Synthetic parametric model - Revision history

Schuetz2 at 05:40, 17 June 2025

2025-06-17T05:40:39Z

← Older revision		Revision as of 05:40, 17 June 2025
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	To cite this benchmark and its data:		To cite this benchmark and its data:
	::The MORwiki Community, '''Synthetic parametric model'''. hosted at MORwiki - Model Order Reduction Wiki, 2005. ~~http~~://modelreduction.org/~~index.php~~/Synthetic_parametric_model		::The MORwiki Community, '''Synthetic parametric model'''. hosted at MORwiki - Model Order Reduction Wiki, 2005. https://modelreduction.org/morwiki/Synthetic_parametric_model

	@MISC{morwiki_synth_pmodel,		@MISC{morwiki_synth_pmodel,
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	title = {Synthetic parametric model},		title = {Synthetic parametric model},
	howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},		howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
	url = <nowiki>{~~http~~://modelreduction.org/~~index.php~~/Synthetic_parametric_model}</nowiki>,		url = <nowiki>{https://modelreduction.org/morwiki/Synthetic_parametric_model}</nowiki>,
	year = 2005		year = 2005
	}		}

Himpe at 12:59, 25 August 2022

2022-08-25T12:59:36Z

← Older revision		Revision as of 12:59, 25 August 2022
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	[[Category:linear]]		[[Category:linear]]
	[[Category:time invariant]]		[[Category:time invariant]]
	[[Category:~~parametric 1 parameter~~]]		[[Category:Parametric]]
	[[Category:first differential order]]		[[Category:first differential order]]
	[[Category:SISO]]		[[Category:SISO]]

Himpe: Fix CrossRefs

2021-08-27T07:47:58Z

Fix CrossRefs

Mlinaric: Add Python code

2021-07-03T18:11:26Z

Add Python code

@@ Line 145: / Line 145: @@
 C(2:2:N)   = d.';
 C          = sparse(C);
 </syntaxhighlight>
@@ Line 193: / Line 224: @@
      '\epsilon = 1/2', ...
      '\epsilon = 1');
 </syntaxhighlight>

Mlinaric: Use syntaxhighlight tag, fix order of plots in code

2021-07-03T18:06:08Z

Use syntaxhighlight tag, fix order of plots in code

← Older revision		Revision as of 18:06, 3 July 2021
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	System data of arbitrary even order <math>N</math> can be generated in MATLAB or Octave by the following script:		System data of arbitrary even order <math>N</math> can be generated in MATLAB or Octave by the following script:

	<~~div class="thumbinner" style="width:25%;text-align:left;" id="scr1"><!--[[Media:ilp.m\|-->~~		<syntaxhighlight lang="octave">
	~~<source~~ lang="~~matlab~~">
	N = 100; % Order of the resulting system.		N = 100; % Order of the resulting system.

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	C(2:2:N) = d.';		C(2:2:N) = d.';
	C = sparse(C);		C = sparse(C);
	</~~source>~~		</syntaxhighlight>
	~~<!--]]--></div~~>

	Beside that, the plots in <xr id="fig1"/> and <xr id="fig2"/> can be generated in MATLAB and Octave using the following script:		Beside that, the plots in <xr id="fig1"/> and <xr id="fig2"/> can be generated in MATLAB and Octave using the following script:

	<~~div class="thumbinner" style="width:25%;text-align:left;"><!--[[Media:ilp.m\|-->~~		<syntaxhighlight lang="octave">
	~~<source~~ lang="~~matlab~~">
	% Get residues of the system.		% Get residues of the system.
	r(1:2:N-1, 1) = c + 1j * d;		r(1:2:N-1, 1) = c + 1j * d;
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	% Plot poles.		% Plot poles.
	figure;		figure;
	~~loglog~~(~~imag~~(jw), ~~abs~~(~~Hjw~~), '~~LineWidth~~', ~~2);~~		plot(real(p), imag(p), '.', 'MarkerSize', 12);
	~~axis tight;~~		xlabel('Re(p)');
	~~xlim([6 1200]~~);		ylabel('Im(p)');
	xlabel('~~frequency~~ (~~rad/sec~~)');
	ylabel('~~magnitude~~');
	legend( ...		legend( ...
	'\epsilon = 1/50', ...		'\epsilon = 1/50', ...
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	% Plot frequency response.		% Plot frequency response.
	figure;		figure;
	~~plot~~(~~real~~(p), ~~imag~~(p), '.', ~~'MarkerSize', 12~~);		loglog(imag(jw), abs(Hjw), 'LineWidth', 2);
	xlabel('Re(p)');		axis tight;
	ylabel('~~Im(p)~~');		xlim([6 1200]);
			xlabel('frequency (rad/sec)');
			ylabel('magnitude');
	legend( ...		legend( ...
	'\epsilon = 1/50', ...		'\epsilon = 1/50', ...
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	'\epsilon = 1/2', ...		'\epsilon = 1/2', ...
	'\epsilon = 1');		'\epsilon = 1');
	</~~source>~~		</syntaxhighlight>
	~~<!--]]--></div~~>

	==Dimensions==		==Dimensions==

Himpe: added procedural cat

2018-08-22T12:26:42Z

added procedural cat

← Older revision		Revision as of 12:26, 22 August 2018
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	[[Category:benchmark]]		[[Category:benchmark]]
			[[Category:procedural]]
	[[Category:parametric]]		[[Category:parametric]]
	[[Category:linear]]		[[Category:linear]]

Werner: Updated BibTeX entry.

2018-04-24T11:29:04Z

Updated BibTeX entry.

← Older revision		Revision as of 11:29, 24 April 2018
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	To cite this benchmark and its data:		To cite this benchmark and its data:
	:: '''Synthetic parametric model'''. hosted at MORwiki - Model Order Reduction Wiki, 2005. http://modelreduction.org/index.php/Synthetic_parametric_model		::The MORwiki Community, '''Synthetic parametric model'''. hosted at MORwiki - Model Order Reduction Wiki, 2005. http://modelreduction.org/index.php/Synthetic_parametric_model

	@MISC{morwiki_synth_pmodel,		@MISC{morwiki_synth_pmodel,
			author = <nowiki>{{The MORwiki Community}}</nowiki>,
	title = {Synthetic parametric model},		title = {Synthetic parametric model},
	howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},		howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
	url = {<nowiki>http://modelreduction.org/index.php/Synthetic_parametric_model</nowiki>},		url = <nowiki>{http://modelreduction.org/index.php/Synthetic_parametric_model}</nowiki>,
	year = 2005		year = 2005
	}		}

Werner: Updated script for plots.

2018-04-06T08:33:51Z

Updated script for plots.

← Older revision		Revision as of 08:33, 6 April 2018
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	<source lang="matlab">		<source lang="matlab">
	% Get residues of the system.		% Get residues of the system.
	r(1:2:n-1, 1) = c + 1j * d;		r(1:2:N-1, 1) = c + 1j * d;
	r(2:2:n, 1) = c - 1j * d;		r(2:2:N, 1) = c - 1j * d;

	ep = [1/50; 1/20; 1/10; 1/5; 1/2; 1]; % Parameter epsilon.		ep = [1/50; 1/20; 1/10; 1/5; 1/2; 1]; % Parameter epsilon.
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	xlabel('frequency (rad/sec)');		xlabel('frequency (rad/sec)');
	ylabel('magnitude');		ylabel('magnitude');
	~~title~~('~~Frequency response for different~~ \epsilon');		legend( ...
			'\epsilon = 1/50', ...
			'\epsilon = 1/20', ...
			'\epsilon = 1/10', ...
			'\epsilon = 1/5', ...
			'\epsilon = 1/2', ...
			'\epsilon = 1');

	% Plot frequency response.		% Plot frequency response.
	figure;		figure;
	plot(real(p), imag(p), '.');		plot(real(p), imag(p), '.', 'MarkerSize', 12);
	~~title~~('~~Poles for different~~ \epsilon');		xlabel('Re(p)');
			ylabel('Im(p)');
			legend( ...
			'\epsilon = 1/50', ...
			'\epsilon = 1/20', ...
			'\epsilon = 1/10', ...
			'\epsilon = 1/5', ...
			'\epsilon = 1/2', ...
			'\epsilon = 1');
	</source>		</source>
	<!--]]--></div>		<!--]]--></div>

Werner: Added comment on MATLAB script for system dimensions.

2018-04-06T08:22:13Z

Added comment on MATLAB script for system dimensions.

← Older revision		Revision as of 08:22, 6 April 2018
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	The matrix name is used as an extension of the matrix file.		The matrix name is used as an extension of the matrix file.

	System data of arbitrary order <math>N</math> can be generated in MATLAB or Octave by the following script:		System data of arbitrary even order <math>N</math> can be generated in MATLAB or Octave by the following script:

	<div class="thumbinner" style="width:25%;text-align:left;"><!--[[Media:ilp.m\|-->		<div class="thumbinner" style="width:25%;text-align:left;" id="scr1"><!--[[Media:ilp.m\|-->
	<source lang="matlab">		<source lang="matlab">
	N = 100; % Order of the resulting system.		N = 100; % Order of the resulting system.
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	System variants:		System variants:

	<tt>Synth_matrices</tt>: <math>N = 100</math>		<tt>Synth_matrices</tt>: <math>N = 100</math>,
			arbitrary even order <math>N</math> by using the [[#scr1\|script]]

	==Citation==		==Citation==

Werner: Replaced old pictures by new, deleted author from citation part.

2018-04-06T08:07:38Z

Replaced old pictures by new, deleted author from citation part.

← Older revision		Revision as of 08:07, 6 April 2018
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	with <math>A_{\varepsilon, k} = \begin{bmatrix} a_{k} & 0 \\ 0 & a_{k} \end{bmatrix}</math>, <math>A_{0, k} = \begin{bmatrix} 0 & b_{k} \\ -b_{k} & 0 \end{bmatrix}</math>, <math>B_{k} = \begin{bmatrix} 2 \\ 0 \end{bmatrix}</math>, <math>C_{k} = \begin{bmatrix} c_{k}, & d_{k} \end{bmatrix}</math>.		with <math>A_{\varepsilon, k} = \begin{bmatrix} a_{k} & 0 \\ 0 & a_{k} \end{bmatrix}</math>, <math>A_{0, k} = \begin{bmatrix} 0 & b_{k} \\ -b_{k} & 0 \end{bmatrix}</math>, <math>B_{k} = \begin{bmatrix} 2 \\ 0 \end{bmatrix}</math>, <math>C_{k} = \begin{bmatrix} c_{k}, & d_{k} \end{bmatrix}</math>.

	<figure id="fig2">[[~~Image~~:synth_freq_resp.png\|~~frame~~\|~~border~~\|right\|Frequency response of synthetic parametric system, for parameter values ~~1/50 (blue), 1/20 (green), 1/10 (red), 1~~/~~5 (teal), 1/2 (purple), 1 (yellow).~~]]</figure>		<figure id="fig2">[[File:synth_freq_resp.png\|600px\|thumb\|right\|<caption>Frequency response of synthetic parametric system for different parameter values.</caption>]]</figure>

	===Numerical Values===		===Numerical Values===

	<figure id="fig3">[[~~Image~~:synth_hsv.png\|~~frame~~\|~~border~~\|right\|Hankel singular values of synthetic parametric system, for parameter values ~~1/50 (blue), 1/20 (green), 1/10 (red), 1~~/~~5 (teal), 1/2 (purple), 1 (yellow).~~]]</figure>		<figure id="fig3">[[File:synth_hsv.png\|600px\|thumb\|right\|<caption>Hankel singular values of synthetic parametric system for different parameter values.</caption>]]</figure>

	We construct a system of order <math>N = 100</math>.		We construct a system of order <math>N = 100</math>.
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	To cite this benchmark and its data:		To cite this benchmark and its data:
	:: ~~The MORwiki Community.~~ '''Synthetic parametric model'''. hosted at MORwiki - Model Order Reduction Wiki, 2005. http://modelreduction.org/index.php/Synthetic_parametric_model		:: '''Synthetic parametric model'''. hosted at MORwiki - Model Order Reduction Wiki, 2005. http://modelreduction.org/index.php/Synthetic_parametric_model

	@MISC{morwiki_synth_pmodel,		@MISC{morwiki_synth_pmodel,
	~~author = {<nowiki>{The MORwiki Community}</nowiki>},~~
	title = {Synthetic parametric model},		title = {Synthetic parametric model},
	howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},		howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},

← Older revision		Revision as of 07:47, 27 August 2021
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	<figure id="fig1">[[File:synth_poles.png\|600px\|thumb\|right\|<caption>System poles for different parameter values.</caption>]]</figure>		<figure id="fig1">[[File:synth_poles.png\|600px\|thumb\|right\|<caption>System poles for different parameter values.</caption>]]</figure>

	On this page you will find a synthetic parametric model with one parameter for which one can easily experiment with different system orders, values of the parameter, as well as different poles and residues (see ~~<xr id="fig1"/>~~).		On this page you will find a synthetic parametric model with one parameter for which one can easily experiment with different system orders, values of the parameter, as well as different poles and residues (see Fig. 1).
	Also, the decay of the Hankel singular values can be changed indirectly through the parameter.		Also, the decay of the Hankel singular values can be changed indirectly through the parameter.

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	The frequency response of the transfer function <math>H(s,\varepsilon) = C(sI_{N}-(\varepsilon A_{\varepsilon} + A_{0}))^{-1}B</math> is plotted for parameter values <math>\varepsilon \in \left[\frac{1}{50}, \frac{1}{20}, \frac{1}{10}, \frac{1}{5}, \frac{1}{2}, 1\right]</math> in ~~<xr id="fig2"/>~~.		The frequency response of the transfer function <math>H(s,\varepsilon) = C(sI_{N}-(\varepsilon A_{\varepsilon} + A_{0}))^{-1}B</math> is plotted for parameter values <math>\varepsilon \in \left[\frac{1}{50}, \frac{1}{20}, \frac{1}{10}, \frac{1}{5}, \frac{1}{2}, 1\right]</math> in Fig. 2.

	Other interesting plots result for small values of the parameter <math>\varepsilon</math>.		Other interesting plots result for small values of the parameter <math>\varepsilon</math>.
	For example, for <math>\varepsilon = \frac{1}{100}</math> or <math>\frac{1}{1000}</math>, the peaks in the frequency response become more pronounced, since the poles move closer to the imaginary axis.		For example, for <math>\varepsilon = \frac{1}{100}</math> or <math>\frac{1}{1000}</math>, the peaks in the frequency response become more pronounced, since the poles move closer to the imaginary axis.

	For <math>\varepsilon \in \left[\frac{1}{50}, \frac{1}{20}, \frac{1}{10}, \frac{1}{5}, \frac{1}{2}, 1\right]</math>, we also plotted the decay of the Hankel singular values in ~~<xr id="fig3"/>~~.		For <math>\varepsilon \in \left[\frac{1}{50}, \frac{1}{20}, \frac{1}{10}, \frac{1}{5}, \frac{1}{2}, 1\right]</math>, we also plotted the decay of the Hankel singular values in Fig. 3.
	Notice that for small values of the parameter, the decay of the Hankel singular values is very slow.		Notice that for small values of the parameter, the decay of the Hankel singular values is very slow.

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	</syntaxhighlight>		</syntaxhighlight>

	Beside that, the plots in ~~<xr id="fig1"/>~~ and ~~<xr id="fig2"/>~~ can be generated in MATLAB and Octave using the following script:		Beside that, the plots in Fig. 1 and Fig. 2 can be generated in MATLAB and Octave using the following script:

	<syntaxhighlight lang="octave">		<syntaxhighlight lang="octave">