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The matrix measure framework for projection-based model order reduction

Von Rosa Castañé-Selga (02.08.2010)

In order to meet the requirements of today's applications and system designs, a common result of accurate modeling in several fields of science, e. g.  engineering, biology, economics, is a large number (few hundred thousands) of ordinary and partial differential equations (ODEs and PDEs). With the current digital computers, it is very diffcult or even impossible to handle these large-scale systems, mainly due to the limitations in memory, accuracy or executing time. In order to be able to handle such systems for the purpose of simulation, optimization, prediction or controller design, it is advisable to find a reduced order model that approximates the behavior of the original system while preserving its original properties (see Fig. 1). The most important of these properties is stability.

Figure 1: The setup of model order reduction[Bildunterschrift / Subline]: Figure 1: The setup of model order reduction.

Stability preservation has been an active and important research field in model order reduction as stability is an important requirement for the analysis, proper simulation and control of any dynamical system. Hence, the reduced-order model is required to be stable to be a valid approximation of the real system. In the work of Rosa Castañé-Selga, sufficient conditions for guaranteeing stability of reduced-order models are established and proven using the concepts of "contractivity" and "matrix measure". The thesis derives a general framework for stability preservation in model order reduction that can be also used to link several other contributions in this field of research. In addition, this work opens new strategies for the solution of unsolved problems in this area. As a particularly interesting case, the results can be applied without any additional numerical effort to second order systems, obtained for example using the Finite Element Method (FEM), which is widely used in industry (see Fig. 2).

Figure 2: Large-scale model of the International Space Station.[Bildunterschrift / Subline]: Figure 2: Large-scale model of the International Space Station.

Since the beginning of this research project in September 2008 the main efforts were focused on developing a new framework to study the stability preservation problem in projection-based model order reduction. This research approach is designed to offer several advantages, which suggest new solutions to several of the main open problems in the field of model order reduction of large scale systems. Among them, the definition of error bounds for the result of the reduction procedure and the problem of stability preservation in model reduction of parametric systems [3].
The preliminary results are currently being considered for publication in two high impact factor journals [1, 2].


[1] Castañé-Selga, Rosa; Lohmann, Boris; Eid, Rudy: "Stability Preservation in Krylov-based Model Order Reduction using Contractivity Properties". In: Systems and Control Letters (submitted) (2009).

[2] Castañé-Selga, Rosa; Lohmann, Boris; Eid, Rudy: "Stability Preservation in Projection-based Model Order Reduction of Large Scale Systems". In: European Journal of Control (submitted) (2010).

[3]  Eid, Rudy; Castañé-Selga, Rosa; Wolf, Thomas; Panzer, Heiko; Lohmann, Boris: "Stability-Preserving Parametric Model Reduction by Matrix Interpolation". In: Journal on Mathematical and Computer Modelling of Dynamical Systems (submitted) (2010).

  • since 2008
  • Ph.D. studies at the TU München on the topic: "Model order reduction of large-scale systems."
  • 2006-2008
  • Ph.D. studies at the Universitat Politècnica de Catalunya (UPC) in Spain on the topic: "Active noise control in motorcycle helmets."
  • 2000-2005
  • M.Sc. degree in Engineering at the UPC, Spain.

Stipendien und Auslandsaufenthalte
  • since May 2009
  • Scholarship of the Elite Network of Bavaria, TU München, Germany.
  • Sept. 2008-April 2009
  • Research fellowship of the Lehrstuhl für Regelungstechnik, TU München, Germany.
  • Aug.-Nov. 2007
  • Visiting Researcher in the Institute of Sound and Vibration Research. Dynamics Group. Southampton University, United Kingdom.
  • Sept. 2006 - Aug. 2008
  • FPI research fellowship on "Robust LPV Control of Lightly Damped Systems". Ministry of Science, Spain.
  • Aug.-Dec. 2005
  • Visiting Graduate Researcher of ARTIST2 (Network of Excellence on Embedded Systems Design). Department of Automatic Control (Reglerteknik, LTH). Lünd Institute of Technology, Sweden.

  • *R. Castañé-Selga, Ricardo S. Sánchez Peña (2007): "Active Noise Control in Motorcycle Helmets" (in spanish).
  • *R. Castañé-Selga, Ricardo S. Sánchez Peña (2009): "Active Noise Hybrid Time-varying Control for Motorcycle Helmets".
  • *R. Castañé-Selga, R. Eid and B. Lohmann (2009): On the stability of Krylov-based order reduction using invariance properties of the controllability subspace.
  • To appear in Roppencker, G. und Lohmann, B. (Hrsg.): Methoden und Anwendungen der Regelungstechnik. Erlangen-Münchener Workshops 2007 und 2008. Shaker Verlag, Aachen.