All pass system - Revision history

Schuetz2 at 08:42, 11 June 2025

2025-06-11T08:42:31Z

← Older revision		Revision as of 08:42, 11 June 2025
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	title = {All-Pass System},		title = {All-Pass System},
	howpublished = {{MORwiki} -- Model Order Reduction Wiki},		howpublished = {{MORwiki} -- Model Order Reduction Wiki},
	url = <nowiki>{~~http~~://modelreduction.org/index.php/All_pass_system}</nowiki>,		url = <nowiki>{https://modelreduction.org/morwiki/index.php/All_pass_system}</nowiki>,
	year = {2020}		year = {2020}
	}		}

Mlinaric: Edit categories, replace source tags

2023-08-29T16:38:19Z

Edit categories, replace source tags

← Older revision		Revision as of 16:38, 29 August 2023
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	[[Category:procedural]]		[[Category:procedural]]
	[[Category:linear]]		[[Category:linear]]
			[[Category:time invariant]]
	[[Category:first differential order]]		[[Category:first differential order]]
	[[Category:~~time invariant~~]]		[[Category:Sparse]]
	[[Category:SISO]]		[[Category:SISO]]

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	<!-- TODO add unbalancing transformation -->		<!-- TODO add unbalancing transformation -->
	:<~~source~~ lang="matlab">		:<syntaxhighlight lang="matlab">
	function [A,B,C,D] = allpass(n)		function [A,B,C,D] = allpass(n)
	% allpass (all-pass system)		% allpass (all-pass system)
Line 62:		Line 63:
	D = 1;		D = 1;
	end		end
	</~~source~~>		</syntaxhighlight>

	The function call requires one argument; the number of states <math>n</math>.		The function call requires one argument; the number of states <math>n</math>.
	The return value consists of four matrices; the system matrix <math>A</math>, the input matrix <math>B</math>, the output matrix <math>C</math>, and the feed-through matrix <math>D</math>.		The return value consists of four matrices; the system matrix <math>A</math>, the input matrix <math>B</math>, the output matrix <math>C</math>, and the feed-through matrix <math>D</math>.

	:<~~source~~ lang="matlab">		:<syntaxhighlight lang="matlab">
	[A,B,C,D] = allpass(n);		[A,B,C,D] = allpass(n);
	</~~source~~>		</syntaxhighlight>

	An equivalent [https://www.python.org/ Python] code is		An equivalent [https://www.python.org/ Python] code is

	:<~~source~~ lang="python">		:<syntaxhighlight lang="python">
	from scipy.sparse import diags, lil_matrix		from scipy.sparse import diags, lil_matrix

Line 86:		Line 87:
	D = 1		D = 1
	return A, B, C, D		return A, B, C, D
	</~~source~~>		</syntaxhighlight>

	==Dimensions==		==Dimensions==

Mlinaric: Fixes, Python code, N->n

2023-08-29T01:00:53Z

Fixes, Python code, N->n

← Older revision		Revision as of 01:00, 29 August 2023
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	For an all-pass system, the transfer function has the property <math>g(s)g(-s) = \sigma^2</math>, <math>\sigma > 0</math>,		For an all-pass system, the transfer function has the property <math>g(s)g(-s) = \sigma^2</math>, <math>\sigma > 0</math>,
	or (equivalently) the controllability and observability Gramians are quasi inverse to each other: <math>W_C W_O = \sigma I</math>,		or (equivalently) the controllability and observability Gramians are quasi inverse to each other: <math>W_C W_O = \sigma I</math>,
	which means this system has a ~~singular~~ Hankel singular value of multiplicity of the system's order.		which means this system has a single Hankel singular value of multiplicity of the system's order.
	The system matrices are ~~constructing~~ based on the scheme:		The system matrices are constructed based on the scheme:

	:<math>		:<math>
	\begin{align}		\begin{align}
	A &= \begin{pmatrix} a_{1,1} & -\alpha_1 \\		A &=
	\alpha_1 & 0 & -\alpha_2 \\		\begin{pmatrix}
	& \~~ddots~~ & ~~\ddots~~ & \ddots \\		a_{1,1} & -\alpha_1 \\
	& & \alpha_{N-1} & 0 & -\alpha_{N-1} \end{pmatrix}, \\		\alpha_1 & 0 & -\alpha_2 \\
	B &= \begin{pmatrix} b_1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \\		& \alpha_2 & 0 & \ddots \\
	C &= \begin{pmatrix} s_1 b_1 & 0 & \~~dots~~ & 0 \end{pmatrix}, \\		& & \ddots & \ddots & -\alpha_{N-1} \\
			& & & \alpha_{N-1} & 0
			\end{pmatrix}, \\
			B &=
			\begin{pmatrix}
			b_1 \\
			0 \\
			\vdots \\
			0
			\end{pmatrix}, \\
			C &=
			\begin{pmatrix}
			s_1 b_1 & 0 & \cdots & 0
			\end{pmatrix}, \\
	D &= -s_1 \sigma.		D &= -s_1 \sigma.
	\end{align}		\end{align}
	</math>		</math>

	We choose <math>s_1 \in \{-1,1\}</math>, to be <math>s_1 \equiv -1</math>, as this makes the system state-space-anti-symmetric.		We choose <math>s_1 \in \{-1,1\}</math> to be <math>s_1 \equiv -1</math>, as this makes the system state-space-anti-symmetric.
	Furthermore, <math>b_1 = 1</math> and <math>\~~sigma_1~~ = 1</math>, which makes <math>a_{1,1} = -\frac{b_1^2}{2 \sigma} = -\frac{1}{2}</math>.		Furthermore, <math>b_1 = 1</math> and <math>\sigma = 1</math>, which makes <math>a_{1,1} = -\frac{b_1^2}{2 \sigma} = -\frac{1}{2}</math>.

	==Data==		==Data==
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	<!-- TODO add unbalancing transformation -->		<!-- TODO add unbalancing transformation -->
			:<source lang="matlab">
	~~<div class="thumbinner" style="width:20%;text-align:left;"><!--[[Media~~:~~allpass.m\|-->~~		function [A,B,C,D] = allpass(n)
	<source lang="matlab">

	function [A,B,C,D] = allpass(N)
	% allpass (all-pass system)		% allpass (all-pass system)
	% by Christian Himpe, 2020		% by Christian Himpe, 2020
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	%*		%*

	A = gallery('tridiag',N,-1,0,1);		A = gallery('tridiag',n,-1,0,1);
	A(1,1) = -0.5;		A(1,1) = -0.5;
	B = sparse(1,1,1,N,1);		B = sparse(1,1,1,n,1);
	C = -B';		C = -B';
	D = 1;		D = 1;
	end		end
	</source>		</source>
	~~<!--]]--></div>~~

	The function call requires one argument; the number of states <math>N</math>.		The function call requires one argument; the number of states <math>n</math>.
	The return value consists of four matrices; the system matrix <math>A</math>, the input matrix <math>B</math>, the output matrix <math>C</math>, and the feed-through matrix <math>D</math>.		The return value consists of four matrices; the system matrix <math>A</math>, the input matrix <math>B</math>, the output matrix <math>C</math>, and the feed-through matrix <math>D</math>.

	:<source lang="matlab">		:<source lang="matlab">
	[A,B,C,D] = allpass(N);		[A,B,C,D] = allpass(n);
			</source>

			An equivalent [https://www.python.org/ Python] code is

			:<source lang="python">
			from scipy.sparse import diags, lil_matrix

			def allpass(n):
			A = diags([-1, 0, 1], offsets=[-1, 0, 1], shape=(n, n), format='lil')
			A[0, 0] = -0.5
			A = A.tocsc()
			B = lil_matrix((n, 1))
			B[0, 0] = 1
			B = B.tocsc()
			C = -B.T
			D = 1
			return A, B, C, D
	</source>		</source>

	==Dimensions==		==Dimensions==


	:<math>		:<math>
	\begin{~~array}{rcl~~}		\begin{align}
	\dot{x}(t) &=& Ax(t) + Bu(t) \\		\dot{x}(t) &= Ax(t) + Bu(t) \\
	y(t) &=& Cx(t) + Du(t)		y(t) &= Cx(t) + Du(t)
	\end{~~array~~}		\end{align}
	</math>		</math>

	System dimensions:		System dimensions:

	<math>A \in \mathbb{R}^{N \times N}</math>,		<math>A \in \mathbb{R}^{n \times n}</math>,
	<math>B \in \mathbb{R}^{N \times 1}</math>,		<math>B \in \mathbb{R}^{n \times 1}</math>,
	<math>C \in \mathbb{R}^{1 \times N}</math>,		<math>C \in \mathbb{R}^{1 \times n}</math>,
	<math>D \in \mathbb{R}</math>.		<math>D \in \mathbb{R}</math>.

Lund: remove preliminary warning

2022-08-26T13:03:49Z

remove preliminary warning

← Older revision		Revision as of 13:03, 26 August 2022
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	~~{{preliminary}} <!-- Do not remove -->~~

	[[Category:benchmark]]		[[Category:benchmark]]
	[[Category:procedural]]		[[Category:procedural]]

Mitchell: typo

2022-03-14T12:59:30Z

typo

← Older revision		Revision as of 12:59, 14 March 2022
Line 12:		Line 12:

	This procedural benchmark generates an all-pass SISO system based on <ref name="Obe87"/>.		This procedural benchmark generates an all-pass SISO system based on <ref name="Obe87"/>.
	For an all-pass system, the ~~trasfer~~ function has the property <math>g(s)g(-s) = \sigma^2</math>, <math>\sigma > 0</math>,		For an all-pass system, the transfer function has the property <math>g(s)g(-s) = \sigma^2</math>, <math>\sigma > 0</math>,
	or (equivalently) the controllability and observability Gramians are quasi inverse to each other: <math>W_C W_O = \sigma I</math>,		or (equivalently) the controllability and observability Gramians are quasi inverse to each other: <math>W_C W_O = \sigma I</math>,
	which means this system has a singular Hankel singular value of multiplicity of the system's order.		which means this system has a singular Hankel singular value of multiplicity of the system's order.

Himpe at 15:26, 8 April 2020

2020-04-08T15:26:29Z

← Older revision		Revision as of 15:26, 8 April 2020
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	* For the benchmark itself and its data:		* For the benchmark itself and its data:
	::The MORwiki Community, '''All-Pass System'''. MORwiki - Model Order Reduction Wiki, ~~2018~~. http://modelreduction.org/index.php/All_pass_system		::The MORwiki Community, '''All-Pass System'''. MORwiki - Model Order Reduction Wiki, 2020. http://modelreduction.org/index.php/All_pass_system

	@MISC{morwiki_allpass,		@MISC{morwiki_allpass,

Himpe: add HSV meaning.

2020-04-08T15:25:55Z

add HSV meaning.

← Older revision		Revision as of 15:25, 8 April 2020
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	This procedural benchmark generates an all-pass SISO system based on <ref name="Obe87"/>.		This procedural benchmark generates an all-pass SISO system based on <ref name="Obe87"/>.
	For an all-pass system, the trasfer function has the property <math>g(s)g(-s) = \sigma^2</math>, <math>\sigma > 0</math>,		For an all-pass system, the trasfer function has the property <math>g(s)g(-s) = \sigma^2</math>, <math>\sigma > 0</math>,
	or (equivalently) the controllability and observability Gramians are quasi inverse to each other: <math>W_C W_O = \sigma I</math>.		or (equivalently) the controllability and observability Gramians are quasi inverse to each other: <math>W_C W_O = \sigma I</math>,
			which means this system has a singular Hankel singular value of multiplicity of the system's order.
	The system matrices are constructing based on the scheme:		The system matrices are constructing based on the scheme:

Himpe: init

2020-04-08T11:27:51Z

init

New page

{{preliminary}} 

[[Category:benchmark]]
[[Category:procedural]]
[[Category:linear]]
[[Category:first differential order]]
[[Category:time invariant]]
[[Category:SISO]]

==Description==

This procedural benchmark generates an all-pass SISO system based on <ref name="Obe87"/>.
For an all-pass system, the trasfer function has the property <math>g(s)g(-s) = \sigma^2</math>, <math>\sigma > 0</math>,
or (equivalently) the controllability and observability Gramians are quasi inverse to each other: <math>W_C W_O = \sigma I</math>.
The system matrices are constructing based on the scheme:

:<math>
\begin{align}
A &= \begin{pmatrix} a_{1,1} & -\alpha_1 \\
\alpha_1 & 0 & -\alpha_2 \\
& \ddots & \ddots & \ddots \\
& & \alpha_{N-1} & 0 & -\alpha_{N-1} \end{pmatrix}, \\
B &= \begin{pmatrix} b_1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \\
C &= \begin{pmatrix} s_1 b_1 & 0 & \dots & 0 \end{pmatrix}, \\
D &= -s_1 \sigma.
\end{align}
</math>

We choose <math>s_1 \in \{-1,1\}</math>, to be <math>s_1 \equiv -1</math>, as this makes the system state-space-anti-symmetric.
Furthermore, <math>b_1 = 1</math> and <math>\sigma_1 = 1</math>, which makes <math>a_{1,1} = -\frac{b_1^2}{2 \sigma} = -\frac{1}{2}</math>.

==Data==

This benchmark is procedural and the state dimensions can be chosen.
Use the following [http://matlab.com MATLAB] code to generate a random system as described above:



<div class="thumbinner" style="width:20%;text-align:left;">
<source lang="matlab">

function [A,B,C,D] = allpass(N)
% allpass (all-pass system)
% by Christian Himpe, 2020
% released under BSD 2-Clause License
%*

A = gallery('tridiag',N,-1,0,1);
A(1,1) = -0.5;
B = sparse(1,1,1,N,1);
C = -B';
D = 1;
end
</source>
</div>

The function call requires one argument; the number of states <math>N</math>.
The return value consists of four matrices; the system matrix <math>A</math>, the input matrix <math>B</math>, the output matrix <math>C</math>, and the feed-through matrix <math>D</math>.

:<source lang="matlab">
[A,B,C,D] = allpass(N);
</source>

==Dimensions==

:<math>
\begin{array}{rcl}
\dot{x}(t) &=& Ax(t) + Bu(t) \\
y(t) &=& Cx(t) + Du(t)
\end{array}
</math>

System dimensions:

<math>A \in \mathbb{R}^{N \times N}</math>,
<math>B \in \mathbb{R}^{N \times 1}</math>,
<math>C \in \mathbb{R}^{1 \times N}</math>,
<math>D \in \mathbb{R}</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''All-Pass System'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/All_pass_system

@MISC{morwiki_allpass,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {All-Pass System},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/All_pass_system}</nowiki>,
year = {2020}
}

==References==

<references>

<ref name="Obe87">R.J. Ober. "<span class="plainlinks">[https://doi.org/10.1016/S1474-6670(17)55030-2 Asymptotically Stable All-Pass Transfer Functions: Canonical Form, Parametrization and Realization]</span>", IFAC Proceedings Volumes, 20(5): 181--185, 1987.</ref>

</references>