MOR Wiki - User contributions [en]

File:Fluid Flow Linearized Open Cavity Model.zip

2022-10-12T10:04:55Z

Poussotvassal: Poussotvassal uploaded a new version of File:Fluid Flow Linearized Open Cavity Model.zip

== Summary ==
Parametric data-driven fluid use-case Matlab files

Fluid Flow Linearized Open Cavity Model

2022-10-12T09:49:48Z

Poussotvassal: /* Description */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way. The approach involves the Loewner framework as in <ref name=GoseaPoussotAntoulas>I.V. Gosea, C. Poussot-Vassal and A.C. Antoulas, "[https://doi.org/10.1016/bs.hna.2021.12.015 Data-driven modeling and control of large-scale dynamical systems in the Loewner framework: Methodology and applications]", in Handbook of Numerical Analysis, 23, 2022, pp. 499-530.</ref>. We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R_+</math> and <math>k=1,\dots,61</math> is computed.
Second, ten intermediate parametric configurations between each <math>l</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math> configurations. Note that this second data interpolation is purely numerical and probably out of real physics.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:Fluid_Flow_Linearized_Open_Cavity_Model.zip|FluidFlowCavityDataParametric.zip]] (36ko) repository contains three files:

* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
** P : the parametric values (real <math>1 \times 21</math> vector).
** S : the complex values where the model is evaluated (real <math>1 \times 123</math> vector). Note that the first element of S is real to ensure a real realisation while the 122 others are complex conjugated.
** H : transfer function matrix evaluation at different S,P couples (complex <math>1 \times 1 \times 123\times 21</math> matrix).

* The <tt>dataONERA_FluidFlowOpenCavity_withMOR.mat</tt> data file, with a parametric ROM obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method.
** Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric descriptor state-space model of the form "Hr_p = @(p) dss(...)", of dimension 40 in s and 10 in p, detailed in the code.

* The <tt>startONERA_FluidFlowOpenCavity.m</tt> script file, used to loads and plots the data for illustration and a solution.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property seems very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For illustration of the Loewner-driven parametric model construction involvin this use-case, see:

@article{GoseaHNA:2022,
author = {V. Gosea and C. Poussot-Vassal and A.C. Antoulas},
title = {Data-driven modeling and control of large-scale dynamical systems in the Loewner framework},
journal = {Handbook of Numerical Analysis},
year = {2022},
volume = {23},
number = {Numerical Control: Part A},
pages = {499-530},
}

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

File:Fluid Flow Linearized Open Cavity Model.zip

2022-10-12T09:48:41Z

Poussotvassal: Parametric data-driven fluid use-case Matlab files

== Summary ==
Parametric data-driven fluid use-case Matlab files

Fluid Flow Linearized Open Cavity Model

2022-10-12T09:41:35Z

Poussotvassal:

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way. The approach involves the Loewner framework as in <ref name=GoseaPoussotAntoulas>I.V. Gosea, C. Poussot-Vassal and A.C. Antoulas, "[https://doi.org/10.1016/bs.hna.2021.12.015 Data-driven modeling and control of large-scale dynamical systems in the Loewner framework: Methodology and applications]", in Handbook of Numerical Analysis, 23, 2022, pp. 499-530.</ref>. We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R_+</math> and <math>k=1,\dots,61</math> is computed.
Second, ten intermediate parametric configurations between each <math>l</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math> configurations. Note that this second data interpolation is purely numerical and probably out of real physics.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FluidFlowCavityDataParametric.zip|FluidFlowCavityDataParametric.zip]] (36ko) repository contains three files:

* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
** P : the parametric values (real <math>1 \times 21</math> vector).
** S : the complex values where the model is evaluated (real <math>1 \times 123</math> vector). Note that the first element of S is real to ensure a real realisation while the 122 others are complex conjugated.
** H : transfer function matrix evaluation at different S,P couples (complex <math>1 \times 1 \times 123\times 21</math> matrix).

* The <tt>dataONERA_FluidFlowOpenCavity_withMOR.mat</tt> data file, with a parametric ROM obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method.
** Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric descriptor state-space model of the form "Hr_p = @(p) dss(...)", of dimension 40 in s and 10 in p, detailed in the code.

* The <tt>startONERA_FluidFlowOpenCavity.m</tt> script file, used to loads and plots the data for illustration and a solution.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property seems very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For illustration of the Loewner-driven parametric model construction involvin this use-case, see:

@article{GoseaHNA:2022,
author = {V. Gosea and C. Poussot-Vassal and A.C. Antoulas},
title = {Data-driven modeling and control of large-scale dynamical systems in the Loewner framework},
journal = {Handbook of Numerical Analysis},
year = {2022},
volume = {23},
number = {Numerical Control: Part A},
pages = {499-530},
}

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2022-10-12T09:38:25Z

Poussotvassal: /* Motivation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way. The approach involves the Loewner framework as in In <ref name=GoseaPoussotAntoulas></ref>. We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R_+</math> and <math>k=1,\dots,61</math> is computed.
Second, ten intermediate parametric configurations between each <math>l</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math> configurations. Note that this second data interpolation is purely numerical and probably out of real physics.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FluidFlowCavityDataParametric.zip|FluidFlowCavityDataParametric.zip]] (36ko) repository contains three files:

* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
** P : the parametric values (real <math>1 \times 21</math> vector).
** S : the complex values where the model is evaluated (real <math>1 \times 123</math> vector). Note that the first element of S is real to ensure a real realisation while the 122 others are complex conjugated.
** H : transfer function matrix evaluation at different S,P couples (complex <math>1 \times 1 \times 123\times 21</math> matrix).

* The <tt>dataONERA_FluidFlowOpenCavity_withMOR.mat</tt> data file, with a parametric ROM obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method.
** Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric descriptor state-space model of the form "Hr_p = @(p) dss(...)", of dimension 40 in s and 10 in p, detailed in the code.

* The <tt>startONERA_FluidFlowOpenCavity.m</tt> script file, used to loads and plots the data for illustration and a solution.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property seems very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For illustration of the Loewner-driven parametric model construction involvin this use-case, see:

@article{GoseaHNA:2022,
author = {V. Gosea and C. Poussot-Vassal and A.C. Antoulas},
title = {Data-driven modeling and control of large-scale dynamical systems in the Loewner framework},
journal = {Handbook of Numerical Analysis},
year = {2022},
volume = {23},
number = {Numerical Control: Part A},
pages = {499-530},
}

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2022-10-12T09:37:20Z

Poussotvassal: /* Citation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R_+</math> and <math>k=1,\dots,61</math> is computed.
Second, ten intermediate parametric configurations between each <math>l</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math> configurations. Note that this second data interpolation is purely numerical and probably out of real physics.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FluidFlowCavityDataParametric.zip|FluidFlowCavityDataParametric.zip]] (36ko) repository contains three files:

* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
** P : the parametric values (real <math>1 \times 21</math> vector).
** S : the complex values where the model is evaluated (real <math>1 \times 123</math> vector). Note that the first element of S is real to ensure a real realisation while the 122 others are complex conjugated.
** H : transfer function matrix evaluation at different S,P couples (complex <math>1 \times 1 \times 123\times 21</math> matrix).

* The <tt>dataONERA_FluidFlowOpenCavity_withMOR.mat</tt> data file, with a parametric ROM obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method.
** Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric descriptor state-space model of the form "Hr_p = @(p) dss(...)", of dimension 40 in s and 10 in p, detailed in the code.

* The <tt>startONERA_FluidFlowOpenCavity.m</tt> script file, used to loads and plots the data for illustration and a solution.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property seems very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For illustration of the Loewner-driven parametric model construction involvin this use-case, see:

@article{GoseaHNA:2022,
author = {V. Gosea and C. Poussot-Vassal and A.C. Antoulas},
title = {Data-driven modeling and control of large-scale dynamical systems in the Loewner framework},
journal = {Handbook of Numerical Analysis},
year = {2022},
volume = {23},
number = {Numerical Control: Part A},
pages = {499-530},
}

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2022-10-12T09:14:08Z

Poussotvassal: /* Citation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R_+</math> and <math>k=1,\dots,61</math> is computed.
Second, ten intermediate parametric configurations between each <math>l</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math> configurations. Note that this second data interpolation is purely numerical and probably out of real physics.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FluidFlowCavityDataParametric.zip|FluidFlowCavityDataParametric.zip]] (36ko) repository contains three files:

* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
** P : the parametric values (real <math>1 \times 21</math> vector).
** S : the complex values where the model is evaluated (real <math>1 \times 123</math> vector). Note that the first element of S is real to ensure a real realisation while the 122 others are complex conjugated.
** H : transfer function matrix evaluation at different S,P couples (complex <math>1 \times 1 \times 123\times 21</math> matrix).

* The <tt>dataONERA_FluidFlowOpenCavity_withMOR.mat</tt> data file, with a parametric ROM obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method.
** Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric descriptor state-space model of the form "Hr_p = @(p) dss(...)", of dimension 40 in s and 10 in p, detailed in the code.

* The <tt>startONERA_FluidFlowOpenCavity.m</tt> script file, used to loads and plots the data for illustration and a solution.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property seems very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For illustration of the Loewner-driven parametric model construction involvin this use-case, see:

@article{GoseaHNA:2022,
author = {V. Gosea and C. Poussot-Vassal and A.C. Antoulas},
title = {Data-driven modeling and control of large-scale dynamical systems in the Loewner framework},
journal = {Handbook of Numerical Analysis},
year = {2022},
month = {January},
volume = {23},
number = {Numerical Control: Part A},
pages = {499-530},
}

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2022-10-12T09:12:57Z

Poussotvassal: /* Citation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R_+</math> and <math>k=1,\dots,61</math> is computed.
Second, ten intermediate parametric configurations between each <math>l</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math> configurations. Note that this second data interpolation is purely numerical and probably out of real physics.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FluidFlowCavityDataParametric.zip|FluidFlowCavityDataParametric.zip]] (36ko) repository contains three files:

* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
** P : the parametric values (real <math>1 \times 21</math> vector).
** S : the complex values where the model is evaluated (real <math>1 \times 123</math> vector). Note that the first element of S is real to ensure a real realisation while the 122 others are complex conjugated.
** H : transfer function matrix evaluation at different S,P couples (complex <math>1 \times 1 \times 123\times 21</math> matrix).

* The <tt>dataONERA_FluidFlowOpenCavity_withMOR.mat</tt> data file, with a parametric ROM obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method.
** Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric descriptor state-space model of the form "Hr_p = @(p) dss(...)", of dimension 40 in s and 10 in p, detailed in the code.

* The <tt>startONERA_FluidFlowOpenCavity.m</tt> script file, used to loads and plots the data for illustration and a solution.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property seems very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For illustration of the parametric Loewner in this use-case:

@article{GoseaHNA:2022,
author = {V. Gosea and C. Poussot-Vassal and A.C. Antoulas},
title = {Data-driven modeling and control of large-scale dynamical systems in the Loewner framework},
journal = {Handbook of Numerical Analysis},
year = {2022},
month = {January},
volume = {23},
number = {Numerical Control: Part A},
pages = {499-530},
}

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2022-10-12T09:12:46Z

Poussotvassal: /* Citation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R_+</math> and <math>k=1,\dots,61</math> is computed.
Second, ten intermediate parametric configurations between each <math>l</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math> configurations. Note that this second data interpolation is purely numerical and probably out of real physics.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FluidFlowCavityDataParametric.zip|FluidFlowCavityDataParametric.zip]] (36ko) repository contains three files:

* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
** P : the parametric values (real <math>1 \times 21</math> vector).
** S : the complex values where the model is evaluated (real <math>1 \times 123</math> vector). Note that the first element of S is real to ensure a real realisation while the 122 others are complex conjugated.
** H : transfer function matrix evaluation at different S,P couples (complex <math>1 \times 1 \times 123\times 21</math> matrix).

* The <tt>dataONERA_FluidFlowOpenCavity_withMOR.mat</tt> data file, with a parametric ROM obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method.
** Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric descriptor state-space model of the form "Hr_p = @(p) dss(...)", of dimension 40 in s and 10 in p, detailed in the code.

* The <tt>startONERA_FluidFlowOpenCavity.m</tt> script file, used to loads and plots the data for illustration and a solution.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property seems very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For illustration of the parametric Loewner in this use-case:

@article{GoseaHNA:2022,
author = {V. Gosea and C. Poussot-Vassal and A.C. Antoulas},
title = {{Data-driven modeling and control of large-scale dynamical systems in the Loewner framework}},
journal = {Handbook of Numerical Analysis},
year = {2022},
month = {January},
volume = {23},
number = {Numerical Control: Part A},
pages = {499-530},
}

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2022-10-12T09:08:17Z

Poussotvassal: /* Objective */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R_+</math> and <math>k=1,\dots,61</math> is computed.
Second, ten intermediate parametric configurations between each <math>l</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math> configurations. Note that this second data interpolation is purely numerical and probably out of real physics.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FluidFlowCavityDataParametric.zip|FluidFlowCavityDataParametric.zip]] (36ko) repository contains three files:

* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
** P : the parametric values (real <math>1 \times 21</math> vector).
** S : the complex values where the model is evaluated (real <math>1 \times 123</math> vector). Note that the first element of S is real to ensure a real realisation while the 122 others are complex conjugated.
** H : transfer function matrix evaluation at different S,P couples (complex <math>1 \times 1 \times 123\times 21</math> matrix).

* The <tt>dataONERA_FluidFlowOpenCavity_withMOR.mat</tt> data file, with a parametric ROM obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method.
** Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric descriptor state-space model of the form "Hr_p = @(p) dss(...)", of dimension 40 in s and 10 in p, detailed in the code.

* The <tt>startONERA_FluidFlowOpenCavity.m</tt> script file, used to loads and plots the data for illustration and a solution.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property seems very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2022-10-12T09:07:45Z

Poussotvassal: /* Description */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R_+</math> and <math>k=1,\dots,61</math> is computed.
Second, ten intermediate parametric configurations between each <math>l</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math> configurations. Note that this second data interpolation is purely numerical and probably out of real physics.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FluidFlowCavityDataParametric.zip|FluidFlowCavityDataParametric.zip]] (36ko) repository contains three files:

* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
** P : the parametric values (real <math>1 \times 21</math> vector).
** S : the complex values where the model is evaluated (real <math>1 \times 123</math> vector). Note that the first element of S is real to ensure a real realisation while the 122 others are complex conjugated.
** H : transfer function matrix evaluation at different S,P couples (complex <math>1 \times 1 \times 123\times 21</math> matrix).

* The <tt>dataONERA_FluidFlowOpenCavity_withMOR.mat</tt> data file, with a parametric ROM obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method.
** Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric descriptor state-space model of the form "Hr_p = @(p) dss(...)", of dimension 40 in s and 10 in p, detailed in the code.

* The <tt>startONERA_FluidFlowOpenCavity.m</tt> script file, used to loads and plots the data for illustration and a solution.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property is very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2022-10-12T09:02:46Z

Poussotvassal: /* Considered data */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R_+</math> and <math>k=1,\dots,61</math> is computed.
Second, ten intermediate parametric configurations between each <math>l</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math> configurations. Note that this second data interpolation is purely numerical and probably out of real physics.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FlexibleAircraft.zip|FlexibleAircraft.zip]] (aaMB) repository contains three files:

* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
** P : the parametric values (real <math>1 \times 21</math> vector).
** S : the complex values where the model is evaluated (real <math>1 \times 123</math> vector). Note that the first element of S is real to ensure a real realisation.
** H : transfer function matrix evaluation at different S,P couples (complex <math>1 \times 1 \times 123\times 21</math> matrix).

* The <tt>dataONERA_FluidFlowOpenCavity_withMOR.mat</tt> data file, with a parametric ROM obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method.
** Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric state-space model of the form "Hr_p = @(p) dss(...)"

* The <tt>startONERA_FluidFlowOpenCavity.m</tt> script file, used to loads and plots the data for illustration and a solution.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property is very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2022-10-12T09:01:48Z

Poussotvassal: /* Motivation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R_+</math> and <math>k=1,\dots,61</math> is computed.
Second, ten intermediate parametric configurations between each <math>l</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math> configurations. Note that this second data interpolation is purely numerical and probably out of real physics.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math>.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FlexibleAircraft.zip|FlexibleAircraft.zip]] (aaMB) repository contains three files:

* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
** P : the parametric values (real <math>1 \times 21</math> vector).
** S : the complex values where the model is evaluated (real <math>1 \times 123</math> vector). Note that the first element of S is real to ensure a real realisation.
** H : transfer function matrix evaluation at different S,P couples (complex <math>1 \times 1 \times 123\times 21</math> matrix).

* The <tt>dataONERA_FluidFlowOpenCavity_withMOR.mat</tt> data file, with a parametric ROM obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method.
** Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric state-space model of the form "Hr_p = @(p) dss(...)"

* The <tt>startONERA_FluidFlowOpenCavity.m</tt> script file, used to loads and plots the data for illustration and a solution.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property is very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2022-10-12T09:00:45Z

Poussotvassal: /* Motivation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R_+</math> and <math>k=1,\dots,61</math> is computed.
Second, ten intermediate parametric configurations between each <math>l</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math> configurations.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math>.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FlexibleAircraft.zip|FlexibleAircraft.zip]] (aaMB) repository contains three files:

* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
** P : the parametric values (real <math>1 \times 21</math> vector).
** S : the complex values where the model is evaluated (real <math>1 \times 123</math> vector). Note that the first element of S is real to ensure a real realisation.
** H : transfer function matrix evaluation at different S,P couples (complex <math>1 \times 1 \times 123\times 21</math> matrix).

* The <tt>dataONERA_FluidFlowOpenCavity_withMOR.mat</tt> data file, with a parametric ROM obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method.
** Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric state-space model of the form "Hr_p = @(p) dss(...)"

* The <tt>startONERA_FluidFlowOpenCavity.m</tt> script file, used to loads and plots the data for illustration and a solution.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property is very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

User:Poussotvassal

2022-10-12T08:56:56Z

Poussotvassal:

Charles Poussot-Vassal 
Co-founder of [http://mordigitalsystems.fr/ MOR Digital Systems] 
Research Director 2 at ONERA - The French Aerospace Lab 
Department of Information Processing and Systems 
2, Avenue Edouard Belin 
France 

e-mail: charles.poussot-vassal@onera.fr 
http://sites.google.com/site/charlespoussotvassal/ 
http://mordigitalsystems.fr/

User:Poussotvassal

2022-10-12T08:55:41Z

Poussotvassal:

Charles Poussot-Vassal 
Co-founder of <a href="http://mordigitalsystems.fr/">MOR Digital Systems</a> 
Research Director 2 at ONERA - The French Aerospace Lab 
Department of Information Processing and Systems 
2, Avenue Edouard Belin 
France 

e-mail: charles.poussot-vassal@onera.fr 
http://sites.google.com/site/charlespoussotvassal/ 
http://mordigitalsystems.fr/

User:Poussotvassal

2022-10-12T08:54:29Z

Poussotvassal:

Charles Poussot-Vassal 
Co-founder of MOR Digital Systems 
Research Director 2 at ONERA - The French Aerospace Lab 
Department of Information Processing and Systems 
2, Avenue Edouard Belin 
France 

e-mail: charles.poussot-vassal@onera.fr 
http://sites.google.com/site/charlespoussotvassal/ 
http://mordigitalsystems.fr/

User:Poussotvassal

2022-10-12T08:54:16Z

Poussotvassal:

Charles Poussot-Vassal 
Co-founder of MOR Digital Systems 
Research Director 2 at ONERA - The French Aerospace Lab 
Department of Information Processing and Systems 
2, Avenue Edouard Belin 
France 

email: charles.poussot-vassal@onera.fr 
http://sites.google.com/site/charlespoussotvassal/ 
http://mordigitalsystems.fr/

User:Poussotvassal

2022-10-12T08:52:39Z

Poussotvassal:

Charles Poussot-Vassal 
Co-founder of MOR Digital Systems 
Research Director 2 at ONERA - The French Aerospace Lab 
Department of Information Processing and Systems 
2, Avenue Edouard Belin 
France 

email: charles.poussot-vassal@onera.fr 
www Personal: http://sites.google.com/site/charlespoussotvassal/ 
www MOR Digital Systems: http://mordigitalsystems.fr/

Fluid Flow Linearized Open Cavity Model

2022-10-11T15:35:06Z

Poussotvassal: /* Motivation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R</math> and <math>k=1,\dots,61</math> is computed.
Second, ten intermediate parametric configurations between each <math>l</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math> configurations.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math>.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FlexibleAircraft.zip|FlexibleAircraft.zip]] (aaMB) repository contains three files:

* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
** P : the parametric values (real <math>1 \times 21</math> vector).
** S : the complex values where the model is evaluated (real <math>1 \times 123</math> vector). Note that the first element of S is real to ensure a real realisation.
** H : transfer function matrix evaluation at different S,P couples (complex <math>1 \times 1 \times 123\times 21</math> matrix).

* The <tt>dataONERA_FluidFlowOpenCavity_withMOR.mat</tt> data file, with a parametric ROM obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method.
** Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric state-space model of the form "Hr_p = @(p) dss(...)"

* The <tt>startONERA_FluidFlowOpenCavity.m</tt> script file, used to loads and plots the data for illustration and a solution.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property is very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2022-10-11T15:15:10Z

Poussotvassal: /* Description */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R</math> and <math>k=1,\dots,61</math> is computed.
Second, ten intermediate parametric configurations between each <math>j</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>, for <math>l=1,\dots,M=21</math> configurations.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math>.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FlexibleAircraft.zip|FlexibleAircraft.zip]] (aaMB) repository contains three files:

* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
** P : the parametric values (real <math>1 \times 21</math> vector).
** S : the complex values where the model is evaluated (real <math>1 \times 123</math> vector). Note that the first element of S is real to ensure a real realisation.
** H : transfer function matrix evaluation at different S,P couples (complex <math>1 \times 1 \times 123\times 21</math> matrix).

* The <tt>dataONERA_FluidFlowOpenCavity_withMOR.mat</tt> data file, with a parametric ROM obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method.
** Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric state-space model of the form "Hr_p = @(p) dss(...)"

* The <tt>startONERA_FluidFlowOpenCavity.m</tt> script file, used to loads and plots the data for illustration and a solution.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property is very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2022-10-11T15:14:27Z

Poussotvassal:

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven way (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{-\imath\omega_k,\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R</math> and <math>k=1,\dots,61</math> is computed.
Second, ten intermediate parametric configurations between each <math>j</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>, for <math>l=1,\dots,M=21</math> configurations.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{-\imath\omega_k,\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=21</math>.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FlexibleAircraft.zip|FlexibleAircraft.zip]] (aaMB) repository contains three files:

* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
** P : the parametric values (real <math>1 \times 21</math> vector).
** S : the complex values where the model is evaluated (real <math>1 \times 123</math> vector). Note that the first element of S is real to ensure a real realisation.
** H : transfer function matrix evaluation at different S,P couples (complex <math>1 \times 1 \times 123\times 21</math> matrix).

* The <tt>dataONERA_FluidFlowOpenCavity_withMOR.mat</tt> data file, with a parametric ROM obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method.
** Hr : linear parametric rational ROM (state-space matrices in Matlab form). This set of matrices leads to the parametric state-space model of the form "Hr_p = @(p) dss(...)"

* The <tt>startONERA_OpenCavityt.m</tt> script file, used to loads and plots the data for illustration.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property is very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2022-10-11T14:54:53Z

Poussotvassal:

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on Fig. 1 and Fig. 2.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_l(s) = C(sE-A_l)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R</math> and <math>k=1,\dots,100</math> is computed.
Second, ten intermediate parametric configurations between each <math>j</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>, for <math>l=1,\dots,M=31</math> configurations.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=31</math>. Specifically, the frequencies are between xxHz until yyHz in steps of zzHz, and parameters are from 1 to 3 in steps of zz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FlexibleAircraft.zip|FlexibleAircraft.zip]] (aaMB) repository contains three files:

* The <tt>dataONERA_OpenCavity.mat</tt> data file, with
** p : the parametric values (real <math>1 \times 31</math> vector).
** s0 : the evaluation of the parametric transfer function at a frozen real (real <math>1 \times 31</math> vector).
** w : the frequency values in rad/s (real <math>1 \times 100</math> vector).
** H : transfer function matrix evaluation at different frequeny points and parametric values (complex <math>1 \times 1 \times 101\times 31</math> matrix).

* The <tt>dataONERA_OpenCavity_withMOR.mat</tt> data file, with 3 ROMs obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method
** Hr1 : linear parametric rational ROMs with varying dimensions (state-space models in Matlab form).

* The <tt>startONERA_OpenCavityt.m</tt> script file, used to loads and plots the data for illustration.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property is very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-09T08:39:50Z

Poussotvassal: /* Description */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_l\}_{l=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_l(t) = A_lx_l(t) + Bu(t) \text{ , } y(t)=Cx_l(t) \text{ where } E,A_l \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE-A_j)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_l(\xi_k)</math> of each <math>l=1,\dots,3</math> configurations for frozen complex values <math>\xi_k = \xi_0 \cup \{\imath\omega_k\}_{i=k}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R</math> and <math>k=1,\dots,100</math> is computed.
Second, ten intermediate parametric configurations between each <math>j</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_l(\xi_k)</math>, leading to <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>, for <math>l=1,\dots,M=31</math> configurations.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_k,\rho_l,H_{kl}\}_{k=1,l=1}^{N+1,M},
</math>

where <math>H_{kl}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_k = 0.1 \cup \{\imath\omega_k\}_{k=1}^N</math> and parameters <math>\{\rho_l\}_{l=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=31</math>. Specifically, the frequencies are between xxHz until yyHz in steps of zzHz, and parameters are from 1 to 3 in steps of zz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FlexibleAircraft.zip|FlexibleAircraft.zip]] (aaMB) repository contains three files:

* The <tt>dataONERA_OpenCavity.mat</tt> data file, with
** p : the parametric values (real <math>1 \times 31</math> vector).
** s0 : the evaluation of the parametric transfer function at a frozen real (real <math>1 \times 31</math> vector).
** w : the frequency values in rad/s (real <math>1 \times 100</math> vector).
** H : transfer function matrix evaluation at different frequeny points and parametric values (complex <math>1 \times 1 \times 101\times 31</math> matrix).

* The <tt>dataONERA_OpenCavity_withMOR.mat</tt> data file, with 3 ROMs obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method
** Hr1 : linear parametric rational ROMs with varying dimensions (state-space models in Matlab form).

* The <tt>startONERA_OpenCavityt.m</tt> script file, used to loads and plots the data for illustration.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property is very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T15:59:23Z

Poussotvassal: /* Objective */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE-A_j)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_j(\xi_i)</math> of each <math>j=1,\dots,3</math> configurations for frozen complex values <math>\xi_i = \xi_0 \cup \{\imath\omega_i\}_{i=1}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R</math> and <math>i=1,\dots,100</math> is computed.
Second, ten intermediate parametric configurations between each <math>j</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_j(\xi_i)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=31</math> configurations.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_i,\rho_j,H_{ij}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i = 0.1 \cup \{\imath\omega_i\}_{i=1}^N</math> and parameters <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=31</math>. Specifically, the frequencies are between xxHz until yyHz in steps of zzHz, and parameters are from 1 to 3 in steps of zz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FlexibleAircraft.zip|FlexibleAircraft.zip]] (aaMB) repository contains three files:

* The <tt>dataONERA_OpenCavity.mat</tt> data file, with
** p : the parametric values (real <math>1 \times 31</math> vector).
** s0 : the evaluation of the parametric transfer function at a frozen real (real <math>1 \times 31</math> vector).
** w : the frequency values in rad/s (real <math>1 \times 100</math> vector).
** H : transfer function matrix evaluation at different frequeny points and parametric values (complex <math>1 \times 1 \times 101\times 31</math> matrix).

* The <tt>dataONERA_OpenCavity_withMOR.mat</tt> data file, with 3 ROMs obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method
** Hr1 : linear parametric rational ROMs with varying dimensions (state-space models in Matlab form).

* The <tt>startONERA_OpenCavityt.m</tt> script file, used to loads and plots the data for illustration.

===Objective===
Find a linear (stable?) parametric reduced order model that well approximates the data. Note that stability property is very promising.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T15:58:21Z

Poussotvassal: /* Description */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE-A_j)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_j(\xi_i)</math> of each <math>j=1,\dots,3</math> configurations for frozen complex values <math>\xi_i = \xi_0 \cup \{\imath\omega_i\}_{i=1}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R</math> and <math>i=1,\dots,100</math> is computed.
Second, ten intermediate parametric configurations between each <math>j</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_j(\xi_i)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=31</math> configurations.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_i,\rho_j,H_{ij}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i = 0.1 \cup \{\imath\omega_i\}_{i=1}^N</math> and parameters <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=31</math>. Specifically, the frequencies are between xxHz until yyHz in steps of zzHz, and parameters are from 1 to 3 in steps of zz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

The [[:File:FlexibleAircraft.zip|FlexibleAircraft.zip]] (aaMB) repository contains three files:

* The <tt>dataONERA_OpenCavity.mat</tt> data file, with
** p : the parametric values (real <math>1 \times 31</math> vector).
** s0 : the evaluation of the parametric transfer function at a frozen real (real <math>1 \times 31</math> vector).
** w : the frequency values in rad/s (real <math>1 \times 100</math> vector).
** H : transfer function matrix evaluation at different frequeny points and parametric values (complex <math>1 \times 1 \times 101\times 31</math> matrix).

* The <tt>dataONERA_OpenCavity_withMOR.mat</tt> data file, with 3 ROMs obtained with the [https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/MORE '''MOR toolbox'''] using the parametric Loewner method
** Hr1 : linear parametric rational ROMs with varying dimensions (state-space models in Matlab form).

* The <tt>startONERA_OpenCavityt.m</tt> script file, used to loads and plots the data for illustration.

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T15:19:26Z

Poussotvassal: /* Description */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE-A_j)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). We then construct the data as follows:

First, the frequency response <math>H_j(\xi_i)</math> of each <math>j=1,\dots,3</math> configurations for frozen complex values <math>\xi_i = \xi_0 \cup \{\imath\omega_i\}_{i=1}^N</math>, with <math>\omega_i\in\mathbb R_+</math>, <math>\xi_0\in\mathbb R</math> and <math>i=1,\dots,100</math> is computed.
Second, ten intermediate parametric configurations between each <math>j</math>-th Reynolds numbers configuration are constructed by linear interpolation of the frequency responses <math>H_j(\xi_i)</math>, leading to <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=31</math> configurations.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_i,\rho_j,H_{ij}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i = 0.1 \cup \{\imath\omega_i\}_{i=1}^N</math> and parameters <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=31</math>. Specifically, the frequencies are between xxHz until yyHz in steps of zzHz, and parameters are from 1 to 3 in steps of zz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T15:05:39Z

Poussotvassal: /* Considered data */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE-A_j)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). First, the frequency response <math>H_j(\xi_i)</math> of each <math>j=1,\dots,3</math> configurations for frozen complex values <math>\xi_i=\imath\omega_i</math>, with <math>\omega_i\in\mathbb R_+</math> and <math>i=1,\dots,100</math> is computed. Then, ten intermediate configurations between each Reynolds numbers configuration <math>j</math> are constructed by linear interpolation of the frequency responses <math>H_j(\xi_i)</math>.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_i,\rho_j,H_{ij}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i = 0.1 \cup \{\imath\omega_i\}_{i=1}^N</math> and parameters <math>\{\rho_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=31</math>. Specifically, the frequencies are between xxHz until yyHz in steps of zzHz, and parameters are from xx to yy in steps of zz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T15:04:54Z

Poussotvassal: /* Considered data */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE-A_j)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). First, the frequency response <math>H_j(\xi_i)</math> of each <math>j=1,\dots,3</math> configurations for frozen complex values <math>\xi_i=\imath\omega_i</math>, with <math>\omega_i\in\mathbb R_+</math> and <math>i=1,\dots,100</math> is computed. Then, ten intermediate configurations between each Reynolds numbers configuration <math>j</math> are constructed by linear interpolation of the frequency responses <math>H_j(\xi_i)</math>.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i = 0.1 \cup \{\imath\omega_i\}_{i=1}^N</math> and parameters <math>\{p_j\}_{j=1}^M\in\mathbb R</math>, for <math>j=1,\dots,M=31</math>. Specifically, the frequencies are between xxHz until yyHz in steps of zzHz, and parameters are from xx to yy in steps of zz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T15:04:09Z

Poussotvassal: /* Considered data */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE-A_j)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). First, the frequency response <math>H_j(\xi_i)</math> of each <math>j=1,\dots,3</math> configurations for frozen complex values <math>\xi_i=\imath\omega_i</math>, with <math>\omega_i\in\mathbb R_+</math> and <math>i=1,\dots,100</math> is computed. Then, ten intermediate configurations between each Reynolds numbers configuration <math>j</math> are constructed by linear interpolation of the frequency responses <math>H_j(\xi_i)</math>.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i = 0.1 \cup \{\imath\omega_i\}_{1=1}^N</math> and parameters <math>p_j= \{\omega_i\}_{1=1}^N\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>. Specifically, the frequencies are between xxHz until yyHz in steps of zzHz, and parameters are from xx to yy in steps of zz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T15:03:47Z

Poussotvassal: /* Considered data */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE-A_j)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). First, the frequency response <math>H_j(\xi_i)</math> of each <math>j=1,\dots,3</math> configurations for frozen complex values <math>\xi_i=\imath\omega_i</math>, with <math>\omega_i\in\mathbb R_+</math> and <math>i=1,\dots,100</math> is computed. Then, ten intermediate configurations between each Reynolds numbers configuration <math>j</math> are constructed by linear interpolation of the frequency responses <math>H_j(\xi_i)</math>.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C</math> represents the transfer from the input signal (upward cavity pressure) to the measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i = 0.1 \cup \{\omega_i\}_{1=1}^N</math> and parameters <math>p_j= \{\omega_i\}_{1=1}^N\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>. Specifically, the frequencies are between xxHz until yyHz in steps of zzHz, and parameters are from xx to yy in steps of zz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T15:02:56Z

Poussotvassal: /* Considered data */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE-A_j)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). First, the frequency response <math>H_j(\xi_i)</math> of each <math>j=1,\dots,3</math> configurations for frozen complex values <math>\xi_i=\imath\omega_i</math>, with <math>\omega_i\in\mathbb R_+</math> and <math>i=1,\dots,100</math> is computed. Then, ten intermediate configurations between each Reynolds numbers configuration <math>j</math> are constructed by linear interpolation of the frequency responses <math>H_j(\xi_i)</math>.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values are inspired (but slightly modified) from the reference paper given below and are given as Matlab file. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i = 0.1 \cup \{\omega_i\}_{1=1}^N</math> and parameters <math>p_j= \{\omega_i\}_{1=1}^N\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>. Specifically, the frequencies are between xxHz until yyHz in steps of zzHz, and parameters are from xx to yy in steps of zz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T15:01:22Z

Poussotvassal: /* Considered data */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE-A_j)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). First, the frequency response <math>H_j(\xi_i)</math> of each <math>j=1,\dots,3</math> configurations for frozen complex values <math>\xi_i=\imath\omega_i</math>, with <math>\omega_i\in\mathbb R_+</math> and <math>i=1,\dots,100</math> is computed. Then, ten intermediate configurations between each Reynolds numbers configuration <math>j</math> are constructed by linear interpolation of the frequency responses <math>H_j(\xi_i)</math>.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i = 0.1 \cup \{\omega_i\}_{1=1}^N</math> and parameters <math>p_j= \{\omega_i\}_{1=1}^N\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>. Specifically, the frequencies are between xxHz until yyHz in steps of zzHz, and parameters are from xx to yy in steps of zz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T14:54:00Z

Poussotvassal: /* Considered data */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE-A_j)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). First, the frequency response <math>H_j(\xi_i)</math> of each <math>j=1,\dots,3</math> configurations for frozen complex values <math>\xi_i=\imath\omega_i</math>, with <math>\omega_i\in\mathbb R_+</math> and <math>i=1,\dots,100</math> is computed. Then, ten intermediate configurations between each Reynolds numbers configuration <math>j</math> are constructed by linear interpolation of the frequency responses <math>H_j(\xi_i)</math>.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The proposed benchmark contains a set of complex-domain responses provided at varying real parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, the data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i = 0.1 \cup {\omega_i}_{1=1}^N</math> and parameters <math>p_j= {\omega_i}_{1=1}^N\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>. Specifically, the frequencies are between xxHz until yyHz in steps of zzHz, and parameters are from xx to yy in steps of zz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T14:45:05Z

Poussotvassal: /* Motivation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE-A_j)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). First, the frequency response <math>H_j(\xi_i)</math> of each <math>j=1,\dots,3</math> configurations for frozen complex values <math>\xi_i=\imath\omega_i</math>, with <math>\omega_i\in\mathbb R_+</math> and <math>i=1,\dots,100</math> is computed. Then, ten intermediate configurations between each Reynolds numbers configuration <math>j</math> are constructed by linear interpolation of the frequency responses <math>H_j(\xi_i)</math>.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where <math>n\approx 700,000</math>):

<math>
E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i\in\mathbb C</math> and <math>p_j\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T14:44:08Z

Poussotvassal: /* Motivation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE-A_j)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). First, the frequency response <math>H_j(\xi_i)</math> of each <math>j=1,\dots,3</math> configurations for frozen complex values <math>\xi_i=\imath\omega_i</math>, with <math>\omega_i\in\mathbb R_+</math> and <math>i=1,\dots,\overline n=\underline n=100</math> is computed. Then, ten intermediate configurations between each Reynolds numbers configuration <math>j</math> are constructed by linear interpolation of the frequency responses <math>H_j(\xi_i)</math>.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where <math>n\approx 700,000</math>):

<math>
E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i\in\mathbb C</math> and <math>p_j\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T14:43:42Z

Poussotvassal: /* Motivation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E_j \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E_j,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE_j-A_j)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). First, the frequency response <math>H_j(\xi_i)</math> of each <math>j=1,\dots,3</math> configurations for frozen complex values <math>\xi_i=\imath\omega_i</math>, with <math>\omega_i\in\mathbb R_+</math> and <math>i=1,\dots,\overline n=\underline n=100</math> is computed. Then, ten intermediate configurations between each Reynolds numbers configuration <math>j</math> are constructed by linear interpolation of the frequency responses <math>H_j(\xi_i)</math>.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where <math>n\approx 700,000</math>):

<math>
E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i\in\mathbb C</math> and <math>p_j\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T14:41:41Z

Poussotvassal: /* Motivation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E_j \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E_j,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE-A)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). First, the frequency response <math>H_j(\xi_i)</math> of each <math>j=1,\dots,3</math> configurations for frozen complex values <math>\xi_i=\imath\omega_i</math>, with <math>\omega_i\in\mathbb R_+</math> and <math>i=1,\dots,\overline n=\underline n=100</math> is computed. Then, ten intermediate configurations between each Reynolds numbers configuration <math>j</math> are constructed by linear interpolation of the frequency responses <math>H_j(\xi_i)</math>.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where <math>n\approx 700,000</math>):

<math>
E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i\in\mathbb C</math> and <math>p_j\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T14:40:48Z

Poussotvassal: /* Motivation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E_j \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E_j,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j(s) = C(sE-A)^{-1}B \in \mathbb{C}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric a data-driven approaches (we used the Loewner framework). First, the frequency response <math>H_j(\xi_i)</math> of each <math>j=1,\dots,3</math> configurations for frozen complex values <math>\xi_i=\imath\omega_i</math>, with <math>\omega_i\in\mathbb R_+</math> and <math>i=1,\dots,\overline n=\underline n=100</math> is computed. Then, ten intermediate configurations between each Reynolds numbers configuration <math>j</math> are constructed by linear interpolation of the frequency responses <math>H_j(\xi_i)</math>. We obtain $\{z_k\}_{k=1}^{N=200}$, $\{p_l\}_{l=1}^{N=31}$ and thus $\Phi \in \IC^{200 \times 31}$. Our objective is to come up with a parametrized linear model that is able to faithfully reproduce the original transfer data on a particular range of frequencies as well as on a target parameter range.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where <math>n\approx 700,000</math>):

<math>
E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i\in\mathbb C</math> and <math>p_j\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T14:16:23Z

Poussotvassal: /* Motivation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E_j \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E_j,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j = E \dot x(t) = Ax(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

In <ref name=PoussotSipp></ref>, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric Loewner framework is applied. First, the frequency response of the three configurations for $N=200$ logarithmically spaced frequencies $\{z_k\}_{k=1}^N=\{\imath \omega_{\overline k},-\imath \omega_{\overline k}\}_{k=1}^{N/2}$ are first computed. Then, ten intermediate configurations between each Reynolds numbers $Re=\{4000,5250,6000\}$ are constructed by linear interpolation. We obtain $\{z_k\}_{k=1}^{N=200}$, $\{p_l\}_{l=1}^{N=31}$ and thus $\Phi \in \IC^{200 \times 31}$. Our objective is to come up with a parametrized linear model that is able to faithfully reproduce the original transfer data on a particular range of frequencies as well as on a target parameter range.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where <math>n\approx 700,000</math>):

<math>
E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i\in\mathbb C</math> and <math>p_j\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T14:15:40Z

Poussotvassal: /* Motivation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realization of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E_j \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E_j,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j = E \dot x(t) = Ax(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

In \cite{PoussotLPVS:2015}, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric Loewner framework is applied. First, the frequency response of the three configurations for $N=200$ logarithmically spaced frequencies $\{z_k\}_{k=1}^N=\{\imath \omega_{\overline k},-\imath \omega_{\overline k}\}_{k=1}^{N/2}$ are first computed. Then, ten intermediate configurations between each Reynolds numbers $Re=\{4000,5250,6000\}$ are constructed by linear interpolation. We obtain $\{z_k\}_{k=1}^{N=200}$, $\{p_l\}_{l=1}^{N=31}$ and thus $\Phi \in \IC^{200 \times 31}$. Our objective is to come up with a parametrized linear model that is able to faithfully reproduce the original transfer data on a particular range of frequencies as well as on a target parameter range.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where <math>n\approx 700,000</math>):

<math>
E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i\in\mathbb C</math> and <math>p_j\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T14:14:59Z

Poussotvassal: /* Description */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on the <xr id="fig:cavity"/> and <xr id="fig:cavityScheme"/>.

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realisation of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E_j \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E_j,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j = E \dot x(t) = Ax(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

In \cite{PoussotLPVS:2015}, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric Loewner framework is applied. First, the frequency response of the three configurations for $N=200$ logarithmically spaced frequencies $\{z_k\}_{k=1}^N=\{\imath \omega_{\overline k},-\imath \omega_{\overline k}\}_{k=1}^{N/2}$ are first computed. Then, ten intermediate configurations between each Reynolds numbers $Re=\{4000,5250,6000\}$ are constructed by linear interpolation. We obtain $\{z_k\}_{k=1}^{N=200}$, $\{p_l\}_{l=1}^{N=31}$ and thus $\Phi \in \IC^{200 \times 31}$. Our objective is to come up with a parametrized linear model that is able to faithfully reproduce the original transfer data on a particular range of frequencies as well as on a target parameter range.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where <math>n\approx 700,000</math>):

<math>
E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i\in\mathbb C</math> and <math>p_j\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T14:07:30Z

Poussotvassal: /* Motivation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on the right figures "fig:cavity" and "fig:cavityScheme".

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realisation of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E_j \dot x_j(t) = A_jx_j(t) + Bu(t) \text{ , } y(t)=Cx_j(t) \text{ where } E_j,A_j \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j = E \dot x(t) = Ax(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

In \cite{PoussotLPVS:2015}, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric Loewner framework is applied. First, the frequency response of the three configurations for $N=200$ logarithmically spaced frequencies $\{z_k\}_{k=1}^N=\{\imath \omega_{\overline k},-\imath \omega_{\overline k}\}_{k=1}^{N/2}$ are first computed. Then, ten intermediate configurations between each Reynolds numbers $Re=\{4000,5250,6000\}$ are constructed by linear interpolation. We obtain $\{z_k\}_{k=1}^{N=200}$, $\{p_l\}_{l=1}^{N=31}$ and thus $\Phi \in \IC^{200 \times 31}$. Our objective is to come up with a parametrized linear model that is able to faithfully reproduce the original transfer data on a particular range of frequencies as well as on a target parameter range.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where <math>n\approx 700,000</math>):

<math>
E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i\in\mathbb C</math> and <math>p_j\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T14:06:57Z

Poussotvassal:

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityScheme">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such a configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and also (with a more dynamical system view point) in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on the right figures "fig:cavity" and "fig:cavityScheme".

For simulation purpose, the phenomena modeled by Navier and Stokes equations, is spatially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realisation of order <math>n=680,974</math> given as (matrices, about 100Mo each, can be provided under request to '' [[User:Poussotvassal]] '')

<math>
E_j \dot x_j(t) = A_jx_j(t) + B_ju(t) \text{ , } y(t)=C_jx_j(t) \text{ where } E_j,A_j \in \mathbb R^{n\times n} \text{ and } B_j,C_j^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j = E \dot x(t) = Ax(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

In \cite{PoussotLPVS:2015}, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric Loewner framework is applied. First, the frequency response of the three configurations for $N=200$ logarithmically spaced frequencies $\{z_k\}_{k=1}^N=\{\imath \omega_{\overline k},-\imath \omega_{\overline k}\}_{k=1}^{N/2}$ are first computed. Then, ten intermediate configurations between each Reynolds numbers $Re=\{4000,5250,6000\}$ are constructed by linear interpolation. We obtain $\{z_k\}_{k=1}^{N=200}$, $\{p_l\}_{l=1}^{N=31}$ and thus $\Phi \in \IC^{200 \times 31}$. Our objective is to come up with a parametrized linear model that is able to faithfully reproduce the original transfer data on a particular range of frequencies as well as on a target parameter range.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where <math>n\approx 700,000</math>):

<math>
E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i\in\mathbb C</math> and <math>p_j\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T08:43:56Z

Poussotvassal: /* Motivation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry and flow computed on FreeFem.</caption>]]</figure>
<figure id="fig:cavityFR">[[File:FluidControl.png|350px|thumb|right|<caption>Open cavity schematic view: actuator on the left and sensor on the right of the cavity.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on the right figure "fig:cavity" (bottom right).

For simulation purpose, the phenomena modelled by Navier and Stokes equations, is spacially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realisation of order <math>n=680,974</math> given as

<math>
E_j \dot x_j(t) = A_jx_j(t) + B_ju(t) \text{ , } y(t)=C_jx_j(t) \text{ where } E_j,A_j \in \mathbb R^{n\times n} \text{ and } B_j,C_j^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j = E \dot x(t) = Ax(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

In \cite{PoussotLPVS:2015}, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric Loewner framework is applied. First, the frequency response of the three configurations for $N=200$ logarithmically spaced frequencies $\{z_k\}_{k=1}^N=\{\imath \omega_{\overline k},-\imath \omega_{\overline k}\}_{k=1}^{N/2}$ are first computed. Then, ten intermediate configurations between each Reynolds numbers $Re=\{4000,5250,6000\}$ are constructed by linear interpolation. We obtain $\{z_k\}_{k=1}^{N=200}$, $\{p_l\}_{l=1}^{N=31}$ and thus $\Phi \in \IC^{200 \times 31}$. Our objective is to come up with a parametrized linear model that is able to faithfully reproduce the original transfer data on a particular range of frequencies as well as on a target parameter range.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where <math>n\approx 700,000</math>):

<math>
E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i\in\mathbb C</math> and <math>p_j\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T08:42:40Z

Poussotvassal: /* Considered data */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>
<figure id="fig:cavityFR">[[File:FluidControl.png|350px|thumb|right|<caption>Open parametric frequency response.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on the right figure "fig:cavity" (bottom right).

For simulation purpose, the phenomena modelled by Navier and Stokes equations, is spacially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realisation of order <math>n=680,974</math> given as

<math>
E_j \dot x_j(t) = A_jx_j(t) + B_ju(t) \text{ , } y(t)=C_jx_j(t) \text{ where } E_j,A_j \in \mathbb R^{n\times n} \text{ and } B_j,C_j^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j = E \dot x(t) = Ax(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

In \cite{PoussotLPVS:2015}, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric Loewner framework is applied. First, the frequency response of the three configurations for $N=200$ logarithmically spaced frequencies $\{z_k\}_{k=1}^N=\{\imath \omega_{\overline k},-\imath \omega_{\overline k}\}_{k=1}^{N/2}$ are first computed. Then, ten intermediate configurations between each Reynolds numbers $Re=\{4000,5250,6000\}$ are constructed by linear interpolation. We obtain $\{z_k\}_{k=1}^{N=200}$, $\{p_l\}_{l=1}^{N=31}$ and thus $\Phi \in \IC^{200 \times 31}$. Our objective is to come up with a parametrized linear model that is able to faithfully reproduce the original transfer data on a particular range of frequencies as well as on a target parameter range.

===Considered data===

<figure id="fig:cavityData">[[File:FluidControl_data.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where <math>n\approx 700,000</math>):

<math>
E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i\in\mathbb C</math> and <math>p_j\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-07T08:41:47Z

Poussotvassal: /* Motivation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>
<figure id="fig:cavityFR">[[File:FluidControl.png|350px|thumb|right|<caption>Open parametric frequency response.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on the right figure "fig:cavity" (bottom right).

For simulation purpose, the phenomena modelled by Navier and Stokes equations, is spacially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realisation of order <math>n=680,974</math> given as

<math>
E_j \dot x_j(t) = A_jx_j(t) + B_ju(t) \text{ , } y(t)=C_jx_j(t) \text{ where } E_j,A_j \in \mathbb R^{n\times n} \text{ and } B_j,C_j^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j = E \dot x(t) = Ax(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

In \cite{PoussotLPVS:2015}, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric Loewner framework is applied. First, the frequency response of the three configurations for $N=200$ logarithmically spaced frequencies $\{z_k\}_{k=1}^N=\{\imath \omega_{\overline k},-\imath \omega_{\overline k}\}_{k=1}^{N/2}$ are first computed. Then, ten intermediate configurations between each Reynolds numbers $Re=\{4000,5250,6000\}$ are constructed by linear interpolation. We obtain $\{z_k\}_{k=1}^{N=200}$, $\{p_l\}_{l=1}^{N=31}$ and thus $\Phi \in \IC^{200 \times 31}$. Our objective is to come up with a parametrized linear model that is able to faithfully reproduce the original transfer data on a particular range of frequencies as well as on a target parameter range.

===Considered data===

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where <math>n\approx 700,000</math>):

<math>
E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i\in\mathbb C</math> and <math>p_j\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

File:FluidControl.png

2021-06-07T08:41:25Z

Poussotvassal:

Fluid Flow Linearized Open Cavity Model

2021-06-04T14:35:24Z

Poussotvassal: /* Description */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|350px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>
<figure id="fig:cavityFR">[[File:cavity.png|350px|thumb|right|<caption>Open parametric frequency response.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on the right figure "fig:cavity" (bottom right).

For simulation purpose, the phenomena modelled by Navier and Stokes equations, is spacially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> three dynamical models <math>\{H_j\}_{j=1}^3</math> can be described as a DAE realisation of order <math>n=680,974</math> given as

<math>
E_j \dot x_j(t) = A_jx_j(t) + B_ju(t) \text{ , } y(t)=C_jx_j(t) \text{ where } E_j,A_j \in \mathbb R^{n\times n} \text{ and } B_jC_i^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. The transfer functions read

<math>
H_j = E \dot x(t) = Ax(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

In \cite{PoussotLPVS:2015}, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric Loewner framework is applied. First, the frequency response of the three configurations for $N=200$ logarithmically spaced frequencies $\{z_k\}_{k=1}^N=\{\imath \omega_{\overline k},-\imath \omega_{\overline k}\}_{k=1}^{N/2}$ are first computed. Then, ten intermediate configurations between each Reynolds numbers $Re=\{4000,5250,6000\}$ are constructed by linear interpolation. We obtain $\{z_k\}_{k=1}^{N=200}$, $\{p_l\}_{l=1}^{N=31}$ and thus $\Phi \in \IC^{200 \times 31}$. Our objective is to come up with a parametrized linear model that is able to faithfully reproduce the original transfer data on a particular range of frequencies as well as on a target parameter range.

===Considered data===

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where <math>n\approx 700,000</math>):

<math>
E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i\in\mathbb C</math> and <math>p_j\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''

Fluid Flow Linearized Open Cavity Model

2021-06-04T14:22:25Z

Poussotvassal: /* Description */

{{preliminary}} 

[[Category:benchmark]]
[[Category:parametric]]
[[Category:ODE]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:SISO]]
[[Category:frequency data]]
[[Category:data-driven]]

==Description==

===Motivation===

<figure id="fig:cavity">[[File:VideoPics.png|450px|thumb|right|<caption>Open cavity geometry.</caption>]]</figure>
<figure id="fig:cavityFR">[[File:cavity.png|450px|thumb|right|<caption>Open parametric frequency response.</caption>]]</figure>

This benchmark considers a fluid-flow configuration. It consists of a two-dimensional open square cavity geometry flow problem where air flows from left to right. Such configuration is described in details in the original work of <ref name=Barbagallo>A. Barbagallo and D. Sipp and P.J. Schmid, "Closed-loop control of an open cavity flow using reduced-order models", Journal of Fluid Mechanics, vol. 641, pp. 1-50</ref> and in <ref name=PoussotSipp>C. Poussot-Vassal and D. Sipp, "[https://doi.org/10.1016/j.ifacol.2015.11.126 Parametric reduced order dynamical model construction of a fluid flow control problem]", in Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, 2015, pp. 133-138.</ref>, and illustrated on the right figure "fig:cavity" (bottom right).

For dimulation, the phenomena modelled by Navier and Stokes equations, is spacially discretized along a mesh composed of <math>193,874</math> triangles, corresponding to <math>n=680,974</math> degrees of freedom for the velocity variables along the <math>x</math> and <math>y</math> axis. After linearization around three fixed points for varying Reynolds numbers <math>Re=\{4000,5250,6000\}</math> and discretisation along the flow axis, three dynamical models <math>\{H_i\}_{i=1}^3</math> can be described as a DAE realisation of order <math>n=680,974</math> given as

<math>
E \dot x(t) = Ax(t) + Bu(t) \text{ , } y(t)=Cx(t) \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

where the input <math>u(t)</math> is the vertical pressure actuator located upstream of the cavity and the output <math>y(t)</math> is a shear stress sensor, located downstream of the cavity. In this particular case, the parameter is the Reynolds number <math>Re</math>. In \cite{PoussotLPVS:2015}, the IRKA approach (being a realization based <math>\mathcal H_2</math>-oriented reduction method) is used to sequentially approximate each realization with a low dimensional one. Then, the interpolation along the parameter is done in a second step by interpolating each coefficients in the canonical basis of the obtained realization.

Here instead, we suggest the parametric Loewner framework is applied. First, the frequency response of the three configurations for $N=200$ logarithmically spaced frequencies $\{z_k\}_{k=1}^N=\{\imath \omega_{\overline k},-\imath \omega_{\overline k}\}_{k=1}^{N/2}$ are first computed. Then, ten intermediate configurations between each Reynolds numbers $Re=\{4000,5250,6000\}$ are constructed by linear interpolation. We obtain $\{z_k\}_{k=1}^{N=200}$, $\{p_l\}_{l=1}^{N=31}$ and thus $\Phi \in \IC^{200 \times 31}$. Our objective is to come up with a parametrized linear model that is able to faithfully reproduce the original transfer data on a particular range of frequencies as well as on a target parameter range.

===Considered data===

The benchmark contains a set of complex-domain reponses provided at varying real frozen Reynolds parametric values. The numerical values data as inspired (but slightly modified) from the reference paper given below and are provided as Matlab data. More specifically, This benchmark contains a set of complex-domain/parametric(real)-domain input-output data computed from a high dimensional linear descriptor model given as (where <math>n\approx 700,000</math>):

<math>
E \dot x = Ax + Bu \text{ , } y=Cx \text{ where } E,A \in \mathbb R^{n\times n} \text{ and } B,C^T \in \mathbb R^{n}
</math>

and which matrices have been obtained by linearizing the Navier and Stokes equations at frozen Reynolds numbers, from a high fidelity simulator.

The data provided in this use-case are given as the triple:

<math>
\{\xi_i,p_j,H_{ij}^{n_y\times n_u}\}_{i=1,j=1}^{N+1,M},
</math>

where <math>H_{ij}\in\mathbb C^{n_y\times n_u}</math> represents the transfer from <math>n_u=1</math> input signal (upward cavity pressure) to <math>n_y=1</math> measurement output (downward cavity pressure), evaluated at varying complex values <math>\xi_i\in\mathbb C</math> and <math>p_j\in\mathbb R</math>, for <math>j=1,\dots,M=3</math>.
Specifically, the frequencies are between 0.1Hz until 42.1Hz in steps of 0.1Hz.

==Origin==

Collaboration between [https://www.onera.fr ONERA] DTIS (dynamical systems and information departement) and DAAA (Fluid mechanics departement). The data come from a fluid simulator (coded in FreeFem++). The model is constructed by D. Sipp and post-processing was performed jointly by P. Vuillemin and C. Poussot-Vassal.

==Data==

===Description===

===Objective===

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Fluid Flow Linearized Open Cavity Model'''. MORwiki - Model Order Reduction Wiki, 2021. https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Fluid_Flow_Linearized_Open_Cavity_Model

* For the background on the benchmark with a dynamical and control engineering point of view:

@inproceedings{PoussotLPVS:2015,
author = {C. Poussot-Vassal and D. Sipp},
title = {Parametric reduced order dynamical model construction of a fluid flow control problem},
booktitle = {Proceedings of the 1st IFAC Workshop on Linear Parameter Varying Systems},
address = {Grenoble, France},
month = {October},
year = {2015},
pages = {133-138},
}

* For the background on the benchmark with a fluid-flow point of view:

@article{Barbagallo:2008,
author = {A. Barbagallo and D. Sipp and P.J. Schmid},
journal = {Journal of Fluid Mechanics},
pages = {1-50},
title = {Closed-loop control of an open cavity flow using reduced-order models},
volume = {641},
year = {2008}
}

==References==

<references />

==Contact==

'' [[User:Poussotvassal]] ''