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Robustness analysis and training of recurrent neural networks using dissipativity theory

Robustheitsanalyse und Training von rekurrenten neuronalen Netzen via Dissipativitätstheorie
Patricia Pauli

Patricia Pauli received the Master’s degree in Mechanical Engineering and Computational Engineering from the Technical University of Darmstadt, Germany, in 2019. She has since been a Ph. D. student with the Institute for Systems Theory and Automatic Control under supervision of Prof. Frank Allgöwer and a member of the International Max Planck Research School for Intelligent Systems (IMPRS-IS). Her research interests are in the area of robust machine learning and learning based control.

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, Julian Berberich

Julian Berberich received the Master’s degree in Engineering Cybernetics from the University of Stuttgart, Germany, in 2018. Since 2018, he has been a Ph. D. student at the Institute for Systems Theory and Automatic Control under supervision of Prof. Frank Allgöwer and a member of the International Max Planck Research School (IMPRS). He received the Outstanding Student Paper Award at the 59th Conference on Decision and Control in 2020. His research interests are in the area of data-driven system analysis and control.

and Frank Allgöwer

Frank Allgöwer is professor of mechanical engineering at the University of Stuttgart, Germany, and Director of the Institute for Systems Theory and Automatic Control (IST) there. Frank is active in serving the community in several roles: Among others he was President of the International Federation of Automatic Control (IFAC) for the years 2017–2020, Vice-president for Technical Activities of the IEEE Control Systems Society for 2013/14, and Editor of the journal Automatica from 2001 until 2015. From 2012 until 2020 Frank served in addition as Vice-president for the German Research Foundation (DFG), which is Germany’s most important research funding organization. His research interests include predictive control, data-based control, networked control, cooperative control, and nonlinear control with application to a wide range of fields including systems biology.

Abstract

Neural networks are widely applied in control applications, yet providing safety guarantees for neural networks is challenging due to their highly nonlinear nature. We provide a comprehensive introduction to the analysis of recurrent neural networks (RNNs) using robust control and dissipativity theory. Specifically, we consider H 2 -performance and the 2 -gain to quantify the robustness of dynamic RNNs with respect to input perturbations. First, we analyze the robustness of RNNs using the proposed robustness certificates and then, we present linear matrix inequality constraints to be used in training of RNNs to enforce robustness. Finally, we illustrate in a numerical example that the proposed approach enhances the robustness of RNNs.

Zusammenfassung

Neuronale Netze sind in regelungstechnischen Anwendungen weit verbreitet, doch die Gewährleistung von Sicherheitsgarantien für neuronale Netze ist aufgrund ihrer hochgradig nichtlinearen Natur schwierig. Wir bieten eine umfassende Einführung in die Analyse von rekurrenten neuronalen Netzen (RNNs) mit Hilfe der robusten Regelung und Dissipativitätstheorie. Um das spezifische Problem der Robustheit gegenüber Eingangsstörungen zu addressieren, betrachten wir die H 2 -Performance und die 2 -Verstärkung als Robustheitsmaße für dynamische RNNs. Zunächst analysieren wir die Robustheit von RNNs unter Verwendung der vorgeschlagenen Robustheitszertifikate und stellen anschließend Lineare-Matrix-Ungleichungs-Bedingungen vor, die beim Training von RNNs verwendet werden können, um Robustheit zu erzwingen. Schließlich zeigen wir in einem numerischen Beispiel, dass sich die Robustheit von RNNs durch unseren Ansatz verbessert.

Award Identifier / Grant number: 390740016

Award Identifier / Grant number: 468094890

Funding statement: This work was funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2075 – 390740016 and under grant 468094890. We acknowledge the support by the Stuttgart Center for Simulation Science (SimTech). The authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Patricia Pauli and Julian Berberich.

About the authors

Patricia Pauli

Patricia Pauli received the Master’s degree in Mechanical Engineering and Computational Engineering from the Technical University of Darmstadt, Germany, in 2019. She has since been a Ph. D. student with the Institute for Systems Theory and Automatic Control under supervision of Prof. Frank Allgöwer and a member of the International Max Planck Research School for Intelligent Systems (IMPRS-IS). Her research interests are in the area of robust machine learning and learning based control.

Julian Berberich

Julian Berberich received the Master’s degree in Engineering Cybernetics from the University of Stuttgart, Germany, in 2018. Since 2018, he has been a Ph. D. student at the Institute for Systems Theory and Automatic Control under supervision of Prof. Frank Allgöwer and a member of the International Max Planck Research School (IMPRS). He received the Outstanding Student Paper Award at the 59th Conference on Decision and Control in 2020. His research interests are in the area of data-driven system analysis and control.

Frank Allgöwer

Frank Allgöwer is professor of mechanical engineering at the University of Stuttgart, Germany, and Director of the Institute for Systems Theory and Automatic Control (IST) there. Frank is active in serving the community in several roles: Among others he was President of the International Federation of Automatic Control (IFAC) for the years 2017–2020, Vice-president for Technical Activities of the IEEE Control Systems Society for 2013/14, and Editor of the journal Automatica from 2001 until 2015. From 2012 until 2020 Frank served in addition as Vice-president for the German Research Foundation (DFG), which is Germany’s most important research funding organization. His research interests include predictive control, data-based control, networked control, cooperative control, and nonlinear control with application to a wide range of fields including systems biology.

Acknowledgment

The authors thank Dennis Gramlich for helpful discussions.

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Received: 2022-02-28
Accepted: 2022-07-05
Published Online: 2022-08-04
Published in Print: 2022-08-26

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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