Nonlinear RC Ladder - Revision history

Schuetz2 at 05:36, 17 June 2025

2025-06-17T05:36:43Z

← Older revision		Revision as of 05:36, 17 June 2025
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	title = {Nonlinear RC Ladder},		title = {Nonlinear RC Ladder},
	howpublished = {{MORwiki} -- Model Order Reduction Wiki},		howpublished = {{MORwiki} -- Model Order Reduction Wiki},
	url = <nowiki>{~~http~~://modelreduction.org/~~index.php~~/Nonlinear_RC_Ladder}</nowiki>,		url = <nowiki>{https://modelreduction.org/morwiki/Nonlinear_RC_Ladder}</nowiki>,
	year = 2018		year = 2018
	}		}

Lund: remove link

2023-05-11T12:24:17Z

remove link

← Older revision		Revision as of 12:24, 11 May 2023
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	===Model 2===		===Model 2===
	Second, we discuss the modelling of an RC circuit, where the nonlinear resistors are connected parallel to the capacitors (see Fig. 7 in <ref name = "RewW03"/>). For this, the underlying model is also given by a ~~([[List_of_abbreviations#~~SISO~~\|SISO]])~~ gradient system of the form <ref name="RewW03"/>:		Second, we discuss the modelling of an RC circuit, where the nonlinear resistors are connected parallel to the capacitors (see Fig. 7 in <ref name = "RewW03"/>). For this, the underlying model is also given by a SISO gradient system of the form <ref name="RewW03"/>:

	:<math>		:<math>

Lund: remove link to list of abbreviations

2023-05-11T12:21:56Z

remove link to list of abbreviations

← Older revision		Revision as of 12:21, 11 May 2023
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	===Model 1===		===Model 1===
	First, we discuss the modelling of an RC circuit, where the nonlinear resistors consist of a parallel connected resistor with a diode as shown in the above figure. For this, the underlying model is given by a ~~([[List_of_abbreviations#~~SISO~~\|SISO]])~~ gradient system of the form <ref name="condon04"/>:		First, we discuss the modelling of an RC circuit, where the nonlinear resistors consist of a parallel connected resistor with a diode as shown in the above figure. For this, the underlying model is given by a SISO gradient system of the form <ref name="condon04"/>:

	:<math>		:<math>

Mitchell: typo

2022-03-15T11:34:15Z

typo

← Older revision		Revision as of 11:34, 15 March 2022
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	==Data==		==Data==

	A sample procedural MATLAB implementation ~~fororder~~ <math>N</math> for all three models is given by:		A sample procedural MATLAB implementation for order <math>N</math> for all three models is given by:

	<div class="thumbinner" style="width:540px;text-align:left;">		<div class="thumbinner" style="width:540px;text-align:left;">

Himpe: Added new nonlinearity variant

2020-03-25T12:18:06Z

Added new nonlinearity variant

← Older revision		Revision as of 12:18, 25 March 2020
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	For this benchmark the parameters are selected as: <math>a = 1</math> as in <ref name="chen99"/>.		For this benchmark the parameters are selected as: <math>a = 1</math> as in <ref name="chen99"/>.

			Alternatively, the variant from <ref name="kawano19"/> can be used, featuring the nonlinearity:

			:<math>
			g(x_i) = \frac{x_i^2}{2} + \frac{x_i^3}{3}.
			</math>

			This represents practically a resistor-inductor cascade with nonlinear resistors.

	===Model 3===		===Model 3===
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	C = sparse(1,1,1,1,N); % output matrix		C = sparse(1,1,1,1,N); % output matrix

	switch(model)		switch lower(model)

	case 'shockley'		case 'shockley'
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	f = @(x) Ax - sign(x).(x.*x);		f = @(x) Ax - sign(x).(x.*x);

			case 'ind'

			A = gallery('tridiag',N,1,-2,1);

			f = @(x) A*x - ((x.^2)./2 + (x.^3)./3);

	case 'inv'		case 'inv'
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	otherwise		otherwise

	error('Choose shockley, sign or inv');		error('Choose shockley, sign, ind or inv');
	end		end
	end		end
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	<ref name="condon04a">M. Condon and R. Ivanov, "<span class="plainlinks">[https://doi.org/10.1108/03321640410510730 Model Reduction of Nonlinear Systems]</span>", COMPEL 23(2): 547--557, 2004</ref>		<ref name="condon04a">M. Condon and R. Ivanov, "<span class="plainlinks">[https://doi.org/10.1108/03321640410510730 Model Reduction of Nonlinear Systems]</span>", COMPEL 23(2): 547--557, 2004</ref>

	<ref name="gu12">C. Gu, "<span class="plainlinks">[https://www2.eecs.berkeley.edu/Pubs/TechRpts/2012/EECS-2012-217.pdf Model Order Reduction of Nonlinear ~~DynamicalSystems~~]</span>", PhD Thesis (University of California, Berkeley), 2012.</ref>		<ref name="gu12">C. Gu, "<span class="plainlinks">[https://www2.eecs.berkeley.edu/Pubs/TechRpts/2012/EECS-2012-217.pdf Model Order Reduction of Nonlinear Dynamical Systems]</span>", PhD Thesis (University of California, Berkeley), 2012.</ref>

	<ref name="reis14">T. Reis. "<span class="plainlinks">[https://doi.org/10.1007/978-3-319-08437-4_2 Mathematical Modeling and Analysis of Nonlinear Time-Invariant RLC Circuits]</span>", In: Large-Scale Networks in Engineering and Life Sciences. Modeling and Simulation in Science, Engineering and Technology: 125--198, 2014.</ref>		<ref name="reis14">T. Reis. "<span class="plainlinks">[https://doi.org/10.1007/978-3-319-08437-4_2 Mathematical Modeling and Analysis of Nonlinear Time-Invariant RLC Circuits]</span>", In: Large-Scale Networks in Engineering and Life Sciences. Modeling and Simulation in Science, Engineering and Technology: 125--198, 2014.</ref>

			<ref name="kawano19">Y. Kawano, J.M.A. Scherpen. "<span class="plainlinks">[https://arxiv.org/abs/1902.09836 Empirical Differential Gramians for Nonlinear Model Reduction]</span>", arXiv (cs.SY): 1902.09836, 2019.</ref>

	</references>		</references>

Himpe: Added third model.

2019-10-16T09:32:08Z

Added third model.

← Older revision		Revision as of 09:32, 16 October 2019
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	\dot{x}(t) = \begin{pmatrix} -2 & 1& & \\ 1 & -2 & 1 & \\ & \ddots & \ddots & \ddots\\ & & 1 & -2 \end{pmatrix} x(t) + g(x),		\dot{x}(t) = \begin{pmatrix} -2 & 1& & \\ 1 & -2 & 1 & \\ & \ddots & \ddots & \ddots\\ & & 1 & -2 \end{pmatrix} x(t) + g(x),
	</math>		</math>

			and the output function:

	:<math>		:<math>
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	</math>		</math>

	where the <math>g</math> is a mapping <math>g(x_i):\mathbb{R} \to \mathbb{R}</math>, ~~respreseting~~ the effect of a nonlinear resistor.		where the <math>g</math> is a mapping <math>g(x_i):\mathbb{R} \to \mathbb{R}</math>, respresenting the effect of a nonlinear resistor.

	:<math>		:<math>
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	</math>		</math>

	which represents the effect of a nonlinear resistor, and sgn denotes the [[wikipedia:Sign_function\|~~Sign_function~~]].		which represents the effect of a nonlinear resistor, and sgn denotes the [[wikipedia:Sign_function\|sign function]].

	====Nonlinearity====		====Nonlinearity====
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	</math>		</math>

	which represents the effect of a nonlinear resistor, and sgn denotes the [[wikipedia:Sign_function\|~~Sign_function~~]].		which represents the effect of a nonlinear resistor, and sgn denotes the [[wikipedia:Sign_function\|sign function]].

	For this benchmark the parameters are selected as: <math>a = 1</math> as in <ref name="chen99"/>.		For this benchmark the parameters are selected as: <math>a = 1</math> as in <ref name="chen99"/>.


			===Model 3===
			Third, a circuit of chained [[wikipedia:Inverter_(logic_gate)\|inverter gates]], a so-called inverter chain <ref name="gu12"/>, is presented.
			This is a SISO system, given by:

			:<math>
			\dot{x}(t) = -x(t) + \begin{pmatrix} 0 \\ g(x_1(t)) \\ \vdots \\ g(x_{N-1}(t)) \end{pmatrix} + \begin{pmatrix} u(t) \\ 0 \\ \vdots \\ 0 \end{pmatrix},
			</math>

			and the output function:

			:<math>
			y(t) = x_N(t).
			</math>

			====Nonlinearity====

			The nonlinear function <math>g</math> is a mapping <math>g(x_i):\mathbb{R} \to \mathbb{R}</math> describing the inverter characteristic:

			:<math>
			g(x_i) = v \tanh(a x_i),
			</math>

			being a parameterized [[wikipedia:Hyperbolic_function\|hyperbolic tangent]] with the supply voltage <math>v=1</math> and a physical parameter <math>a</math>, i.e. <math>a=5</math>.


	===Input===		===Input===
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	==Data==		==Data==

	A sample procedural MATLAB implementation ~~of order~~ <math>N</math> for ~~both~~ models is given by:		A sample procedural MATLAB implementation fororder <math>N</math> for all three models is given by:

	<div class="thumbinner" style="width:540px;text-align:left;">		<div class="thumbinner" style="width:540px;text-align:left;">
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	function [f,B,C] = nrc(N,model)		function [f,B,C] = nrc(N,model)
	%% Procedural generation of "Nonlinear RC Ladder" benchmark system		%% Procedural generation of "Nonlinear RC Ladder" benchmark system

	B = sparse(1,1,1,N,1); % input matrix		B = sparse(1,1,1,N,1); % input matrix
	C = sparse(1,1,1,1,N); % output matrix		C = sparse(1,1,1,1,N); % output matrix

	switch(model)		switch(model)

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	A = gallery('tridiag',N,1,-2,1);		A = gallery('tridiag',N,1,-2,1);

	f = @(x) Ax - sign(x).(x.*x);		f = @(x) Ax - sign(x).(x.*x);

			case 'inv'

			f = @(x) [0;tanh(x(2:end))] - x;

			C = sparse(1,N,1,1,N);

	otherwise		otherwise

	error('Choose ~~either~~ shockley or ~~sign~~');		error('Choose shockley, sign or inv');
	end		end
	end		end
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	}		}

	* For the background on the ~~benchmark~~: for Model 1 <span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morChe99 morChe99]</span> (<span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morChe99 BibTeX]</span>) ~~and~~ for Model 2 <span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morRew03 morRew03]</span> (<span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morRew03 BibTeX]</span>)		* For the background on the benchmarks:
			** for Model 1 <span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morChe99 morChe99]</span> (<span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morChe99 BibTeX]</span>)
			** for Model 2 <span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morRew03 morRew03]</span> (<span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morRew03 BibTeX]</span>)
			** for Model 3 <span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morGu12 morGu12]</span> (<span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morGu12 BibTeX]</span>)

	==References==		==References==
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	<ref name="condon04a">M. Condon and R. Ivanov, "<span class="plainlinks">[https://doi.org/10.1108/03321640410510730 Model Reduction of Nonlinear Systems]</span>", COMPEL 23(2): 547--557, 2004</ref>		<ref name="condon04a">M. Condon and R. Ivanov, "<span class="plainlinks">[https://doi.org/10.1108/03321640410510730 Model Reduction of Nonlinear Systems]</span>", COMPEL 23(2): 547--557, 2004</ref>

			<ref name="gu12">C. Gu, "<span class="plainlinks">[https://www2.eecs.berkeley.edu/Pubs/TechRpts/2012/EECS-2012-217.pdf Model Order Reduction of Nonlinear DynamicalSystems]</span>", PhD Thesis (University of California, Berkeley), 2012.</ref>

	<ref name="reis14">T. Reis. "<span class="plainlinks">[https://doi.org/10.1007/978-3-319-08437-4_2 Mathematical Modeling and Analysis of Nonlinear Time-Invariant RLC Circuits]</span>", In: Large-Scale Networks in Engineering and Life Sciences. Modeling and Simulation in Science, Engineering and Technology: 125--198, 2014.</ref>		<ref name="reis14">T. Reis. "<span class="plainlinks">[https://doi.org/10.1007/978-3-319-08437-4_2 Mathematical Modeling and Analysis of Nonlinear Time-Invariant RLC Circuits]</span>", In: Large-Scale Networks in Engineering and Life Sciences. Modeling and Simulation in Science, Engineering and Technology: 125--198, 2014.</ref>

Himpe: added procedural cat

2018-08-22T12:26:51Z

added procedural cat

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

Werner: Updated BibTeX entry.

2018-04-24T09:10:28Z

Updated BibTeX entry.

← Older revision		Revision as of 09:10, 24 April 2018
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	* For the benchmark itself and its data:		* For the benchmark itself and its data:
	::The MORwiki Community. '''Nonlinear RC Ladder'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Nonlinear_RC_Ladder		::The MORwiki Community, '''Nonlinear RC Ladder'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Nonlinear_RC_Ladder

	@MISC{morwiki_modNonRCL,		@MISC{morwiki_modNonRCL,
	author = {The {MORwiki} ~~Community~~},		author = <nowiki>{{The MORwiki Community}}</nowiki>,
	title = {Nonlinear RC Ladder},		title = {Nonlinear RC Ladder},
	howpublished = {{MORwiki} -- Model Order Reduction Wiki},		howpublished = {{MORwiki} -- Model Order Reduction Wiki},
	url = {<nowiki>http://modelreduction.org/index.php/Nonlinear_RC_Ladder</nowiki>},		url = <nowiki>{http://modelreduction.org/index.php/Nonlinear_RC_Ladder}</nowiki>,
	year = {2018}		year = 2018
	}		}

	* For the background on the benchmark: for Model 1 <span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morChe99 morChe99]</span> (<span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morChe99 BibTeX]</span>) and for Model 2 <span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morRew03 morRew03]</span> (<span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morRew03 BibTeX]</span>)		* For the background on the benchmark: for Model 1 <span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morChe99 morChe99]</span> (<span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morChe99 BibTeX]</span>) and for Model 2 <span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morRew03 morRew03]</span> (<span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morRew03 BibTeX]</span>)

Himpe: /* Data */ code update

2018-04-23T11:23:14Z

Data: code update

← Older revision		Revision as of 11:23, 23 April 2018
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	%% Procedural generation of "Nonlinear RC Ladder" benchmark system		%% Procedural generation of "Nonlinear RC Ladder" benchmark system

	~~% input matrix~~		B = sparse(1,1,1,N,1); % input matrix
	B = sparse(N,1);		C = sparse(1,1,1,1,N); % output matrix
	B(1,1) ~~= 1~~;

	% ~~output~~ matrix
	C = sparse(1,~~N);~~
	C(1,1) ~~= 1~~;

	if model == '~~model1~~'		switch(model)
	~~% nonlinearity~~
			case 'shockley'

	g = @(x) exp(40.0*x) + x - 1.0;		g = @(x) exp(40.0*x) + x - 1.0;

	A0 = sparse(N,~~N);~~		A0 = sparse(1,1,1,N,N);
	~~A0(~~1,1) ~~= 1~~;

	A1 = spdiags(ones(N-1,1),-1,N,N) - speye(N);		A1 = spdiags(ones(N-1,1),-1,N,N) - speye(N);
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	A2 = spdiags([ones(N-1,1);0],0,N,N) - spdiags(ones(N,1),1,N,N);		A2 = spdiags([ones(N-1,1);0],0,N,N) - spdiags(ones(N,1),1,N,N);

	~~% vector field and output functional~~
	f = @(x) -g(A0x) + g(A1x) - g(A2*x);		f = @(x) -g(A0x) + g(A1x) - g(A2*x);
	~~else~~
	~~if model ==~~ '~~model2~~'		case 'sign'
	A = ~~spdiags(ones~~(N-1,1~~),-1,N~~,N) - 2*speye(N) +spdiags(ones(N,1),1~~,N,N~~) ;
			A = gallery('tridiag',N,1,-2,1);

	f = @(x) Ax - sign(x).x.^2;		f = @(x) Ax - sign(x).(x.*x);
	~~else~~
	~~msg =~~ '~~Input: Error.~~ Choose either ~~model1~~ or ~~model2~~';		otherwise
	~~error(msg)~~
	~~end~~		error('Choose either shockley or sign');
	end		end
	end		end

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	</div>		</div>

	Here the ~~nonlinear part of the vectorfield~~ is realized in a vectorized form as a [[wikipedia:Closure_(computer_programming)\|closure]].		Here the vector field is realized in a vectorized form as a [[wikipedia:Closure_(computer_programming)\|closure]].

	==Dimensions==		==Dimensions==

Goyal: /* Citation */

2018-04-23T09:23:47Z

Citation

← Older revision		Revision as of 09:23, 23 April 2018
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	}		}

	* For the background on the benchmark: for Model 1 <span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morChe99 morChe99]</span> (<span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morChe99 BibTeX]</span>)		* For the background on the benchmark: for Model 1 <span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morChe99 morChe99]</span> (<span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morChe99 BibTeX]</span>) and for Model 2 <span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morRew03 morRew03]</span> (<span class="plainlinks">[https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morRew03 BibTeX]</span>)

	==References==		==References==