MOR Wiki - User contributions [en]

FitzHugh-Nagumo System

2018-05-28T15:27:54Z

Pontes: /* Dimensions */

[[Category:benchmark]]
[[Category:nonlinear]]
[[Category:time invariant]]
[[Category:first differential order]]

==Description==

The [[:Wikipedia:FitzHugh–Nagumo_model|FitzHugh-Nagumo]] system describes a prototype of an excitable system, e.g., a neuron.
If the external stimulus exceeds a certain threshold value, then the system will exhibit a characteristic excursion in phase space, representing activation and deactivation of the neuron. This behaviour is typical for spike generations (=short elevation of membrane voltage) in a neuron after stimulation by an external input current.

<figure id="fig:fhn">
[[File:FHN.png|frame|<caption>FitzHugh-Nagumo System</caption>]]
</figure>

===Model Equations===
Here, we present the setting from <ref name="chat10"/>, where the dynamics of the system is governed by the following coupled nonlinear PDEs:

:<math>
\epsilon v_t(x,t)=\epsilon^2v_{xx}(x,t)+f(v(x,t))-w(x,t)+g,
</math>

:<math>
w_t(x,t)=hv(x,t)-\gamma w(x,t)+g,
</math>

with <math>f(v)=v(v-0.1)(1-v)</math> and initial and boundary conditions

:<math>
v(x,0)=0,\quad w(x,0)=0, \quad x\in [0,1],
</math>

:<math>
v_x(0,t)=-i_0(t), \quad v_x(1,t)\quad =0, \quad t \geq 0,
</math>

where <math>\epsilon=0.015,\;h=0.5,\;\gamma=2,\;g=0.05;\;i_0(t)=5\cdot
10^4t^3 \exp(-15t)</math> is the external stimulus, and the variables <math>v</math> and <math>w</math> are the voltage and the recovery of the voltage, respectively. <xr id="fig:fhn"/> shows the typical limit cycle behaviour described above.

==Reformulation as a quadratic-bilinear system==

Instead of the cubic system of ODEs, one can alternatively study a so-called quadratic-bilinear control system of the form

:<math>
E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + \sum_{j=1}^2 N_j x(t) u_j(t) + B u(t),
</math>

with <math>E,A,N_j \in \mathbb R^{n\times n}, j \in\{1,2\}, \ H \in \mathbb R^{n\times n^2} </math> and <math> B \in \mathbb R^{n\times 2}</math>, and the input function is <math>u(t)=[i_0(t),1]</math>. The idea relies on artificially introducing a new state variable defined as <math>z(t)=v(t)^2</math> and subsequently computing the dynamics of the new variable, i.e., specifying <math>\dot{z}(t).</math> The technique goes back to <ref name="gu11"/>, where it is successfully applied to several smooth nonlinear control-affine systems. As discussed in <ref name="benner12"/>, introducing <math>z</math> as an addictional variable yields a quadratic-bilinear control of dimension <math> n = 3\cdot k</math>, where <math>k</math> denotes the number of discretization nodes for each PDE, with state vector <math>x = [v,w,z]^T.</math> The increase of the state dimension has the advantage of reducing the nonlinearity from cubic to quadratic. This, however, opens up the possibility to reduce the system by model reduction schemes developed for quadratic-bilinear systems such as balanced truncation <ref name="morBenG17"/>, or interpolation-based approaches, e.g., given in <ref name="benner12"/> and <ref name="gu11"/>.

==Data==

In <ref name="chat10"/>, the previous system of coupled nonlinear PDEs is spatially discretized by means of a finite difference scheme with <math>k=512 </math> nodes for each PDE. Hence, one obtains a nonlinear (cubic) system of ODEs with state dimension <math>2\cdot k=1024</math>. As discussed above, the nonlinear system can be rewritten as a quadratic-bilinear system of dimension <math>3\cdot k=1536</math>, and all matrices of the quadratic-bilinear formulation discretized are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format. The matrix name is used as an extension of the matrix file and can be found at:

[[Media:FitzHughNagumo.tar.gz|FitzNagumo.tar.gz]].

The archive contains the matrices <math>E</math>, <math>A</math>, <math>B</math>, <math>H</math> and <math>N_1</math>; The matrix <math>N_2</math> is a zero matrix of appropiate size.

For more information on the discretization details, see <ref name="chat10"/>.

In addition, one can have an output
:<math>
y(t) = Cx(t),
</math>
where the matrix <math>C</math> is a <math>2\times N</math> matrix such that <math>C(1,1) = 1,
C(2,1+k) = 1 </math> and the other elements are zero. This output <math>y(t)</math> corresponds to the left boundary of the limit cycles. For more information on the output, see <ref name="benner12"/>.

==Dimensions==
System structure:
:<math>
\begin{align}
E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + N x(t) u_(t) + B u(t)
\end{align}
</math>

System dimensions:

<math>E \in \mathbb{R}^{1536 \times 1536}</math>,
<math>A \in \mathbb{R}^{1536 \times 1536}</math>,
<math>B \in \mathbb{R}^{1536 \times 2}</math>,
<math>H \in \mathbb{R}^{1536 \times 2359296}</math>,
<math>N_1 \in \mathbb{R}^{1536 \times 1536}</math>, <math>N_2 \in \mathbb{R}^{1536 \times 1536}</math>, <math>C \in \mathbb{R}^{2 \times 1536}</math>.

==Citation==
To cite this benchmark, use the following references:

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

@MISC{morwiki_modFHN,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {FitzHugh-Nagumo System},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/FitzHugh-Nagumo_System}</nowiki>,
year = 2018
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morChaS10 morChaS10] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#ChaS10 BibTeX])

==References==

<references>

<ref name="chat10">S. Chaturantabut and D.C. Sorensen, "[https://doi.org/10.1137/090766498 Nonlinear Model Reduction via Discrete Empirical Interpolation]", SIAM J. Sci. Comput., 32: 2737--2764, 2010.</ref>

<ref name="gu11">C. Gu, "[https://doi.org/10.1109/TCAD.2011.2142184 QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems]", IEEE T. Comput. Aid. D., 30: 1307--1320, 2011.</ref>

<ref name="benner12">P. Benner and T. Breiten, "[https://doi.org/10.1137/14097255X Two-Sided Moment Matching Methods for Nonlinear Model Reduction]", SIAM J. Sci. Comput., 37(2): B239--B260, 2015.</ref>

<ref name="morBenG17">P. Benner and P. Goyal, "[https://arxiv.org/abs/1705.00160 Balanced Truncation Model Order Reduction for Quadratic-Bilinear Systems]", arXiv e-prints, 2017</ref>

</references>

==Contact==

'' [[User:Breiten]]''

FitzHugh-Nagumo System

2018-05-28T15:26:20Z

Pontes: /* Data */

[[Category:benchmark]]
[[Category:nonlinear]]
[[Category:time invariant]]
[[Category:first differential order]]

==Description==

The [[:Wikipedia:FitzHugh–Nagumo_model|FitzHugh-Nagumo]] system describes a prototype of an excitable system, e.g., a neuron.
If the external stimulus exceeds a certain threshold value, then the system will exhibit a characteristic excursion in phase space, representing activation and deactivation of the neuron. This behaviour is typical for spike generations (=short elevation of membrane voltage) in a neuron after stimulation by an external input current.

<figure id="fig:fhn">
[[File:FHN.png|frame|<caption>FitzHugh-Nagumo System</caption>]]
</figure>

===Model Equations===
Here, we present the setting from <ref name="chat10"/>, where the dynamics of the system is governed by the following coupled nonlinear PDEs:

:<math>
\epsilon v_t(x,t)=\epsilon^2v_{xx}(x,t)+f(v(x,t))-w(x,t)+g,
</math>

:<math>
w_t(x,t)=hv(x,t)-\gamma w(x,t)+g,
</math>

with <math>f(v)=v(v-0.1)(1-v)</math> and initial and boundary conditions

:<math>
v(x,0)=0,\quad w(x,0)=0, \quad x\in [0,1],
</math>

:<math>
v_x(0,t)=-i_0(t), \quad v_x(1,t)\quad =0, \quad t \geq 0,
</math>

where <math>\epsilon=0.015,\;h=0.5,\;\gamma=2,\;g=0.05;\;i_0(t)=5\cdot
10^4t^3 \exp(-15t)</math> is the external stimulus, and the variables <math>v</math> and <math>w</math> are the voltage and the recovery of the voltage, respectively. <xr id="fig:fhn"/> shows the typical limit cycle behaviour described above.

==Reformulation as a quadratic-bilinear system==

Instead of the cubic system of ODEs, one can alternatively study a so-called quadratic-bilinear control system of the form

:<math>
E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + \sum_{j=1}^2 N_j x(t) u_j(t) + B u(t),
</math>

with <math>E,A,N_j \in \mathbb R^{n\times n}, j \in\{1,2\}, \ H \in \mathbb R^{n\times n^2} </math> and <math> B \in \mathbb R^{n\times 2}</math>, and the input function is <math>u(t)=[i_0(t),1]</math>. The idea relies on artificially introducing a new state variable defined as <math>z(t)=v(t)^2</math> and subsequently computing the dynamics of the new variable, i.e., specifying <math>\dot{z}(t).</math> The technique goes back to <ref name="gu11"/>, where it is successfully applied to several smooth nonlinear control-affine systems. As discussed in <ref name="benner12"/>, introducing <math>z</math> as an addictional variable yields a quadratic-bilinear control of dimension <math> n = 3\cdot k</math>, where <math>k</math> denotes the number of discretization nodes for each PDE, with state vector <math>x = [v,w,z]^T.</math> The increase of the state dimension has the advantage of reducing the nonlinearity from cubic to quadratic. This, however, opens up the possibility to reduce the system by model reduction schemes developed for quadratic-bilinear systems such as balanced truncation <ref name="morBenG17"/>, or interpolation-based approaches, e.g., given in <ref name="benner12"/> and <ref name="gu11"/>.

==Data==

In <ref name="chat10"/>, the previous system of coupled nonlinear PDEs is spatially discretized by means of a finite difference scheme with <math>k=512 </math> nodes for each PDE. Hence, one obtains a nonlinear (cubic) system of ODEs with state dimension <math>2\cdot k=1024</math>. As discussed above, the nonlinear system can be rewritten as a quadratic-bilinear system of dimension <math>3\cdot k=1536</math>, and all matrices of the quadratic-bilinear formulation discretized are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format. The matrix name is used as an extension of the matrix file and can be found at:

[[Media:FitzHughNagumo.tar.gz|FitzNagumo.tar.gz]].

The archive contains the matrices <math>E</math>, <math>A</math>, <math>B</math>, <math>H</math> and <math>N_1</math>; The matrix <math>N_2</math> is a zero matrix of appropiate size.

For more information on the discretization details, see <ref name="chat10"/>.

In addition, one can have an output
:<math>
y(t) = Cx(t),
</math>
where the matrix <math>C</math> is a <math>2\times N</math> matrix such that <math>C(1,1) = 1,
C(2,1+k) = 1 </math> and the other elements are zero. This output <math>y(t)</math> corresponds to the left boundary of the limit cycles. For more information on the output, see <ref name="benner12"/>.

==Dimensions==
System structure:
:<math>
\begin{align}
E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + N x(t) u_(t) + B u(t)
\end{align}
</math>

System dimensions:

<math>E \in \mathbb{R}^{1536 \times 1536}</math>,
<math>A \in \mathbb{R}^{1536 \times 1536}</math>,
<math>B \in \mathbb{R}^{1536 \times 2}</math>,
<math>H \in \mathbb{R}^{1536 \times 2359296}</math>,
<math>N_? \in \mathbb{R}^{1536 \times 1536}</math>.

==Citation==
To cite this benchmark, use the following references:

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

@MISC{morwiki_modFHN,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {FitzHugh-Nagumo System},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/FitzHugh-Nagumo_System}</nowiki>,
year = 2018
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morChaS10 morChaS10] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#ChaS10 BibTeX])

==References==

<references>

<ref name="chat10">S. Chaturantabut and D.C. Sorensen, "[https://doi.org/10.1137/090766498 Nonlinear Model Reduction via Discrete Empirical Interpolation]", SIAM J. Sci. Comput., 32: 2737--2764, 2010.</ref>

<ref name="gu11">C. Gu, "[https://doi.org/10.1109/TCAD.2011.2142184 QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems]", IEEE T. Comput. Aid. D., 30: 1307--1320, 2011.</ref>

<ref name="benner12">P. Benner and T. Breiten, "[https://doi.org/10.1137/14097255X Two-Sided Moment Matching Methods for Nonlinear Model Reduction]", SIAM J. Sci. Comput., 37(2): B239--B260, 2015.</ref>

<ref name="morBenG17">P. Benner and P. Goyal, "[https://arxiv.org/abs/1705.00160 Balanced Truncation Model Order Reduction for Quadratic-Bilinear Systems]", arXiv e-prints, 2017</ref>

</references>

==Contact==

'' [[User:Breiten]]''

Flexible Aircraft

2018-03-29T09:31:39Z

Pontes:

{{preliminary}} 

[[Category:benchmark]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:frequency data]]
[[Category:data-driven]]
==Description==

<figure id="fig:flexibleAC">[[File:Flexible1.png|450px|thumb|right|<caption>View of the aerostructure of the flexible aircraft.</caption>]]</figure>

This benchmark contains a set of frequency-domain data
:<math>
\{\omega_i,\Phi_i\}_{i=1}^N
</math>
representing the output responses (accelerations and moments at different coordinates of a flexible aircraft wings and tail) in response to an input signal (gust disturbance) <math>\Phi_i</math>, evaluated at varying frequencies <math>\omega_i</math> [rad/s], for <math>i=1,\dots,N</math>.

==Motivation==

Computing responses to discrete gusts are sizing steps when designing and optimizing a new aircraft structure and geometry. Indeed, this is part of the imposed clearance certifications requested by the flight authorities. During the aircraft preliminary design phase, this clearance is done by intensive simulations, however, due to the involved model's complexity, these latter are time-consuming and imply an important computational burden. Moreover, these simulations are involved at different steps of the aircraft optimization process, e.g., by aeroelastic, flight and control engineers. In <ref name=PousstVassal2018>C. Poussot-Vassal, D. Quero, and P. Vuillemin, "Data-driven approximation of a high fidelity gust-oriented flexible aircraft dynamical model", in Proceedings of the 9th Vienna International Conference on Mathematical Modelling (MATHMOD), Vienna, Austria, 2018.</ref>, a systematic way to fasten the gust simulation step and simplify the analysis by mean of data-driven model approximation in the Loewner framework is proposed, as well as a description of this model.

Flexible aircraft models are very challenging the civil aeronautics due to their lighter structure. Models are largely used to optimize and analyze critical phenomena in the pre-design phase. Such model can, e.g., be used to monitor some dimensioning critical stress of the aircraft in response to discrete and continuous gust situations.

==Origin==

[https://www.onera.fr ONERA] - The French Aerospace Lab.

==Data==

The data is contained in the [[:File:FlexibleAircraft.zip|FlexibleAircraft.zip]] (585KB) which holds two files.

The <tt>data.mat</tt> files contains :
* W : the frequency values in rad/s (real <math>1 \times 421</math> vector).
* H : transfer function matrix evaluation at different output measurements points of the aircraft (complex <math>92 \times 1 \times 421</math> matrix).

The transfer function matrix H represents the transfer from the
* gust input
to the 92 measurements gathering from
* 1--44: the local aerodynamic lift on the aerodynamic strips.
* 45--88: the local aerodynamic pitch moment on the aerodynamic strips.
* 89--92: the four generalized coordinates derivative (heave and pitch derivatives) and the first two flexible modes.

The <tt>startData.m</tt> file loads and plots the data for illustration.



==Citation==


==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Flexible Aircraft

2018-03-29T09:28:10Z

Pontes:

{{preliminary}} 

[[Category:benchmark]]
[[Category:ODE]]
[[Category:DAE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:frequency data]]

==Description==

<figure id="fig:flexibleAC">[[File:Flexible1.png|450px|thumb|right|<caption>View of the aerostructure of the flexible aircraft.</caption>]]</figure>

This benchmark contains a set of frequency-domain data
:<math>
\{\omega_i,\Phi_i\}_{i=1}^N
</math>
representing the output responses (accelerations and moments at different coordinates of a flexible aircraft wings and tail) in response to an input signal (gust disturbance) <math>\Phi_i</math>, evaluated at varying frequencies <math>\omega_i</math> [rad/s], for <math>i=1,\dots,N</math>.

==Motivation==

Computing responses to discrete gusts are sizing steps when designing and optimizing a new aircraft structure and geometry. Indeed, this is part of the imposed clearance certifications requested by the flight authorities. During the aircraft preliminary design phase, this clearance is done by intensive simulations, however, due to the involved model's complexity, these latter are time-consuming and imply an important computational burden. Moreover, these simulations are involved at different steps of the aircraft optimization process, e.g., by aeroelastic, flight and control engineers. In <ref name=PousstVassal2018>C. Poussot-Vassal, D. Quero, and P. Vuillemin, "Data-driven approximation of a high fidelity gust-oriented flexible aircraft dynamical model", in Proceedings of the 9th Vienna International Conference on Mathematical Modelling (MATHMOD), Vienna, Austria, 2018.</ref>, a systematic way to fasten the gust simulation step and simplify the analysis by mean of data-driven model approximation in the Loewner framework is proposed, as well as a description of this model.

Flexible aircraft models are very challenging the civil aeronautics due to their lighter structure. Models are largely used to optimize and analyze critical phenomena in the pre-design phase. Such model can, e.g., be used to monitor some dimensioning critical stress of the aircraft in response to discrete and continuous gust situations.

==Origin==

[https://www.onera.fr ONERA] - The French Aerospace Lab.

==Data==

The data is contained in the [[:File:FlexibleAircraft.zip|FlexibleAircraft.zip]] (585KB) which holds two files.

The <tt>data.mat</tt> files contains :
* W : the frequency values in rad/s (real <math>1 \times 421</math> vector).
* H : transfer function matrix evaluation at different output measurements points of the aircraft (complex <math>92 \times 1 \times 421</math> matrix).

The transfer function matrix H represents the transfer from the
* gust input
to the 92 measurements gathering from
* 1--44: the local aerodynamic lift on the aerodynamic strips.
* 45--88: the local aerodynamic pitch moment on the aerodynamic strips.
* 89--92: the four generalized coordinates derivative (heave and pitch derivatives) and the first two flexible modes.

The <tt>startData.m</tt> file loads and plots the data for illustration.



==Citation==


==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Flexible Aircraft

2018-03-29T09:27:08Z

Pontes:

{{preliminary}} 

[[Category:benchmark]]
[[Category:ODE]]
[[Category:DAE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:frequency data]]

==Description==

<figure id="fig:flexibleAC">[[File:Flexible1.png|450px|thumb|right|<caption>View of the aerostructure of the flexible aircraft.</caption>]]</figure>

This benchmark contains a set of frequency-domain data
:<math>
\{\omega_i,\Phi_i\}_{i=1}^N
</math>
representing the output responses (accelerations and moments at different coordinates of a flexible aircraft wings and tail) in response to an input signal (gust disturbance) <math>\Phi_i</math>, evaluated at varying frequencies <math>\omega_i</math> [rad/s], for <math>i=1,\dots,N</math>.

==Motivation==

Computing responses to discrete gusts are sizing steps when designing and optimizing a new aircraft structure and geometry. Indeed, this is part of the imposed clearance certifications requested by the flight authorities. During the aircraft preliminary design phase, this clearance is done by intensive simulations, however, due to the involved model's complexity, these latter are time-consuming and imply an important computational burden. Moreover, these simulations are involved at different steps of the aircraft optimization process, e.g., by aeroelastic, flight and control engineers. In <ref name=PousstVassal2018>C. Poussot-Vassal, D. Quero, and P. Vuillemin, "Data-driven approximation of a high fidelity gust-oriented flexible aircraft dynamical model", in Proceedings of the 9th Vienna International Conference on Mathematical Modelling (MATHMOD), Vienna, Austria, 2018.</ref>, a systematic way to fasten the gust simulation step and simplify the analysis by mean of data-driven model approximation in the Loewner framework is proposed, as well as a description of this model.

Flexible aircraft models are very challenging the civil aeronautics due to their lighter structure. Models are largely used to optimize and analyze critical phenomena in the pre-design phase. Such model can, e.g., be used to monitor some dimensioning critical stress of the aircraft in response to discrete and continuous gust situations.

==Origin==

[https://www.onera.fr ONERA] - The French Aerospace Lab.

==Data==

The data is contained in the [[:File:FlexibleAircraft.zip|FlexibleAircraft.zip]] (585KB) which holds two files.

The <tt>data.mat</tt> files contains :
* W : the frequency values in rad/s (real <math>1 \times 421</math> vector).
* H : transfer function matrix evaluation at different output measurements points of the aircraft (complex <math>92 \times 1 \times 421</math> matrix).

The transfer function matrix H represents the transfer from the
* gust input
to the 92 measurements gathering from
* 1--44: the local aerodynamic lift on the aerodynamic strips.
* 45--88: the local aerodynamic pitch moment on the aerodynamic strips.
* 89--92: the four generalized coordinates derivative (heave and pitch derivatives) and the first two flexible modes.

The <tt>startData.m</tt> file loads and plots the data for illustration.



==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

FitzHugh-Nagumo System

2018-03-29T09:12:59Z

Pontes: /* Citation */

[[Category:benchmark]]
[[Category:nonlinear]]
[[Category:time invariant]]
[[Category:first differential order]]

==Description==

The [[:Wikipedia:FitzHugh–Nagumo_model|FitzHugh-Nagumo]] system describes a prototype of an excitable system, e.g., a neuron.
If the external stimulus exceeds a certain threshold value, then the system will exhibit a characteristic excursion in phase space, representing activation and deactivation of the neuron. This behaviour is typical for spike generations (=short elevation of membrane voltage) in a neuron after stimulation by an external input current.

<figure id="fig:fhn">
[[File:FHN.png|frame|<caption>FitzHugh-Nagumo System</caption>]]
</figure>

===Model Equations===
Here, we present the setting from <ref name="chat10">S. Chaturantabut and D.C. Sorensen, "[http://dx.doi.org/10.1137/090766498 Nonlinear Model Reduction via Discrete Empirical Interpolation]", SIAM J. Sci. Comput., 32 (2010), pp. 2737-2764.</ref>, where the dynamics of the system is governed by the following coupled nonlinear PDEs:

:<math>
\epsilon v_t(x,t)=\epsilon^2v_{xx}(x,t)+f(v(x,t))-w(x,t)+g,
</math>

:<math>
w_t(x,t)=hv(x,t)-\gamma w(x,t)+g,
</math>

with <math>f(v)=v(v-0.1)(1-v)</math> and initial and boundary conditions

:<math>
v(x,0)=0,\quad w(x,0)=0, \quad x\in [0,1],
</math>

:<math>
v_x(0,t)=-i_0(t), \quad v_x(1,t)\quad =0, \quad t \geq 0,
</math>

where <math>\epsilon=0.015,\;h=0.5,\;\gamma=2,\;g=0.05;\;i_0(t)=5\cdot
10^4t^3 \exp(-15t)</math> is the external stimulus, and the variables <math>v</math> and <math>w</math> are the voltage and the recovery of the voltage, respectively. <xr id="fig:fhn"/> shows the typical limit cycle behaviour described above.

==Reformulation as a quadratic-bilinear system==

Instead of the cubic system of ODEs, one can alternatively study a so-called quadratic-bilinear control system of the form

:<math>
E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + \sum_{j=1}^2 N_j x(t) u_j(t) + B u(t),
</math>

with <math>E,A,N_j \in \mathbb R^{n\times n}, j \in\{1,2\}, \ H \in \mathbb R^{n\times n^2} </math> and <math> B \in \mathbb R^{n\times 2}</math>, and the input function is <math>u(t)=[i_0(t),1]</math>. The idea relies on artificially introducing a new state variable defined as <math>z(t)=v(t)^2</math> and subsequently computing the dynamics of the new variable, i.e., specifying <math>\dot{z}(t).</math> The technique goes back to <ref name="gu11">C. Gu, "[http://dx.doi.org/10.1109/TCAD.2011.2142184 QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems]", IEEE T. Comput. Aid. D., 30 (2011), pp. 1307-1320.</ref>, where it is successfully applied to several smooth nonlinear control-affine systems. As discussed in <ref name="benner12">P. Benner and T. Breiten, "[http://www.mpi-magdeburg.mpg.de/preprints/2012/12/ Two-Sided Moment Matching Methods for Nonlinear Model Reduction]", 2012, Preprint MPIMD/12-12.</ref>, introducing <math>z</math> as an addictional variable yields a quadratic-bilinear control of dimension <math> n = 3\cdot k</math>, where <math>k</math> denotes the number of discretization nodes for each PDE, with state vector <math>x = [v,w,z]^T.</math> The increase of the state dimension has the advantage of reducing the nonlinearity from cubic to quadratic. This however opens up the possibility to reduce the system by the generalized moment-matching approach from <ref name="gu11"/>, see also <ref name="benner12"/> for more details on the implementation.

==Data==

In <ref name="chat10"/>, the previous system of coupled nonlinear PDEs is spatially discretized by means of a finite difference scheme with <math>k=512 </math> nodes for each PDE. Hence, one obtains a nonlinear (cubic) system of ODEs with state dimension <math>2\cdot k=1024</math>. As discussed above, the nonlinear system can be rewritten as a quadratic-bilinear system of dimension <math>3\cdot k=1536</math>, and all matrices of the quadratic-bilinear formulation discretized are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format. The matrix name is used as an extension of the matrix file and can be found at:

[[Media:FitzHughNagumo.tar.gz|FitzNagumo.tar.gz]].

For more information on the discretization details, see <ref name="chat10"/>.

==Citation==
To cite this benchmark, use the following references:

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

@MISC{morwiki_modFHN,
author = {The {MORwiki} Community},
title = {FitzHugh-Nagumo System},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = {<nowiki>http://modelreduction.org/index.php/FitzHugh-Nagumo_System</nowiki>},
year = {2018}
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morChaS10 morChaS10] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#ChaS10 BibTeX])

==References==

<references/>

==Contact==

'' [[User:Breiten]]''

FitzHugh-Nagumo System

2018-03-29T09:12:01Z

Pontes: /* Citation */

[[Category:benchmark]]
[[Category:nonlinear]]
[[Category:time invariant]]
[[Category:first differential order]]

==Description==

The [[:Wikipedia:FitzHugh–Nagumo_model|FitzHugh-Nagumo]] system describes a prototype of an excitable system, e.g., a neuron.
If the external stimulus exceeds a certain threshold value, then the system will exhibit a characteristic excursion in phase space, representing activation and deactivation of the neuron. This behaviour is typical for spike generations (=short elevation of membrane voltage) in a neuron after stimulation by an external input current.

<figure id="fig:fhn">
[[File:FHN.png|frame|<caption>FitzHugh-Nagumo System</caption>]]
</figure>

===Model Equations===
Here, we present the setting from <ref name="chat10">S. Chaturantabut and D.C. Sorensen, "[http://dx.doi.org/10.1137/090766498 Nonlinear Model Reduction via Discrete Empirical Interpolation]", SIAM J. Sci. Comput., 32 (2010), pp. 2737-2764.</ref>, where the dynamics of the system is governed by the following coupled nonlinear PDEs:

:<math>
\epsilon v_t(x,t)=\epsilon^2v_{xx}(x,t)+f(v(x,t))-w(x,t)+g,
</math>

:<math>
w_t(x,t)=hv(x,t)-\gamma w(x,t)+g,
</math>

with <math>f(v)=v(v-0.1)(1-v)</math> and initial and boundary conditions

:<math>
v(x,0)=0,\quad w(x,0)=0, \quad x\in [0,1],
</math>

:<math>
v_x(0,t)=-i_0(t), \quad v_x(1,t)\quad =0, \quad t \geq 0,
</math>

where <math>\epsilon=0.015,\;h=0.5,\;\gamma=2,\;g=0.05;\;i_0(t)=5\cdot
10^4t^3 \exp(-15t)</math> is the external stimulus, and the variables <math>v</math> and <math>w</math> are the voltage and the recovery of the voltage, respectively. <xr id="fig:fhn"/> shows the typical limit cycle behaviour described above.

==Reformulation as a quadratic-bilinear system==

Instead of the cubic system of ODEs, one can alternatively study a so-called quadratic-bilinear control system of the form

:<math>
E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + \sum_{j=1}^2 N_j x(t) u_j(t) + B u(t),
</math>

with <math>E,A,N_j \in \mathbb R^{n\times n}, j \in\{1,2\}, \ H \in \mathbb R^{n\times n^2} </math> and <math> B \in \mathbb R^{n\times 2}</math>, and the input function is <math>u(t)=[i_0(t),1]</math>. The idea relies on artificially introducing a new state variable defined as <math>z(t)=v(t)^2</math> and subsequently computing the dynamics of the new variable, i.e., specifying <math>\dot{z}(t).</math> The technique goes back to <ref name="gu11">C. Gu, "[http://dx.doi.org/10.1109/TCAD.2011.2142184 QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems]", IEEE T. Comput. Aid. D., 30 (2011), pp. 1307-1320.</ref>, where it is successfully applied to several smooth nonlinear control-affine systems. As discussed in <ref name="benner12">P. Benner and T. Breiten, "[http://www.mpi-magdeburg.mpg.de/preprints/2012/12/ Two-Sided Moment Matching Methods for Nonlinear Model Reduction]", 2012, Preprint MPIMD/12-12.</ref>, introducing <math>z</math> as an addictional variable yields a quadratic-bilinear control of dimension <math> n = 3\cdot k</math>, where <math>k</math> denotes the number of discretization nodes for each PDE, with state vector <math>x = [v,w,z]^T.</math> The increase of the state dimension has the advantage of reducing the nonlinearity from cubic to quadratic. This however opens up the possibility to reduce the system by the generalized moment-matching approach from <ref name="gu11"/>, see also <ref name="benner12"/> for more details on the implementation.

==Data==

In <ref name="chat10"/>, the previous system of coupled nonlinear PDEs is spatially discretized by means of a finite difference scheme with <math>k=512 </math> nodes for each PDE. Hence, one obtains a nonlinear (cubic) system of ODEs with state dimension <math>2\cdot k=1024</math>. As discussed above, the nonlinear system can be rewritten as a quadratic-bilinear system of dimension <math>3\cdot k=1536</math>, and all matrices of the quadratic-bilinear formulation discretized are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format. The matrix name is used as an extension of the matrix file and can be found at:

[[Media:FitzHughNagumo.tar.gz|FitzNagumo.tar.gz]].

For more information on the discretization details, see <ref name="chat10"/>.

==Citation==
To cite this benchmark, use the following references:

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

@MISC{morwiki_modFHN,
author = {The {MORwiki} Community},
title = {FitzHugh-Nagumo System},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = {<nowiki>http://modelreduction.org/index.php/FitzHugh-Nagumo_System</nowiki>},
year = {2018}
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morCheS10 morCheS10] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#CheS10 BibTeX])

==References==

<references/>

==Contact==

'' [[User:Breiten]]''

FitzHugh-Nagumo System

2018-03-29T09:11:12Z

Pontes: /* Citation */

[[Category:benchmark]]
[[Category:nonlinear]]
[[Category:time invariant]]
[[Category:first differential order]]

==Description==

The [[:Wikipedia:FitzHugh–Nagumo_model|FitzHugh-Nagumo]] system describes a prototype of an excitable system, e.g., a neuron.
If the external stimulus exceeds a certain threshold value, then the system will exhibit a characteristic excursion in phase space, representing activation and deactivation of the neuron. This behaviour is typical for spike generations (=short elevation of membrane voltage) in a neuron after stimulation by an external input current.

<figure id="fig:fhn">
[[File:FHN.png|frame|<caption>FitzHugh-Nagumo System</caption>]]
</figure>

===Model Equations===
Here, we present the setting from <ref name="chat10">S. Chaturantabut and D.C. Sorensen, "[http://dx.doi.org/10.1137/090766498 Nonlinear Model Reduction via Discrete Empirical Interpolation]", SIAM J. Sci. Comput., 32 (2010), pp. 2737-2764.</ref>, where the dynamics of the system is governed by the following coupled nonlinear PDEs:

:<math>
\epsilon v_t(x,t)=\epsilon^2v_{xx}(x,t)+f(v(x,t))-w(x,t)+g,
</math>

:<math>
w_t(x,t)=hv(x,t)-\gamma w(x,t)+g,
</math>

with <math>f(v)=v(v-0.1)(1-v)</math> and initial and boundary conditions

:<math>
v(x,0)=0,\quad w(x,0)=0, \quad x\in [0,1],
</math>

:<math>
v_x(0,t)=-i_0(t), \quad v_x(1,t)\quad =0, \quad t \geq 0,
</math>

where <math>\epsilon=0.015,\;h=0.5,\;\gamma=2,\;g=0.05;\;i_0(t)=5\cdot
10^4t^3 \exp(-15t)</math> is the external stimulus, and the variables <math>v</math> and <math>w</math> are the voltage and the recovery of the voltage, respectively. <xr id="fig:fhn"/> shows the typical limit cycle behaviour described above.

==Reformulation as a quadratic-bilinear system==

Instead of the cubic system of ODEs, one can alternatively study a so-called quadratic-bilinear control system of the form

:<math>
E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + \sum_{j=1}^2 N_j x(t) u_j(t) + B u(t),
</math>

with <math>E,A,N_j \in \mathbb R^{n\times n}, j \in\{1,2\}, \ H \in \mathbb R^{n\times n^2} </math> and <math> B \in \mathbb R^{n\times 2}</math>, and the input function is <math>u(t)=[i_0(t),1]</math>. The idea relies on artificially introducing a new state variable defined as <math>z(t)=v(t)^2</math> and subsequently computing the dynamics of the new variable, i.e., specifying <math>\dot{z}(t).</math> The technique goes back to <ref name="gu11">C. Gu, "[http://dx.doi.org/10.1109/TCAD.2011.2142184 QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems]", IEEE T. Comput. Aid. D., 30 (2011), pp. 1307-1320.</ref>, where it is successfully applied to several smooth nonlinear control-affine systems. As discussed in <ref name="benner12">P. Benner and T. Breiten, "[http://www.mpi-magdeburg.mpg.de/preprints/2012/12/ Two-Sided Moment Matching Methods for Nonlinear Model Reduction]", 2012, Preprint MPIMD/12-12.</ref>, introducing <math>z</math> as an addictional variable yields a quadratic-bilinear control of dimension <math> n = 3\cdot k</math>, where <math>k</math> denotes the number of discretization nodes for each PDE, with state vector <math>x = [v,w,z]^T.</math> The increase of the state dimension has the advantage of reducing the nonlinearity from cubic to quadratic. This however opens up the possibility to reduce the system by the generalized moment-matching approach from <ref name="gu11"/>, see also <ref name="benner12"/> for more details on the implementation.

==Data==

In <ref name="chat10"/>, the previous system of coupled nonlinear PDEs is spatially discretized by means of a finite difference scheme with <math>k=512 </math> nodes for each PDE. Hence, one obtains a nonlinear (cubic) system of ODEs with state dimension <math>2\cdot k=1024</math>. As discussed above, the nonlinear system can be rewritten as a quadratic-bilinear system of dimension <math>3\cdot k=1536</math>, and all matrices of the quadratic-bilinear formulation discretized are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format. The matrix name is used as an extension of the matrix file and can be found at:

[[Media:FitzHughNagumo.tar.gz|FitzNagumo.tar.gz]].

For more information on the discretization details, see <ref name="chat10"/>.

==Citation==
To cite this benchmark, use the following references:

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

@MISC{morwiki_modFHN,
author = {The {MORwiki} Community},
title = {FitzHugh-Nagumo System},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = {<nowiki>http://modelreduction.org/index.php/FitzHugh-Nagumo_System</nowiki>},
year = {2018}
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morChe99 morChe99] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morChe99 BibTeX])

==References==

<references/>

==Contact==

'' [[User:Breiten]]''

FitzHugh-Nagumo System

2018-03-29T09:08:18Z

Pontes: /* Citation */

[[Category:benchmark]]
[[Category:nonlinear]]
[[Category:time invariant]]
[[Category:first differential order]]

==Description==

The [[:Wikipedia:FitzHugh–Nagumo_model|FitzHugh-Nagumo]] system describes a prototype of an excitable system, e.g., a neuron.
If the external stimulus exceeds a certain threshold value, then the system will exhibit a characteristic excursion in phase space, representing activation and deactivation of the neuron. This behaviour is typical for spike generations (=short elevation of membrane voltage) in a neuron after stimulation by an external input current.

<figure id="fig:fhn">
[[File:FHN.png|frame|<caption>FitzHugh-Nagumo System</caption>]]
</figure>

===Model Equations===
Here, we present the setting from <ref name="chat10">S. Chaturantabut and D.C. Sorensen, "[http://dx.doi.org/10.1137/090766498 Nonlinear Model Reduction via Discrete Empirical Interpolation]", SIAM J. Sci. Comput., 32 (2010), pp. 2737-2764.</ref>, where the dynamics of the system is governed by the following coupled nonlinear PDEs:

:<math>
\epsilon v_t(x,t)=\epsilon^2v_{xx}(x,t)+f(v(x,t))-w(x,t)+g,
</math>

:<math>
w_t(x,t)=hv(x,t)-\gamma w(x,t)+g,
</math>

with <math>f(v)=v(v-0.1)(1-v)</math> and initial and boundary conditions

:<math>
v(x,0)=0,\quad w(x,0)=0, \quad x\in [0,1],
</math>

:<math>
v_x(0,t)=-i_0(t), \quad v_x(1,t)\quad =0, \quad t \geq 0,
</math>

where <math>\epsilon=0.015,\;h=0.5,\;\gamma=2,\;g=0.05;\;i_0(t)=5\cdot
10^4t^3 \exp(-15t)</math> is the external stimulus, and the variables <math>v</math> and <math>w</math> are the voltage and the recovery of the voltage, respectively. <xr id="fig:fhn"/> shows the typical limit cycle behaviour described above.

==Reformulation as a quadratic-bilinear system==

Instead of the cubic system of ODEs, one can alternatively study a so-called quadratic-bilinear control system of the form

:<math>
E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + \sum_{j=1}^2 N_j x(t) u_j(t) + B u(t),
</math>

with <math>E,A,N_j \in \mathbb R^{n\times n}, j \in\{1,2\}, \ H \in \mathbb R^{n\times n^2} </math> and <math> B \in \mathbb R^{n\times 2}</math>, and the input function is <math>u(t)=[i_0(t),1]</math>. The idea relies on artificially introducing a new state variable defined as <math>z(t)=v(t)^2</math> and subsequently computing the dynamics of the new variable, i.e., specifying <math>\dot{z}(t).</math> The technique goes back to <ref name="gu11">C. Gu, "[http://dx.doi.org/10.1109/TCAD.2011.2142184 QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems]", IEEE T. Comput. Aid. D., 30 (2011), pp. 1307-1320.</ref>, where it is successfully applied to several smooth nonlinear control-affine systems. As discussed in <ref name="benner12">P. Benner and T. Breiten, "[http://www.mpi-magdeburg.mpg.de/preprints/2012/12/ Two-Sided Moment Matching Methods for Nonlinear Model Reduction]", 2012, Preprint MPIMD/12-12.</ref>, introducing <math>z</math> as an addictional variable yields a quadratic-bilinear control of dimension <math> n = 3\cdot k</math>, where <math>k</math> denotes the number of discretization nodes for each PDE, with state vector <math>x = [v,w,z]^T.</math> The increase of the state dimension has the advantage of reducing the nonlinearity from cubic to quadratic. This however opens up the possibility to reduce the system by the generalized moment-matching approach from <ref name="gu11"/>, see also <ref name="benner12"/> for more details on the implementation.

==Data==

In <ref name="chat10"/>, the previous system of coupled nonlinear PDEs is spatially discretized by means of a finite difference scheme with <math>k=512 </math> nodes for each PDE. Hence, one obtains a nonlinear (cubic) system of ODEs with state dimension <math>2\cdot k=1024</math>. As discussed above, the nonlinear system can be rewritten as a quadratic-bilinear system of dimension <math>3\cdot k=1536</math>, and all matrices of the quadratic-bilinear formulation discretized are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format. The matrix name is used as an extension of the matrix file and can be found at:

[[Media:FitzHughNagumo.tar.gz|FitzNagumo.tar.gz]].

For more information on the discretization details, see <ref name="chat10"/>.

==Citation==
To cite this benchmark, use the following references:

* 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

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

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morChe99 morChe99] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morChe99 BibTeX])

==References==

<references/>

==Contact==

'' [[User:Breiten]]''

FitzHugh-Nagumo System

2018-03-29T09:07:28Z

Pontes: Citation added

[[Category:benchmark]]
[[Category:nonlinear]]
[[Category:time invariant]]
[[Category:first differential order]]

==Description==

The [[:Wikipedia:FitzHugh–Nagumo_model|FitzHugh-Nagumo]] system describes a prototype of an excitable system, e.g., a neuron.
If the external stimulus exceeds a certain threshold value, then the system will exhibit a characteristic excursion in phase space, representing activation and deactivation of the neuron. This behaviour is typical for spike generations (=short elevation of membrane voltage) in a neuron after stimulation by an external input current.

<figure id="fig:fhn">
[[File:FHN.png|frame|<caption>FitzHugh-Nagumo System</caption>]]
</figure>

===Model Equations===
Here, we present the setting from <ref name="chat10">S. Chaturantabut and D.C. Sorensen, "[http://dx.doi.org/10.1137/090766498 Nonlinear Model Reduction via Discrete Empirical Interpolation]", SIAM J. Sci. Comput., 32 (2010), pp. 2737-2764.</ref>, where the dynamics of the system is governed by the following coupled nonlinear PDEs:

:<math>
\epsilon v_t(x,t)=\epsilon^2v_{xx}(x,t)+f(v(x,t))-w(x,t)+g,
</math>

:<math>
w_t(x,t)=hv(x,t)-\gamma w(x,t)+g,
</math>

with <math>f(v)=v(v-0.1)(1-v)</math> and initial and boundary conditions

:<math>
v(x,0)=0,\quad w(x,0)=0, \quad x\in [0,1],
</math>

:<math>
v_x(0,t)=-i_0(t), \quad v_x(1,t)\quad =0, \quad t \geq 0,
</math>

where <math>\epsilon=0.015,\;h=0.5,\;\gamma=2,\;g=0.05;\;i_0(t)=5\cdot
10^4t^3 \exp(-15t)</math> is the external stimulus, and the variables <math>v</math> and <math>w</math> are the voltage and the recovery of the voltage, respectively. <xr id="fig:fhn"/> shows the typical limit cycle behaviour described above.

==Reformulation as a quadratic-bilinear system==

Instead of the cubic system of ODEs, one can alternatively study a so-called quadratic-bilinear control system of the form

:<math>
E\dot{x}(t) = A x(t) + H x(t) \otimes x(t) + \sum_{j=1}^2 N_j x(t) u_j(t) + B u(t),
</math>

with <math>E,A,N_j \in \mathbb R^{n\times n}, j \in\{1,2\}, \ H \in \mathbb R^{n\times n^2} </math> and <math> B \in \mathbb R^{n\times 2}</math>, and the input function is <math>u(t)=[i_0(t),1]</math>. The idea relies on artificially introducing a new state variable defined as <math>z(t)=v(t)^2</math> and subsequently computing the dynamics of the new variable, i.e., specifying <math>\dot{z}(t).</math> The technique goes back to <ref name="gu11">C. Gu, "[http://dx.doi.org/10.1109/TCAD.2011.2142184 QLMOR: A Projection-Based Nonlinear Model Order Reduction Approach Using Quadratic-Linear Representation of Nonlinear Systems]", IEEE T. Comput. Aid. D., 30 (2011), pp. 1307-1320.</ref>, where it is successfully applied to several smooth nonlinear control-affine systems. As discussed in <ref name="benner12">P. Benner and T. Breiten, "[http://www.mpi-magdeburg.mpg.de/preprints/2012/12/ Two-Sided Moment Matching Methods for Nonlinear Model Reduction]", 2012, Preprint MPIMD/12-12.</ref>, introducing <math>z</math> as an addictional variable yields a quadratic-bilinear control of dimension <math> n = 3\cdot k</math>, where <math>k</math> denotes the number of discretization nodes for each PDE, with state vector <math>x = [v,w,z]^T.</math> The increase of the state dimension has the advantage of reducing the nonlinearity from cubic to quadratic. This however opens up the possibility to reduce the system by the generalized moment-matching approach from <ref name="gu11"/>, see also <ref name="benner12"/> for more details on the implementation.

==Data==

In <ref name="chat10"/>, the previous system of coupled nonlinear PDEs is spatially discretized by means of a finite difference scheme with <math>k=512 </math> nodes for each PDE. Hence, one obtains a nonlinear (cubic) system of ODEs with state dimension <math>2\cdot k=1024</math>. As discussed above, the nonlinear system can be rewritten as a quadratic-bilinear system of dimension <math>3\cdot k=1536</math>, and all matrices of the quadratic-bilinear formulation discretized are in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format. The matrix name is used as an extension of the matrix file and can be found at:

[[Media:FitzHughNagumo.tar.gz|FitzNagumo.tar.gz]].

For more information on the discretization details, see <ref name="chat10"/>.

==Citation==
To cite this benchmark, use the following references:

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection '''Steel Profile'''. hosted at MORwiki - Model Order Reduction Wiki, 2005. http://modelreduction.org/index.php/Steel_Profile

@MISC{morwiki_steel,
author = {Oberwolfach Benchmark Collection},
title = {Steel Profile},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = {<nowiki>http://modelreduction.org/index.php/Steel_Profile</nowiki>},
year = 2005
}

* For the background on the benchmark:

@TechReport{BenS05b,
title = {Linear-Quadratic Regulator Design for Optimal Cooling of Steel
Profiles},
author = {P. Benner and J. Saak},
institution = {Sonderforschungsbereich 393 {\itshape Parallele Numerische
Simulation f\"ur Physik und Kontinuumsmechanik}, TU
Chem\-nitz},
year = 2005,
address = {D-09107 Chem\-nitz (Germany)},
number = {SFB393/05-05},
url = {<nowiki>http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601597</nowiki>}
}

==References==

<references/>

==Contact==

'' [[User:Breiten]]''

FitzHugh-Nagumo System

2018-03-29T08:54:20Z

Pontes: /* Reformulation as a quadratic-bilinear system */

FitzHugh-Nagumo System

2018-03-29T08:50:18Z

Pontes: /* Reformulation as a quadratic-bilinear system */

FitzHugh-Nagumo System

2018-03-29T08:48:46Z

Pontes: /* Reformulation as a quadratic-bilinear system */

FitzHugh-Nagumo System

2018-03-29T08:44:30Z

Pontes: /* Correction of input*/

FitzHugh-Nagumo System

2018-03-29T08:41:10Z

Pontes: /* Description */

FitzHugh-Nagumo System

2018-03-29T08:40:53Z

Pontes: /* Data update*/

FitzHugh-Nagumo System

2018-03-29T08:39:32Z

Pontes: /* Moved details to data information */

FitzHugh-Nagumo System

2018-03-29T08:38:17Z

Pontes: /* Description equation details*/

FitzHugh-Nagumo System

2018-03-29T08:34:45Z

Pontes: /* Description */

FitzHugh-Nagumo System

2018-03-29T08:34:19Z

Pontes: /* Description update */

FitzHugh-Nagumo System

2018-03-29T08:26:55Z

Pontes: Added "Model Equation"

FitzHugh-Nagumo System

2018-03-29T08:21:57Z

Pontes: /* Reference */

FitzHugh-Nagumo System

2018-03-29T08:17:49Z

Pontes: /*Reference correction */

FitzHugh-Nagumo System

2018-03-17T19:33:09Z

Pontes: /* Reformulation as a quadratic-bilinear system */

Flexible Aircraft

2018-03-17T19:26:17Z

Pontes: /* Motivation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:ODE]]
[[Category:DAE]]
[[Category:linear]]
[[Category:time invariant]]

==Description==

<figure id="fig:flexibleAC">[[File:Flexible1.png|450px|thumb|right|<caption>View of the aerostructure of the flexible aircraft.</caption>]]</figure>

This benchmark contains a set of frequency-domain data
:<math>
\{\omega_i,\Phi_i\}_{i=1}^N
</math>
representing the output responses (accelerations and moments at different coordinates of a flexible aircraft wings and tail) in response to an input signal (gust disturbance) <math>\Phi_i</math>, evaluated at varying frequencies <math>\omega_i</math> [rad/s], for <math>i=1,\dots,N</math>.

==Motivation==

Computing responses to discrete gusts are sizing steps when designing and optimizing a new aircraft structure and geometry. Indeed, this is part of the imposed clearance certifications requested by the flight authorities. During the aircraft preliminary design phase, this clearance is done by intensive simulations, however, due to the involved model's complexity, these latter are time-consuming and imply an important computational burden. Moreover, these simulations are involved at different steps of the aircraft optimization process, e.g., by aeroelastic, flight and control engineers. In <ref name=PousstVassal2018>C. Poussot-Vassal, D. Quero, and P. Vuillemin, "Data-driven approximation of a high fidelity gust-oriented flexible aircraft dynamical model", in Proceedings of the 9th Vienna International Conference on Mathematical Modelling (MATHMOD), Vienna, Austria, 2018.</ref>, a systematic way to fasten the gust simulation step and simplify the analysis by mean of data-driven model approximation in the Loewner framework is proposed, as well as a description of this model.

Flexible aircraft models are very challenging the civil aeronautics due to their lighter structure. Models are largely used to optimize and analyze critical phenomena in the pre-design phase. Such model can, e.g., be used to monitor some dimensioning critical stress of the aircraft in response to discrete and continuous gust situations.

==Origin==

[https://www.onera.fr ONERA] - The French Aerospace Lab.

==Data==

The data is contained in the [[:File:FlexibleAircraft.zip|FlexibleAircraft.zip]] (585KB) which holds two files.

The <tt>data.mat</tt> files contains :
* W : the frequency values in rad/s (real <math>1 \times 421</math> vector).
* H : transfer function matrix evaluation at different output measurements points of the aircraft (complex <math>92 \times 1 \times 421</math> matrix).

The transfer function matrix H represents the transfer from the
* gust input
to the 92 measurements gathering from
* 1--44: the local aerodynamic lift on the aerodynamic strips.
* 45--88: the local aerodynamic pitch moment on the aerodynamic strips.
* 89--92: the four generalized coordinates derivative (heave and pitch derivatives) and the first two flexible modes.

The <tt>startData.m</tt> file loads and plots the data for illustration.



==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''