Uncertainty quantification of nonlinear distributed parameter systems using generalized polynomial chaos
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Chettapong Janya-anurak
Dr.-Ing. Chettapong Janya-anurak is a lecturer in a Faculty of Engineering at King Mongkut’s University of Technology Thonburi. He studied mechanical engineering at Karlsruhe Institute of Technology. He received his PhD in the Faculty of Informatics from the Karlsruhe Institute of Technology (KIT) in 2016. From 2011 until 2016, he was a research assistant at Fraunhofer Institute of Optronics, System Technologies and Image Exploitation (IOSB) in Karlsruhe, Germany. His main research areas are: simulation, control and data analysis under uncertainties and uncertainty quantification.
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Thomas Bernard
Dr. Thomas Bernard is Physicist. He received his PhD from the University Karlsruhe (TH) in 2000 with a thesis in the field of automatic control. Since 1996, he has been a scientific assistant at Fraunhofer Institute of Optronics, System Technologies and Image Exploitation (IOSB) in Karlsruhe, Germany. Since 2007 he has been a leader of the research group Process Control and Data Analysis at IOSB. His main research areas are: Simulation, control and data analysis of environmental and industrial processes.
Prof. Dr.-Ing. habil. Jürgen Beyerer has been a full professor for informatics at the Institute for Anthropomatics and Robotics at the Karlsruhe Institute of Technology KIT since March 2004 and director of the Fraunhofer Institute of Optronics, System Technologies and Image Exploitation IOSB in Ettlingen, Karlsruhe, Ilmenau and Lemgo. His research interests include automated visual inspection, signal and image processing, variable image acquisition and processing, active vision, metrology, information theory, fusion of data and information from heterogeneous sources, system theory, autonomous systems and automaton.
Abstract
Many industrial and environmental processes are characterized as complex spatio-temporal systems. Such systems known as distributed parameter systems (DPSs) are usually highly complex and it is difficult to establish the relation between model inputs, model outputs and model parameters. Moreover, the solutions of physics-based models commonly differ somehow from the measurements. In this work, appropriate Uncertainty Quantification (UQ) approaches are selected and combined systematically to analyze and identify systems. However, there are two main challenges when applying the UQ approaches to nonlinear distributed parameter systems. These are: (1) how uncertainties are modeled and (2) the computational effort, as the conventional methods require numerous evaluations of the model to compute the probability density function of the response. This paper presents a framework to solve these two issues. Within the Bayesian framework, incomplete knowledge about the system is considered as uncertainty of the system. The uncertainties are represented by random variables, whose probability density function can be achieved by converting the knowledge of the parameters using the Principle of Maximum Entropy. The generalized Polynomial Chaos (gPC) expansion is employed to reduce the computational effort. The framework using gPC based on Bayesian UQ proposed in this work is capable of analyzing systems systematically and reducing the disagreement between model predictions and measurements of the real processes to fulfill user defined performance criteria. The efficiency of the framework is assessed by applying it to a benchmark model (neutron diffusion equation) and to a model of a complex rheological forming process. These applications illustrate that the framework is capable of systematically analyzing the system and optimally calibrating the model parameters.
Zusammenfassung
Viele Industrie- und Umweltprozesse sind komplexe örtlich-zeitliche Systeme, die auch als verteilt-parametrische Systeme bezeichnet werden. Sie können oft durch nichtlineare gekoppelte partielle Differentialgleichungen (PDE) beschrieben werden. Solche Systeme sind oft komplex und es ist schwierig, die Beziehung zwischen Modelleingang, Modellausgang und den Modellparametern herzustellen. Zusätzlich weicht die Prädiktion der physikalischen Modelle in der Regel von den realen Messungen ab. Im vorliegenden Beitrag wird ein systematischer Ansatz vorgestellt, um das Systemverhalten bei solchen Prozessen zu verstehen und die Abweichung zwischen der Modellprädiktion und dem realen Prozess zu reduzieren. Unvollständiges Wissen über den Prozess kann als Unsicherheit im Bayes’schen Sinn interpretiert werden. Die Unsicherheiten werden durch Zufallsvariablen oder ein Zufallsfeld beschrieben und mit der Methode der Unsicherheitsquantifizierung (UQ) quantifiziert. Die Hauptschwierigkeiten der Anwendung dieser UQ Ansätze auf nichtlineare PDE sind: (1) Modellierung der Unsicherheiten, (2) Reduktion des Rechenaufwandes. Herkömmliche Verfahren, wie beispielsweise sampling-basierte Verfahren, benötigen zahlreiche Modellberechnungen. Dieser Rechenaufwand kann durch die Methode des verallgemeinerten polynomialen Chaos (generalized polynomial chaos, gPC) vermindert werden. Die gPC Ansätze sind als effiziente Methode zur Quantifizierung der Unsicherheit bekannt. Als ein neues numerisches Verfahren wird die Polynomial Chaos Expansion (PCE) auf die rekursive Bayes’sche Schätzung angewendet. Die Effizienz und Leistungsfähigkeit des vorgeschlagenen Frameworks wird anhand der Diffusionsgleichung als Benchmark-Modell sowie anhand eines komplexen rheologischen Prozesses zur Herstellung von Glasstäben gezeigt. Mit Hilfe des beschriebenen Vorgehens kann der Prozess systematisch analysiert und die Modellparameter optimal kalibriert werden.
About the authors
Dr.-Ing. Chettapong Janya-anurak is a lecturer in a Faculty of Engineering at King Mongkut’s University of Technology Thonburi. He studied mechanical engineering at Karlsruhe Institute of Technology. He received his PhD in the Faculty of Informatics from the Karlsruhe Institute of Technology (KIT) in 2016. From 2011 until 2016, he was a research assistant at Fraunhofer Institute of Optronics, System Technologies and Image Exploitation (IOSB) in Karlsruhe, Germany. His main research areas are: simulation, control and data analysis under uncertainties and uncertainty quantification.
Dr. Thomas Bernard is Physicist. He received his PhD from the University Karlsruhe (TH) in 2000 with a thesis in the field of automatic control. Since 1996, he has been a scientific assistant at Fraunhofer Institute of Optronics, System Technologies and Image Exploitation (IOSB) in Karlsruhe, Germany. Since 2007 he has been a leader of the research group Process Control and Data Analysis at IOSB. His main research areas are: Simulation, control and data analysis of environmental and industrial processes.
Prof. Dr.-Ing. habil. Jürgen Beyerer has been a full professor for informatics at the Institute for Anthropomatics and Robotics at the Karlsruhe Institute of Technology KIT since March 2004 and director of the Fraunhofer Institute of Optronics, System Technologies and Image Exploitation IOSB in Ettlingen, Karlsruhe, Ilmenau and Lemgo. His research interests include automated visual inspection, signal and image processing, variable image acquisition and processing, active vision, metrology, information theory, fusion of data and information from heterogeneous sources, system theory, autonomous systems and automaton.
Acknowledgment
This work was developed in Fraunhofer Cluster of Excellence “Cognitive Internet Technologies”.
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© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Methoden
- Stabilitätsgesicherte Stellgrößenbegrenzungsstrategie für beliebige zeitdiskrete PI-Zustandsregler bei linearen zeitinvarianten Systemen
- Uncertainty quantification of nonlinear distributed parameter systems using generalized polynomial chaos
- Control of NCSs with energy efficient channel assignment and power allocation
- Verification of hybrid systems using Kaucher arithmetic
- Anwendungen
- Umsetzung einer echtzeitfähigen modellprädiktiven Trajektorienplanung für eine mehrachsige Hybridkinematik auf einer Industriesteuerung
- Vergleich Impedanzregelung (IC) – Parallele Impedanzregelung (PIC) zur Kraft-/Impedanzregelung dynamischer Systeme
Articles in the same Issue
- Frontmatter
- Methoden
- Stabilitätsgesicherte Stellgrößenbegrenzungsstrategie für beliebige zeitdiskrete PI-Zustandsregler bei linearen zeitinvarianten Systemen
- Uncertainty quantification of nonlinear distributed parameter systems using generalized polynomial chaos
- Control of NCSs with energy efficient channel assignment and power allocation
- Verification of hybrid systems using Kaucher arithmetic
- Anwendungen
- Umsetzung einer echtzeitfähigen modellprädiktiven Trajektorienplanung für eine mehrachsige Hybridkinematik auf einer Industriesteuerung
- Vergleich Impedanzregelung (IC) – Parallele Impedanzregelung (PIC) zur Kraft-/Impedanzregelung dynamischer Systeme
