception parameters • a random noise i.i.d. at each time tk, with the sensor j and the repetition i
obs i , j ,k
Y
M (t k ) j (t k ) i , j ,k
i , j ,k ~ N (0, ) 2
The model parameters are then estimated by likelihood maximisation 11
Observed trajectories and simulated trajectories Considering the statistical model estimated, Monte Carlo simulations are performed to compute trajectories of the sensors
Blue plots: 2 trajectories picked from the 20 sensor outputs, Red plots: 2 trajectories picked from the set of simulated trajectories
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Observed trajectories and simulated trajectories With 5 trajectories picked from the set of simulated trajectories
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Observed trajectories and simulated trajectories With 20 trajectories picked from the set of simulated trajectories
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Observed trajectories and simulated trajectories With 100 trajectories picked from the set of simulated trajectories
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Observed trajectories and simulated trajectories With the all set of simulated trajectories (10000 trajectories)
What is the uncertainty associated to the response of the sensors ?
How to compute a coverage region associated to these trajectories ?
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Coverage region Defining a coverage region at a given level
is a coverage region, defined such that :
This region must contain % of the trajectories
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Proposal definition of the confidence region Non unique way to define a coverage region Proposed process :
CI at each instant t 2.To define coverage region of level based on these intervals 1.To obtain confidence intervals
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Mathematical definition of the coverage region R is a coverage region if,
RCI U tT CI (t )
t T , P (Y (t ) R ) constant
defines a coverage region of level
g ( )
• Empirical estimation of the function g() • Inversion of the function leads to find, for a desired level , the value of to be used The red star indicates the value of for = 95%
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Construction of the coverage region R
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Coverage region of level = 95%
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Adequacy between observed trajectories and R
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Quality control of the sensors Among all simulations of sensors responses according to their own time constant, let’s focus on two particular sensors A and B The trajectory of sensor A lies within the coverage region previously estimated The trajectory of sensor B lies outside the coverage region
May we conclude that A is a “good” sensor and B a “bad” one ?
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Identification of sensors A and B The cut-off frequency is the inverse of the time constant of the sensor Sensor A has an expected frequency in the intermediate range such as all the sensors in green whose trajectories belong to the coverage region
On the contrary, sensor B has a low frequency, indicating an inaccurate behaviour, like almost all the sensors outside the coverage region 24
Conclusion
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Proposal to characterize measurement uncertainty for dynamic measurement based on the concept of coverage region
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The estimation of the coverage region requires a statistical model of the sensor output
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The statistical model developed allows to catch both the mean behaviour and the magnitude of the departures from the mean behaviour
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A prior model of the sensor is not required
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Gaussian process is a suitable statistical tool to take into account the time dependence
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Non utile ? Question : besoin d’exemples de réalisations de processus gaussien ?
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Non utile ? Estimated mean behavior. Additional supplied information: confidence area for the mean behavior
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