Enhancement of the GUM method to dynamical systems · 8th Workshop on Analysis of Dynamic...
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8 th Workshop on Analysis of Dynamic Measurements, Turin, Italy – May 5-6 2014 Enhancement of the GUM method to dynamical systems Dr. Marco Wegener, Prof. Eckehard Schnieder, Federico Grasso Toro
Transcript of Enhancement of the GUM method to dynamical systems · 8th Workshop on Analysis of Dynamic...
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8th Workshop on Analysis of Dynamic Measurements, Turin, Italy – May 5-6 2014
Enhancement of the GUM method to dynamical systems Dr. Marco Wegener, Prof. Eckehard Schnieder, Federico Grasso Toro
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May 5 2014 | Federico Grasso Toro | Enhancement of the GUM method to dynamical systems | Slide 2
Technische Universität Braunschweig
Technische Universität Braunschweig
Braunschweig
NFF Braunschweig
Carl Friedrich Gauss
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May 5 2014 | Federico Grasso Toro | Enhancement of the GUM method to dynamical systems | Slide 3
Technische Universität Braunschweig
PTB Headquarter Braunschweig
Physikalisch-Technische Bundesanstalt (PTB) The institution responsible for measurement standards and scientific metrology in Germany.
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May 5 2014 | Federico Grasso Toro | Enhancement of the GUM method to dynamical systems | Slide 4
Overview
Motivation
Summary of the GUM method
Adaptation of the GUM method to dynamical systems
Analysis of the system behaviour
Examples
Summary
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May 5 2014 | Federico Grasso Toro | Enhancement of the GUM method to dynamical systems | Slide 5
Motivation
Situation: Real-time uncertainty information required in transportation
Focussed (safety-relevant) applications:
Advanced driver assistance systems GPS-based vehicle localisation with intelligent maps
Track selective localisation Measured values are directly processed by dynamical systems
Safety case demands uncertainty evaluation according to standards ⇒ Use of the “Guide to the Expression of Uncertainty in Measurement” (GUM)
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May 5 2014 | Federico Grasso Toro | Enhancement of the GUM method to dynamical systems | Slide 6
Motivation
Problems:
Time-dependent quantities Continuously varying measurement conditions Systems for data processing with internal memory
⇒ Not covered by the GUM
Goal:
Real-time uncertainty evaluation: With very limited on-board computing power In line with the GUM
Approach:
Adaptation of the GUM algorithm to dynamical systems in state-space
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May 5 2014 | Federico Grasso Toro | Enhancement of the GUM method to dynamical systems | Slide 7
1) Modelling of the measurement
2) Determination of the input estimates
3) Evaluation of the input uncertainties and covariances
4) Calculation of the output estimates
5) Calculation of the output uncertainty matrix
Summary of the ISO GUM method From static to dynamical measuring systems
Steps of the GUM algorithm Static Dynamic
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May 5 2014 | Federico Grasso Toro | Enhancement of the GUM method to dynamical systems | Slide 8
Adaptation of the ISO GUM method to dynamical systems
Step IV: Calculation of the output estimates: Step V: Calculation of the output uncertainties and covariances:
Problem: Methods from systems theory for an analytical examination of system’s
properties cannot be applied ⇒ Transformation of the difference equation into common state-space required form
Calculation of output estimates and output uncertainty matrix
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May 5 2014 | Federico Grasso Toro | Enhancement of the GUM method to dynamical systems | Slide 9
Difference equation to be transformed:
Transformed difference equation:
Matrix vectorisation
Analysis of the system behaviour Transformation of the state uncertainty equation
Kronecker product
Transformation Theorem:
, with
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May 5 2014 | Federico Grasso Toro | Enhancement of the GUM method to dynamical systems | Slide 10
Stability of the state uncertainty matrix
The system represented by
is asymptotically stable if holds of
Equilibrium of the state uncertainty matrix Assuming asymptotic stability Assuming a constant input uncertainty
Analysis of the system behaviour Examination of system properties
Stability Theorem:
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May 5 2014 | Federico Grasso Toro | Enhancement of the GUM method to dynamical systems | Slide 11
Examples
Input of first order low-pass filter
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May 5 2014 | Federico Grasso Toro | Enhancement of the GUM method to dynamical systems | Slide 12
Examples
Output of first order low-pass filter
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May 5 2014 | Federico Grasso Toro | Enhancement of the GUM method to dynamical systems | Slide 13
Examples
Input of double integrator
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May 5 2014 | Federico Grasso Toro | Enhancement of the GUM method to dynamical systems | Slide 14
Examples
Output of double integrator
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May 5 2014 | Federico Grasso Toro | Enhancement of the GUM method to dynamical systems | Slide 15
Summary
Problems: Increasing demand on standard compliant evaluations of time-varying measurement
uncertainties
Real-time uncertainty evaluation with very low computing complexity required
Approach: Adaptation of the GUM algorithm to dynamical systems in state-space
Results: Behaviour of uncertainties attributed to values processed by a dynamical system can be
described by a dynamical system again
Its transformation into state-space form allows analytical calculation of uncertainty matrix’s evolution over time
Approach suitable for advanced applications in transportation e.g. on-board uncertainty evaluation of vehicle localisation
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May 5 2014 | Federico Grasso Toro | Enhancement of the GUM method to dynamical systems | Slide 16
Thank you
Contact: Ing. Federico Grasso Toro
Institute for Traffic Safety and Automation Engineering Technische Universität Braunschweig, Germany
[email protected] www.iva.ing.tu-braunschweig.de Research project QualiSaR is funded by:
www.qualisar.eu
mailto:[email protected]://www.iva.ing.tu-bs.de/http://www.qualisar.eu/
Enhancement of the GUM method to dynamical systemsTechnische Universität BraunschweigTechnische Universität BraunschweigOverviewMotivationMotivationSummary of the ISO GUM methodAdaptation of the ISO GUM method to dynamical systemsAnalysis of the system behaviourAnalysis of the system behaviourExamplesExamplesExamplesExamplesSummaryThank you
