Fluid Flow Linearized Open Cavity Model - Revision history

Poussotvassal: /* Description */

2022-10-12T09:49:48Z

Description

← Older revision		Revision as of 09:49, 12 October 2022
Line 59:		Line 59:
	===Description===		===Description===

	The [[:File:~~FluidFlowCavityDataParametric~~.zip\|FluidFlowCavityDataParametric.zip]] (36ko) repository contains three files:		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		* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with

Poussotvassal at 09:41, 12 October 2022

2022-10-12T09:41:35Z

← Older revision		Revision as of 09:41, 12 October 2022
Line 34:		Line 34:
	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.		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:		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, "<span class="plainlinks">[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]</span>", 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.		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.

Poussotvassal: /* Motivation */

2022-10-12T09:38:25Z

Motivation

← Older revision		Revision as of 09:38, 12 October 2022
Line 34:		Line 34:
	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.		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:		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.		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.

Poussotvassal: /* Citation */

2022-10-12T09:37:20Z

Citation

← Older revision		Revision as of 09:37, 12 October 2022
Line 88:		Line 88:
	journal = {Handbook of Numerical Analysis},		journal = {Handbook of Numerical Analysis},
	year = {2022},		year = {2022},
	~~month = {January},~~
	volume = {23},		volume = {23},
	number = {Numerical Control: Part A},		number = {Numerical Control: Part A},

Poussotvassal: /* Citation */

2022-10-12T09:14:08Z

Citation

← Older revision		Revision as of 09:14, 12 October 2022
Line 81:		Line 81:
	::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		::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:		* For illustration of the Loewner-driven parametric model construction involvin this use-case, see:

	@article{GoseaHNA:2022,		@article{GoseaHNA:2022,

Poussotvassal: /* Citation */

2022-10-12T09:12:57Z

Citation

← Older revision		Revision as of 09:12, 12 October 2022
Line 85:		Line 85:
	@article{GoseaHNA:2022,		@article{GoseaHNA:2022,
	author = {V. Gosea and C. Poussot-Vassal and A.C. Antoulas},		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}},		title = {Data-driven modeling and control of large-scale dynamical systems in the Loewner framework},
	journal = {Handbook of Numerical Analysis},		journal = {Handbook of Numerical Analysis},
	year = {2022},		year = {2022},

Poussotvassal: /* Citation */

2022-10-12T09:12:46Z

Citation

@@ Line 80: / Line 80: @@
 * 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:

Poussotvassal: /* Objective */

2022-10-12T09:08:17Z

Objective

← Older revision		Revision as of 09:08, 12 October 2022
Line 72:		Line 72:

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

	==Citation==		==Citation==

Poussotvassal: /* Description */

2022-10-12T09:07:45Z

Description

← Older revision		Revision as of 09:07, 12 October 2022
Line 59:		Line 59:
	===Description===		===Description===

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

	* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with		* The <tt>dataONERA_FluidFlowOpenCavity.mat</tt> data file, with
	** P : the parametric values (real <math>1 \times 21</math> vector).		** 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.		** 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).		** 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.		* 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(...)"		** 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.		* The <tt>startONERA_FluidFlowOpenCavity.m</tt> script file, used to loads and plots the data for illustration and a solution.

Poussotvassal: /* Considered data */

2022-10-12T09:02:46Z

Considered data

← Older revision		Revision as of 09:02, 12 October 2022
Line 49:		Line 49:
	</math>		</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>.		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==		==Origin==