Modal truncation - Revision history

Himpe at 12:57, 25 August 2022

2022-08-25T12:57:16Z

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	[[Category:method]]		[[Category:method]]
	[[Category:linear]]		[[Category:linear]]
			[[Category:Differential_algebraic_system]]
	[[Category:time invariant]]		[[Category:time invariant]]
	[[Category:first differential order]]		[[Category:first differential order]]

Himpe at 12:54, 25 August 2022

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	[[Category:method]]		[[Category:method]]
	~~[[Category:DAE order unspecified]]~~
	[[Category:linear]]		[[Category:linear]]
	[[Category:time invariant]]		[[Category:time invariant]]

Himpe: Fixed refs and CrossRefs

2021-08-27T08:05:27Z

Fixed refs and CrossRefs

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	==Description==		==Description==
	Model truncation<ref name="Dav66"></~~ref~~> is one of the oldest MOR methods for linear time invariant systems		Model truncation<ref name="Dav66"/> is one of the oldest MOR methods for linear time invariant systems

	:<equation id="gensys" shownumber>		:<equation id="gensys" shownumber>
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	</math>		</math>

	which occur frequently in vibration analysis for mechanical systems. There, <math>M,D,K</math> being referred to as Mass, Damping, and Stiffness matrix are represent a finite element discretization of the mechanical system. In fact, a large variety of modal truncation type approaches originated in this application<ref name="KouB08"></~~ref~~>, e.g., Condensation (Guyan reduction)<ref name="Guy65"></~~ref~~> and Component Mode Synthesis (Craig-Bampton)<ref name="CraB68"></~~ref~~>.		which occur frequently in vibration analysis for mechanical systems. There, <math>M,D,K</math> being referred to as Mass, Damping, and Stiffness matrix are represent a finite element discretization of the mechanical system. In fact, a large variety of modal truncation type approaches originated in this application<ref name="KouB08"/>, e.g., Condensation (Guyan reduction)<ref name="Guy65"/> and Component Mode Synthesis (Craig-Bampton)<ref name="CraB68"/>.
	Modal truncation methods can also be generalized to handle linear, time-invariant system with higher time derivatives and even with time-delays, e.g.		Modal truncation methods can also be generalized to handle linear, time-invariant system with higher time derivatives and even with time-delays, e.g.

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	== Dominant pole based modal truncation ==		== Dominant pole based modal truncation ==

	<figure id="~~bode~~">		<figure id="bode1">
	~~<subfigure~~>
	[[File:Transferf.jpg\|350px\|thumb\|right\|<caption>3D Bode plot of transfer function.</caption>]]		[[File:Transferf.jpg\|350px\|thumb\|right\|<caption>3D Bode plot of transfer function.</caption>]]
	</~~subfigure~~>		</figure>
	<~~subfigure~~>		<figure id="bode2">
	[[File:Bode_newengland.jpg\|350px\|thumb\|right\|<caption>2D Bode plot of transfer function.</caption>]]		[[File:Bode_newengland.jpg\|350px\|thumb\|right\|<caption>2D Bode plot of transfer function.</caption>]]
	~~</subfigure>~~
	</figure>		</figure>

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	\frac{\\|R_j\\|}{\|\real{\lambda_j}\|}		\frac{\\|R_j\\|}{\|\real{\lambda_j}\|}
	</math>		</math>
	is larger than the ones of the other eigentriples which corresponds in some sense to the largest summand in the residue expansion of <math>H(s)</math>. Dominant poles can be observed in the Bode, or sigma plot. ~~<xr id="bode"/> on the right~~ illustrates this phenomenon for the [[Power_system_examples\|New England]] test system.		is larger than the ones of the other eigentriples which corresponds in some sense to the largest summand in the residue expansion of <math>H(s)</math>. Dominant poles can be observed in the Bode, or sigma plot. Fig. 1 and Fig. 2 illustrates this phenomenon for the [[Power_system_examples\|New England]] test system.

	The upper figure shows a three dimensional surface plot of <math>H(s)</math> in a region in the left half plane. The poles of <math>H(s)</math> (eigenvalues of		The upper figure shows a three dimensional surface plot of <math>H(s)</math> in a region in the left half plane. The poles of <math>H(s)</math> (eigenvalues of
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	Note that <math>\tilde{H}(s)</math> constructed in that way is often called modal equivalent of <math>H(s)</math>.		Note that <math>\tilde{H}(s)</math> constructed in that way is often called modal equivalent of <math>H(s)</math>.
	A specially tailored eigenvalue algorithm to compute the sought dominant eigentriplets for this task is the [[DPA\|(Subspace Accelerated) Dominant Pole Algorithm]]		A specially tailored eigenvalue algorithm to compute the sought dominant eigentriplets for this task is the [[DPA\|(Subspace Accelerated) Dominant Pole Algorithm]]
	<ref name="MarLP96"></~~ref~~><ref name="RomM06a"></~~ref~~><ref name="RomM06b"></~~ref~~><ref name="Rom07"></~~ref~~><ref name="RomS08"></~~ref~~><ref name="Kue10"></~~ref~~>		<ref name="MarLP96"/><ref name="RomM06a"/><ref name="RomM06b"/><ref name="Rom07"/><ref name="RomS08"/><ref name="Kue10"/>.
	.

	A MATLAB implementation of this algorithms and certain variants thereof can be found at [https://sites.google.com/site/rommes/software https://sites.google.com/site/rommes/software].		A MATLAB implementation of this algorithms and certain variants thereof can be found at [https://sites.google.com/site/rommes/software https://sites.google.com/site/rommes/software].
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	<ref name="Kue10">P. Kürschner, "<span class="plainlinks">[http://www.mpi-magdeburg.mpg.de/mpcsc/mitarbeiter/kuerschner/docs/masterthesis.pdf Two-sided eigenvalue methods for modal approximation]</span>", Master’s thesis, Chemnitz University of Technology,		<ref name="Kue10">P. Kürschner, "<span class="plainlinks">[http://www.mpi-magdeburg.mpg.de/mpcsc/mitarbeiter/kuerschner/docs/masterthesis.pdf Two-sided eigenvalue methods for modal approximation]</span>", Master’s thesis, Chemnitz University of Technology,
	Department of Mathematics, Germany, 2010.</ref>		Department of Mathematics, Germany, 2010.</ref>
	</ references>		</references>

Kuerschner at 17:33, 11 January 2017

2017-01-11T17:33:38Z

← Older revision		Revision as of 17:33, 11 January 2017
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	.		.

	A MATLAB implementation of this algorithms and certain variants of thereof can be found at [https://sites.google.com/site/rommes/software https://sites.google.com/site/rommes/software].		A MATLAB implementation of this algorithms and certain variants thereof can be found at [https://sites.google.com/site/rommes/software https://sites.google.com/site/rommes/software].

	==References==		==References==

Himpe: equation indent

2013-04-30T23:07:18Z

equation indent

← Older revision		Revision as of 23:07, 30 April 2013
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	Model truncation<ref name="Dav66"></ref> is one of the oldest MOR methods for linear time invariant systems		Model truncation<ref name="Dav66"></ref> is one of the oldest MOR methods for linear time invariant systems

	<equation id="gensys" shownumber>		:<equation id="gensys" shownumber>
	<math>		<math>
	E\dot{x}(t)=A x(t)+B u(t), \quad		E\dot{x}(t)=A x(t)+B u(t), \quad
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	certain eigenvalues <math>\lambda_i\in\Lambda(A,E)</math>. The eigentriples <math>(\lambda_i,x_i,y_i)</math> satisfy.		certain eigenvalues <math>\lambda_i\in\Lambda(A,E)</math>. The eigentriples <math>(\lambda_i,x_i,y_i)</math> satisfy.

	<math>		:<math>
	Ax_i=\lambda_iEx_i,\quad A^Hy_i=\overline{\lambda_i}E^Hy_i,\quad i=1,\ldots,r.		Ax_i=\lambda_iEx_i,\quad A^Hy_i=\overline{\lambda_i}E^Hy_i,\quad i=1,\ldots,r.
	</math>		</math>
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	One advantage of such eigenvalue based MOR methods is that they can be easily adapted to more general time invariant systems, for instance, systems in second order form		One advantage of such eigenvalue based MOR methods is that they can be easily adapted to more general time invariant systems, for instance, systems in second order form

	<math>		:<math>
	M\ddot{x}(t)+D\dot{x}(t)+K x(t)=B u(t), \quad		M\ddot{x}(t)+D\dot{x}(t)+K x(t)=B u(t), \quad
	y(t)=C_px(t)+C_v\dot x(t)+Du(t)		y(t)=C_px(t)+C_v\dot x(t)+Du(t)
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	Modal truncation methods can also be generalized to handle linear, time-invariant system with higher time derivatives and even with time-delays, e.g.		Modal truncation methods can also be generalized to handle linear, time-invariant system with higher time derivatives and even with time-delays, e.g.

	<math>		:<math>
	E\dot{x}(t)=A x(t)+Fx(t-\tau)+B u(t), \quad		E\dot{x}(t)=A x(t)+Fx(t-\tau)+B u(t), \quad
	y(t)=Cx(t)+Du(t),		y(t)=Cx(t)+Du(t),
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	This modal truncation variant aims at the identification of eigentriplets <math>(\lambda_i,x_i,y_i)</math> which have a strong contribution to the input-output behavior of the dynamical. For this purpose, let all eigenvalues be semisimple and consider the residue expansion of the transfer function matrix (exemplary of (1))		This modal truncation variant aims at the identification of eigentriplets <math>(\lambda_i,x_i,y_i)</math> which have a strong contribution to the input-output behavior of the dynamical. For this purpose, let all eigenvalues be semisimple and consider the residue expansion of the transfer function matrix (exemplary of (1))

	<equation id="residue" shownumber>		:<equation id="residue" shownumber>
	<math>H(s)=C(sE-A)^{-1}B+D=\sum\limits_{j=1}^{n_f}\frac{(Cx_j)(y_j^HB)}{s-\lambda_j}+D+R_{\infty},\qquad(2)</math>		<math>H(s)=C(sE-A)^{-1}B+D=\sum\limits_{j=1}^{n_f}\frac{(Cx_j)(y_j^HB)}{s-\lambda_j}+D+R_{\infty},\qquad(2)</math>
	</equation>		</equation>
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	take the associated right and left eigenvectors as columns of the truncation matrices <math>V, W</math>. Equivalently, the reduced order model is obtained by truncating the residue expansion (2):		take the associated right and left eigenvectors as columns of the truncation matrices <math>V, W</math>. Equivalently, the reduced order model is obtained by truncating the residue expansion (2):

	<math>H(s)\approx\tilde{H}(s)=\sum\limits_{j=1}^{r}\frac{(Cx_j)(y_j^HB)}{s-\lambda_j}+D</math>.		:<math>H(s)\approx\tilde{H}(s)=\sum\limits_{j=1}^{r}\frac{(Cx_j)(y_j^HB)}{s-\lambda_j}+D</math>.

	Note that <math>\tilde{H}(s)</math> constructed in that way is often called modal equivalent of <math>H(s)</math>.		Note that <math>\tilde{H}(s)</math> constructed in that way is often called modal equivalent of <math>H(s)</math>.

Kuerschner at 11:47, 29 April 2013

2013-04-29T11:47:06Z

← Older revision		Revision as of 11:47, 29 April 2013
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	Note that <math>\tilde{H}(s)</math> constructed in that way is often called modal equivalent of <math>H(s)</math>.		Note that <math>\tilde{H}(s)</math> constructed in that way is often called modal equivalent of <math>H(s)</math>.
	A specially tailored eigenvalue algorithm to compute the sought dominant eigentriplets for this task is the (Subspace Accelerated) Dominant Pole Algorithm		A specially tailored eigenvalue algorithm to compute the sought dominant eigentriplets for this task is the [[DPA\|(Subspace Accelerated) Dominant Pole Algorithm]]
	<ref name="MarLP96"></ref><ref name="RomM06a"></ref><ref name="RomM06b"></ref><ref name="Rom07"></ref><ref name="RomS08"></ref><ref name="Kue10"></ref>		<ref name="MarLP96"></ref><ref name="RomM06a"></ref><ref name="RomM06b"></ref><ref name="Rom07"></ref><ref name="RomS08"></ref><ref name="Kue10"></ref>
	.		.

Kuerschner at 10:52, 29 April 2013

2013-04-29T10:52:40Z

← Older revision		Revision as of 10:52, 29 April 2013
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	==References==		==References==
	<references>		<references>
	<ref name="Dav66">E. J. Davison, "A method for simplifying linear dynamic systems"		<ref name="Dav66">E. J. Davison, "<span class="plainlinks">[http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1098264 A method for simplifying linear dynamic systems]</span>"
	, IEEE Transaction on Automated Control, vol. 11, no. 1, 93–101, 1966</ref>		, IEEE Transaction on Automated Control, vol. 11, no. 1, 93–101, 1966</ref>

	<ref name="KouB08">P. Koutsovasilis and M. Beitelschmidt, "Comparison of Model Reduction Techniques for Large Mechanical Systems", Multibody		<ref name="KouB08">P. Koutsovasilis and M. Beitelschmidt, "<span class="plainlinks">[http://link.springer.com/article/10.1007%2Fs11044-008-9116-4# Comparison of Model Reduction Techniques for Large Mechanical Systems]</span>", Multibody
	System Dynamics, vol.20, no.2, pp.111-128, 2008</ref>		System Dynamics, vol.20, no.2, pp.111-128, 2008</ref>

	<ref name="Guy65">R.J. Guyan, "Reduction of Stiffness and Mass Matrices", AIAA Journal, vol.3, no.2, pp.380, 1965</ref>		<ref name="Guy65">R.J. Guyan, "<span class="plainlinks">[http://arc.aiaa.org/doi/pdf/10.2514/3.2874 Reduction of Stiffness and Mass Matrices]</span>", AIAA Journal, vol.3, no.2, pp.380, 1965</ref>

	<ref name="CraB68">R. Craig and M. Bampton, "Coupling of Substructures for Dynamic Analyses", AIAA Journal, vol.6, no.7, pp.1313-1319, 1968</ref>		<ref name="CraB68">R. Craig and M. Bampton, "<span class="plainlinks">[http://arc.aiaa.org/doi/pdf/10.2514/3.4741 Coupling of Substructures for Dynamic Analyses]</span>", AIAA Journal, vol.6, no.7, pp.1313-1319, 1968</ref>

	<ref name="MarLP96">N. Martins, L. Lima, and H. Pinto, "Computing dominant poles		<ref name="MarLP96">N. Martins, L. Lima, and H. Pinto, "<span class="plainlinks">[http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=486093 Computing dominant poles of power system transfer functions]</span>", IEEE Transactions on
	of power system transfer functions", IEEE Transactions on
	Power Systems, vol.11, no.1, pp.162-170, 1996</ref>		Power Systems, vol.11, no.1, pp.162-170, 1996</ref>

	<ref name="RomM06a">J. Rommes and N. Martins, "<span class="plainlinks">[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=34850&arnumber=1664957&count=60&index=22 Efficient computation of transfer		<ref name="RomM06a">J. Rommes and N. Martins, "<span class="plainlinks">[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=34850&arnumber=1664957&count=60&index=22 Efficient computation of transfer function dominant poles using subspace acceleration]</span>", IEEE Transactions on
	function dominant poles using subspace acceleration]</span>", IEEE Transactions on
	Power Systems, vol.21, no.3, pp.1218-1226, 2006</ref>		Power Systems, vol.21, no.3, pp.1218-1226, 2006</ref>

Kuerschner at 10:45, 29 April 2013

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← Older revision		Revision as of 10:45, 29 April 2013
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	Power Systems, vol.11, no.1, pp.162-170, 1996</ref>		Power Systems, vol.11, no.1, pp.162-170, 1996</ref>

	<ref name="RomM06a">J. Rommes and N. Martins, "Efficient computation of transfer		<ref name="RomM06a">J. Rommes and N. Martins, "<span class="plainlinks">[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=34850&arnumber=1664957&count=60&index=22 Efficient computation of transfer
	function dominant poles using subspace acceleration", IEEE Transactions on		function dominant poles using subspace acceleration]</span>", IEEE Transactions on
	Power Systems, vol.21, no.3, pp.1218-1226, 2006</ref>		Power Systems, vol.21, no.3, pp.1218-1226, 2006</ref>

	<ref name="RomM06b">J. Rommes and N. Martins, "Efficient computation of multivariable		<ref name="RomM06b">J. Rommes and N. Martins, "<span class="plainlinks">[http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isnumber=36135&arnumber=1717547&count=61&index=0 Efficient computation of multivariable transfer function dominant poles using subspace acceleration]</span>", IEEE Transactions on
	transfer function dominant poles using subspace acceleration", IEEE Transactions on
	Power Systems, vol.21, no.4, pp.1471-1483, 2006</ref>		Power Systems, vol.21, no.4, pp.1471-1483, 2006</ref>

	<ref name="Rom07">J. Rommes, "Methods for eigenvalue problems with applications		<ref name="Rom07">J. Rommes, "<span class="plainlinks">[http://igitur-archive.library.uu.nl/dissertations/2007-0626-202553/index.htm Methods for eigenvalue problems with applications in model order reduction]</span>", Ph.D. dissertation, Universiteit
	in model order reduction", Ph.D. dissertation, Universiteit
	Utrecht, 2007.</ref>		Utrecht, 2007.</ref>

	<ref name="RomS08">J. Rommes and G. L. G. Sleijpen, ~~“Convergence~~ of the dominant		<ref name="RomS08">J. Rommes and G. L. G. Sleijpen, "<span class="plainlinks">[http://epubs.siam.org/doi/abs/10.1137/060671401 Convergence of the dominant pole algorithm and Rayleigh quotient iteration]</span>", SIAM
	pole algorithm and Rayleigh quotient ~~iteration”~~, SIAM
	Journal on Matrix Analysis and Applications, vol. 30, no. 1,		Journal on Matrix Analysis and Applications, vol. 30, no. 1,
	pp. 346–363, 2008.</ref>		pp. 346–363, 2008.</ref>

	<ref name="Kue10">P. Kürschner, "<span class="plainlinks">[http://www.mpi-magdeburg.mpg.de/mpcsc/mitarbeiter/kuerschner/docs/masterthesis.pdf Two-sided eigenvalue methods for modal approximation]</span>”, Master’s thesis, Chemnitz University of Technology,		<ref name="Kue10">P. Kürschner, "<span class="plainlinks">[http://www.mpi-magdeburg.mpg.de/mpcsc/mitarbeiter/kuerschner/docs/masterthesis.pdf Two-sided eigenvalue methods for modal approximation]</span>", Master’s thesis, Chemnitz University of Technology,
	Department of Mathematics, Germany, 2010.</ref>		Department of Mathematics, Germany, 2010.</ref>
	</ references>		</ references>

Kuerschner at 10:33, 29 April 2013

2013-04-29T10:33:33Z

← Older revision		Revision as of 10:33, 29 April 2013
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	Note that <math>\tilde{H}(s)</math> constructed in that way is often called modal equivalent of <math>H(s)</math>.		Note that <math>\tilde{H}(s)</math> constructed in that way is often called modal equivalent of <math>H(s)</math>.
	A specially tailored eigenvalue algorithm to compute the sought dominant eigentriplets for this task is the (Subspace Accelerated) Dominant Pole Algorithm		A specially tailored eigenvalue algorithm to compute the sought dominant eigentriplets for this task is the (Subspace Accelerated) Dominant Pole Algorithm
	<ref name="MarLP96"></ref><ref name="RomM06a"></ref><ref name="RomM06b"></ref><ref name="Rom07"></ref><ref name="Kue10"></ref>		<ref name="MarLP96"></ref><ref name="RomM06a"></ref><ref name="RomM06b"></ref><ref name="Rom07"></ref><ref name="RomS08"></ref><ref name="Kue10"></ref>
	.		.

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	in model order reduction", Ph.D. dissertation, Universiteit		in model order reduction", Ph.D. dissertation, Universiteit
	Utrecht, 2007.</ref>		Utrecht, 2007.</ref>

			<ref name="RomS08">J. Rommes and G. L. G. Sleijpen, “Convergence of the dominant
			pole algorithm and Rayleigh quotient iteration”, SIAM
			Journal on Matrix Analysis and Applications, vol. 30, no. 1,
			pp. 346–363, 2008.</ref>

	<ref name="Kue10">P. Kürschner, "<span class="plainlinks">[http://www.mpi-magdeburg.mpg.de/mpcsc/mitarbeiter/kuerschner/docs/masterthesis.pdf Two-sided eigenvalue methods for modal approximation]</span>”, Master’s thesis, Chemnitz University of Technology,		<ref name="Kue10">P. Kürschner, "<span class="plainlinks">[http://www.mpi-magdeburg.mpg.de/mpcsc/mitarbeiter/kuerschner/docs/masterthesis.pdf Two-sided eigenvalue methods for modal approximation]</span>”, Master’s thesis, Chemnitz University of Technology,
	Department of Mathematics, Germany, 2010.</ref>		Department of Mathematics, Germany, 2010.</ref>
	</ references>		</ references>

Kuerschner at 18:02, 26 April 2013

2013-04-26T18:02:04Z

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	<math>		<math>
	E\dot{x}(t)=A x(t)+B u(t), \quad		E\dot{x}(t)=A x(t)+B u(t), \quad
	y(t)=Cx(t)+Du(t).\qquad (1)		y(t)=Cx(t)+Du(t),\quad E,A\in\mathbb{R}^{n\times n},~B\in\mathbb{R}^{n\times m},~C\in\mathbb{R}^{p\times n},~D\in\mathbb{R}^{p\times m}.\qquad (1)
	</math>		</math>
	</equation>		</equation>