ℌ2 and ℌ∞ error bounds for model order reduction of second order systems by Krylov subspace methods

2013 ◽  
Author(s):  
Heiko K. F. Panzer ◽  
Thomas Wolf ◽  
Boris Lohmann
Keyword(s):  
Error Bounds ◽  
Model Order Reduction ◽  
Krylov Subspace ◽  
Order Reduction ◽  
Krylov Subspace Methods ◽  
Second Order ◽  
Subspace Methods ◽  
Model Order ◽  
Second Order Systems

Model order reduction by Krylov subspace methods with global error bounds and automatic choice of parameters

2015 ◽  
Vol 63 (8) ◽  
Author(s):  
Heiko Peuscher
Keyword(s):  
Error Bounds ◽  
Model Order Reduction ◽  
Krylov Subspace ◽  
Order Reduction ◽  
Krylov Subspace Methods ◽  
Global Error ◽  
Subspace Methods ◽  
Model Order ◽  
Automatic Choice

ZusammenfassungDie Dissertation stellt rigorose Fehlerschranken und Verfahren zur automatischen Entwicklungspunktwahl bei der Modellordnungsreduktion linearer, zeitinvarianter Systeme mittels Krylow-Unterraum-Methoden vor.Die örtliche Diskretisierung partieller Differentialgleichungen, welche zur Beschreibung dynamischer Systeme in diversen ingenieurwissenschaftlichen Bereichen zum Einsatz kommen, führt meist zu sehr großen Systemen gewöhnlicher Differentialgleichungen, deren Anzahl mit steigenden Ansprüchen an die Modellgenauigkeit zunimmt. Zur Erfüllung von Simulations-, Regelungs- oder Optimierungsaufgaben ist eine Vereinfachung des Modells daher oft unumgänglich; hierzu wurden zahlreiche Methoden mit spezifischen Vor- und Nachteilen beschrieben. Krylow-Unterraum-Methoden, die im Zentrum dieser Arbeit stehen, erfordern verhältnismäßig geringen numerischen Aufwand und sind daher zur Reduktion auch sehr großer Modelle geeignet. Allerdings erhalten sie nicht zwangsläufig die Stabilität des Modells, bieten keine Information über die Reduktionsgüte und erfordern die günstige Wahl gewisser Parameter, der sogenannten Entwicklungspunkte (,,Shifts“) sowie der Ordnung des reduzierten Modells.Ausgehend von einer neuen Formulierung des Fehlersystems werden neue Zugänge zu diesen Problemstellungen aufgezeigt. Ein kumulatives Reduktionsvorgehen, währenddessen das reduzierte Modell iterativ aufgebaut wird, ermöglicht die adaptive Wahl der reduzierten Ordnung und der Entwicklungspunkte. Letztere erfolgt mittels Optimierung in einem Abstiegsverfahren, das oft nur wenige Schritte benötigt. Schließlich werden globale Fehlerschranken für eine Klasse von Zustandsraummodellen eingeführt; der verursachten Überschätzung wird durch Umformulierung des Optimierungsproblems begegnet. Die vorgestellten Methoden können z.B. effizient auf viele Systeme zweiter Ordnung angewandt werden.Fallstudien anhand von Modellen aus der Strukturmechanik, Elektrothermik, Akustik u. a. belegen ihre Effektivität.

scholarly journals Stability Preserving, Adaptive Model Reduction of DAEs by Krylov Subspace Methods

2016 ◽  
Author(s):  
Alessandro Castagnotto ◽  
Heiko Panzer ◽  
Boris Lohmann
Keyword(s):  
Model Reduction ◽  
Model Order Reduction ◽  
Krylov Subspace ◽  
Order Reduction ◽  
Krylov Subspace Methods ◽  
Differential Algebraic Equations ◽  
Design Parameters ◽  
Subspace Methods ◽  
Model Order ◽  
Reduction Strategies

Model order reduction based on Krylov subspace methods stands out due to its generality and low computational cost, making it a predestined candidate for the reduction of truly-large-scale systems. Even so, the inherent flexibility of the method can lead to quite unsatisfactory results as well. In particular, the preservation of stability is not guaranteed per se, attaching even more importance to the careful selection of free design parameters. Whenever a given system is modeled by a set of linear ordinary differential equations (ODE), some remedies for stability preservation are available, such as the one presented in [4] for strictly dissipative realizations or the H2-pseudooptimal reduction strategy introduced in [3, 5]. Oftentimes the object oriented, computerized modelling of dynamical systems yields a system of differential algebraic equations (DAE), which present characteristics not covered by standard ODE theory. In particular, the transfer behavior might be improper and in general, model reduction involves the approximaton of the dynamical and preservation of the algebraic part [1]. Even though in recent years many publications addressed DAE-aware reduction strategies for different indices and structures, the problem of stability preservation is hardly covered. In this contribution, we consider index-1 DAEs in semiexplicit form and propose two reduction strategies that guarantee the stability of the reduced model. In this context, we will take special care in effectively reducing the underlying ODE while operating on the DAE. We will show in theory and through numerical examples that this is not always granted when extending the DAE-aware procedure described in [1] to the case of one-sided reduction. Moreover, we will show that also in the DAE case H2-pseudooptimal reduction has a series of advantages. The resulting stategy, adapted from [2], will preserve stability and select adaptively both the expansion points and the order of the Krylov subspace. The case of improper DAEs retaining an implicit feedthrough will be considered both in theory and examples. [1] S. Gugercin, T. Stykel, and S. Wyatt. Model reduction of descriptor systems by interpolatory projection methods. SIAM J. Sci. Comput., 35(5):B1010–B1033, 2013. [2] H. K. F. Panzer. Model Order Reduction by Krylov Subspace Methods with Global Error Bounds and Automatic Choice of Parameters. PhD thesis, Technische Universität München, 2014. [3] H. K. F. Panzer, S. Jaensch, T. Wolf, and B. Lohmann. A greedy rational Krylov method for H2-pseudooptimal model order reduction with preservation of stability. In American Control Conference, pages 5532–5537, 2013. [4] L. M. Silveira, M. Kamon, I. Elfadel, and J. White. A coordinate-transformed Arnoldi algorithm for generating guaranteed stable reduced-order models of RLC circuits. Computer Methods in Applied Mechanics and Engineering, 169(3-4):377–389, 1999. [5] T. Wolf, H. K. F. Panzer, and B. Lohmann. H2 pseudo-optimality in model order reduction by Krylov subspace methods. In European Control Conference, 2013.

The Error Bound of Timing Domain in Model Order Reduction by Krylov Subspace Methods

2018 ◽  
Vol 27 (06) ◽  
pp. 1850093 ◽  
Author(s):  
Xinsheng Wang ◽  
Mingyan Yu
Keyword(s):  
Error Bound ◽  
Time Domain ◽  
Model Order Reduction ◽  
Krylov Subspace ◽  
Order Reduction ◽  
Auxiliary Variable ◽  
Krylov Subspace Methods ◽  
Subspace Methods ◽  
Model Order ◽  
Error System

In this paper, we present four different error bound estimates of timing domain in model order reduction by Krylov subspace methods. Firstly, we give integral method based on the impulse response in time domain. The second method is to use small sample statistical method to estimate the error bound based on an error system. The error induced by model order reduction process is constructed by an independent system output. We next present the error bound based on frequency domain error bound transformed into time domain method. The final method is reconstructing an error system, which is factorized to the sum of two parts, resulting from model order reduction by Krylov subspace. It is shown that the first factor is of the reduced order system except for subtracting an auxiliary variable, while the second factor is of the original system except for adding an auxiliary variable. In addition, we also give the analysis of the four methods. A few numerical examples are used to show the efficiency of the four different error bound estimate methods.

scholarly journals Model Reduction of Structured Dynamical Systems by Projecting onto the Dominant Eigenspace of the Gramians

2018 ◽  
Vol 10 (2) ◽  
pp. 94
Author(s):  
Shazzad Hasan ◽  
M. Monir Uddin
Keyword(s):  
Model Order Reduction ◽  
Krylov Subspace ◽  
Order Reduction ◽  
Second Order ◽  
Low Rank ◽  
Order Model ◽  
Model Order ◽  
Reduced Order ◽  
Second Order Systems ◽  
Theoretical Results

This paper studies the structure preserving (second-order to second-order) model order reduction of second-order systems applying the projection onto the dominant eigenspace of the Gramians of the systems. The projectors which create the reduced order model are generated cheaply from the low-rank Gramian factors. The low-rank Gramian factors are computed efficiently by solving the corresponding Lyapunov equations of the system using the rational Krylov subspace method. The efficiency of the theoretical results are then illustrated by numerical experiments.

Sign in / Sign up

Export Citation Format

Share Document