[[Category:Software]]
[[Category:Dense]]
[[Category:Linear algebra]]
[[Category:MATLAB]]
[[Category:Octave]]

[[file:MORLAB_Logo.png|220px|right|MORLAB Logo]]

[http://www.mpi-magdeburg.mpg.de/projects/morlab MORLAB], the '''M'''odel '''O'''rder '''R'''eduction '''LAB'''oratory toolbox, is a collection of [https://de.mathworks.com/products/matlab.html MATLAB] and [https://www.gnu.org/software/octave/ Octave] routines for model order reduction of dense linear time-invariant continuous-time systems.
The toolbox contains model reduction methods for standard, descriptor and second-order systems based on the solution of matrix equations.
Therefore, also spectral projection based methods for the solution of the corresponding matrix equations are included.

== Features ==
The following main features are provided in the latest release of the software (version '''4.0'''):

'''Model reduction methods:'''
* Modal truncation method ([[Modal truncation|MT]])
* Balancing related methods ([[Balanced Truncation|BT]], FLBT, BST, LQGBT, PRBT, BRBT, HINFBT, TLBT)
* Hankel-norm approximation method ([[Hankel-Norm Approximation|HNA]])

'''Matrix equation solvers:'''
* Matrix sign function based solvers for continuous-time algebraic Lyapunov, Sylvester, Bernoulli and Riccati equations
* Newton-Kleinman type solvers for continuous-time algebraic Riccati equations with a negative quadratic term
* Newton type solvers for continuous-time algebraic Riccati equations with a positive quadratic term
* Riccati iteration based solver for continuous-time algebraic Riccati equations with an indefinite quadratic term

'''Further methods:'''
* Routines for the additive decomposition of transfer functions of linear systems
* Partial stabilization of linear systems
* Newton iteration to compute the matrix sign function
* Inverse-free iteration to compute the right matrix pencil disk function

== References ==

* P. Benner, S. W. R. Werner, [https://doi.org/10.1016/j.ifacol.2018.03.092 Model reduction of descriptor systems with the MORLAB toolbox], IFAC-PapersOnLine 9th Vienna International Conference on Mathematical Modelling MATHMOD 2018, Vienna, Austria, 21--23 February 2018 51 (2) (2018) 547--552.
* P. Benner, S. W. R. Werner, [http://www.control.tf.uni-kiel.de/files/gma/2017/Tagungsband2017.pdf MORLAB - Modellreduktion in MATLAB], in: T. Meurer, F. Woittennek (Eds.), Tagungsband GMA-FA 1.30 ’Modellierung, Identifikation und Simulation in der Automatisierungstechnik’ und GMA-FA 1.40 ’Theoretische Verfahren der Regelungstechnik’, Workshop in Anif, Salzburg, 18.-22.09.2017, 2017, pp. 508--517.
* P. Benner, [https://doi.org/10.1109/CACSD-CCA-ISIC.2006.4776618 A MATLAB repository for model reduction based on spectral projection], in: 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control, 2006, pp. 19--24.
* P. Benner, E. S. Quintana-Ortı́, [https://doi.org/10.1007/3-540-27909-1_1 Model reduction based on spectral projection methods], in: P. Benner, V. Mehrmann, D. Sorensen (Eds.), Dimension Reduction of Large-Scale Systems, Vol. 45 of Lect. Notes Comput. Sci. Eng., Springer, Berlin/Heidelberg, Germany, 2005, pp. 5--45.

== Links ==

* Official project website: https://www.mpi-magdeburg.mpg.de/projects/morlab
* Latest upload on Zenodo: https://zenodo.org/record/1574083

== Contact ==
[[User:Werner| Steffen Werner]]

File:MORLAB Logo.png

2018-11-06T14:26:06Z

Werner: Werner uploaded a new version of "File:MORLAB Logo.png"

MORLAB

2018-09-14T08:30:30Z

Werner: Added reference.

[[Category:Software]]
[[Category:Dense]]
[[Category:Linear algebra]]
[[Category:MATLAB]]
[[Category:Octave]]

[[file:MORLAB_Logo.png|220px|right|MORLAB Logo]]

[http://www.mpi-magdeburg.mpg.de/projects/morlab MORLAB], the '''M'''odel '''O'''rder '''R'''eduction '''LAB'''oratory toolbox, is a collection of [https://de.mathworks.com/products/matlab.html MATLAB] and [https://www.gnu.org/software/octave/ Octave] routines for model order reduction of dense linear time-invariant continuous-time systems.
The toolbox contains model reduction methods for standard and descriptor systems based on the solution of matrix equations.
Therefore, also spectral projection based methods for the solution of the corresponding matrix equations are included.

== Features ==
The following main features are provided in the latest release of the software (version '''3.0'''):

'''Model reduction methods:'''
* Modal truncation method ([[Modal truncation|MT]])
* Balancing related methods ([[Balanced Truncation|BT]], BST, LQGBT, PRBT, BRBT, HinfBT)
* Hankel-norm approximation method ([[Hankel-Norm Approximation|HNA]])

'''Matrix equation solvers:'''
* Matrix sign function based solvers for continuous-time Lyapunov, Sylvester and algebraic Bernoulli equations
* Newton-Kleinman type solvers for the continuous-time algebraic Riccati equations with negative quadratic term
* Newton type solvers for continuous-time algebraic Riccati equation with positive quadratic term

'''Further methods:'''
* Routines for the additive decomposition of transfer functions of linear systems
* Partial stabilization of linear systems
* Newton iteration to compute the matrix sign function
* Inverse-free iteration to compute the right matrix pencil disk function

== References ==

* P. Benner, S. W. R. Werner, [https://doi.org/10.1016/j.ifacol.2018.03.092 Model reduction of descriptor systems with the MORLAB toolbox], IFAC-PapersOnLine 9th Vienna International Conference on Mathematical Modelling MATHMOD 2018, Vienna, Austria, 21--23 February 2018 51 (2) (2018) 547--552.
* P. Benner, S. W. R. Werner, [http://www.control.tf.uni-kiel.de/files/gma/2017/Tagungsband2017.pdf MORLAB - Modellreduktion in MATLAB], in: T. Meurer, F. Woittennek (Eds.), Tagungsband GMA-FA 1.30 ’Modellierung, Identifikation und Simulation in der Automatisierungstechnik’ und GMA-FA 1.40 ’Theoretische Verfahren der Regelungstechnik’, Workshop in Anif, Salzburg, 18.-22.09.2017, 2017, pp. 508--517.
* P. Benner, [https://doi.org/10.1109/CACSD-CCA-ISIC.2006.4776618 A MATLAB repository for model reduction based on spectral projection], in: 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control, 2006, pp. 19--24.
* P. Benner, E. S. Quintana-Ortı́, [https://doi.org/10.1007/3-540-27909-1_1 Model reduction based on spectral projection methods], in: P. Benner, V. Mehrmann, D. Sorensen (Eds.), Dimension Reduction of Large-Scale Systems, Vol. 45 of Lect. Notes Comput. Sci. Eng., Springer, Berlin/Heidelberg, Germany, 2005, pp. 5--45.

== Links ==

* Official website: https://www.mpi-magdeburg.mpg.de/projects/morlab
* Latest upload on Zenodo: https://zenodo.org/record/842659

== Contact ==
[[User:Werner| Steffen Werner]]

MORLAB

2018-09-13T16:35:13Z

Werner: Added new references, increased Logo.

Category:Method

2018-05-07T12:11:11Z

Werner: Added link to HNA.

All pages describing a model order reduction method are part of this category.

A comparison of model order reduction methods can be found for example in<ref name="antoulas00">A.C. Antoulas; S. Gugercin, "[http://www.math.vt.edu/people/gugercin/ttpapers/GugercinC3.pdf A comparative study of 7 algorithms for model reduction]", Proceedings of the 39th IEEE Conference on Decision and Control, vol.3, pp. 2367--2372, 2000.</ref>, <ref name="antoulas01">
A.C. Antoulas; D.C. Sorensen; S. Gugercin, "[http://www.math.vt.edu/people/gugercin/papers/survey.pdf A survey of model reduction methods for large-scale systems]", Contemporary mathematics, vol.280, pp. 193--220, 2001.</ref> and cover:

* [[Balanced Truncation]]
* [[Balanced Truncation|Approximate Balancing]]
* [[Hankel-Norm Approximation|Hankel-Norm Approximation]]
* Singular Perturbation
* [[IRKA|Rational Krylov Method]]
* Lanczos Method
* Arnoldi Method

A comparison of parametric model order reduction methods is conducted in <ref name="baur15">U. Baur; P. Benner; B. Haasdonk; C. Himpe; I. Martini; M. Ohlberger, "[http://doi.org/10.1137/1.9781611974829.ch9 Comparison of Methods for Parametric Model Order Reduction of Time-Dependent Problems]", In: Model Reduction and Approximation: Theory and Algorithms, Editors: P. Benner, A. Cohen, M. Ohlberger and K. Willcox, SIAM, 2017.</ref> and covers:

* POD
* POD-Greedy
* Matrix Interpolation
* [[Transfer Function Interpolation]]
* [[Piecewise H2 Tangential Interpolation]]
* [[Moment-matching PMOR method|Multi Parameter Moment Matching]]
* [[Emgr|Empirical Linear Cross Gramian]]

Preliminary results of a PMOR comparison are published on the posters:

[[File:poster_Baur_MoRePaS.pdf]] [[File:Poster_Baur.pdf]]

== References ==

<references/>

----

Hankel-Norm Approximation

2018-05-03T09:04:08Z

Werner:

[[Category:method]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:linear algebra]]

The '''Hankel-norm approximation''' method is a model reduction approach that solves the best-approximation problem in the Hankel semi-norm<ref name="morGlo84"></ref>.

== Description ==
Consider the standard linear-time invariant system

::<math>G:\left\{ \begin{align} \dot{x}(t) & = Ax(t) + Bu(t),\\ y(t) & = Cx(t) + Du(t), \end{align} \right.</math>

with the matrices <math style="vertical-align: top;">A \in \mathbb{R}^{n \times n}</math>, <math style="vertical-align: top;">B \in \mathbb{R}^{n \times m}</math>, <math style="vertical-align: top;">C \in \mathbb{R}^{p \times n}</math> and <math style="vertical-align: top;">D \in \mathbb{R}^{p \times m}</math>.
For a system <math style="vertical-align: top;">G</math>, the Hankel operator <math style="vertical-align: top;">\mathcal{H}</math> maps past inputs <math>u_{-}</math> to future outputs <math>y_{+}</math> of the system, i.e., <math>y_{+} = \mathcal{H}u_{-}</math>.
Then, the Hankel semi-norm of the system <math>G</math> is defined as the <math>\mathcal{L}_{2}</math>-induced norm of the Hankel opertor

::<math>\lVert G \rVert_{H} := \sup\limits_{u_{-} \in \mathcal{L}_{2}\left(-\infty, 0\right]}\frac{\lVert y_{+} \rVert_{\mathcal{L}_{2}}}{\lVert u_{-} \rVert_{\mathcal{L}_{2}}}.</math>

If the system <math>G</math> is stable, the controllability and observability Gramians <math>\mathcal{G}_{c}</math> and <math>\mathcal{G}_{o}</math> of the system above are given as the unique positive semidefinite solutions of the two Lyapunov equations

::<math>\begin{align} A\mathcal{G}_{c} + \mathcal{G}_{c}A^{T} + BB^{T} & = 0,\\ A^{T}\mathcal{G}_{o} + \mathcal{G}_{o}A + C^{T}C & = 0. \end{align}</math>

The Hankel singular values of the system <math>G</math> are then defined as the square-roots of the eigenvalues of the multiplied system Gramians, i.e., <math>\sqrt{\Lambda(\mathcal{G}_{c}\mathcal{G}_{o})} = \{ \varsigma_{1}, \ldots, \varsigma_{n} \}</math>.
It can be shown, that the Hankel semi-norm of a system is given by the largest Hankel singular value <math>\lVert G \rVert_{H} = \varsigma_{\text{max}}</math>.

The idea of the Hankel-norm approximation method is, to construct a reduced-order model <math>G_{r}</math> of order <math>r</math> such that the error system <math>\mathcal{E} = G - G_{r}</math> has a scaled all-pass transfer function

::<math>\mathcal{E}(s)\mathcal{E}^{T}(-s) = \varsigma_{r + 1}^{2} I_{p},</math>

with <math>\varsigma_{r + 1}</math> the <math>(r + 1)</math>-st Hankel singular value of the system <math>G</math>.

For such error systems, the Hankel semi-norm is known to be <math>\lVert \mathcal{E} \rVert_{H} = \varsigma_{r + 1}.</math>

== Algorithm ==
Here, the algorithm of the Hankel-norm approximation method is shortly described <ref name="morBenQQ04"></ref>:

1. Compute a minimal balanced realization <math>(\check{A}, \check{B}, \check{C}, D)</math> using the [[Balanced Truncation#Balancing and Truncation|balanced truncation square-root method]].
2. Choose the Hankel singular value <math>\varsigma_{r + 1}</math>.
3. Permute the balanced realization such that the Gramians have the form
<math>\begin{align}\check{\mathcal{G}}_{c} = \check{\mathcal{G}}_{o} & = \mathrm{diag}(\varsigma_{1}, \ldots, \varsigma_{r}, \varsigma_{r + k + 1}, \ldots, \varsigma_{n}, \varsigma_{r + 1}I_{k})\\ & = \mathrm{diag}(\Sigma, \varsigma_{r + 1}I_{k}).\end{align}</math>
4. Partition the resulting permuted system according to the Gramians
<math>\check{A} = \begin{bmatrix} A_{11} & A_{12}\\ A_{21} & A_{22}\end{bmatrix}, ~~~ \check{B} = \begin{bmatrix} B_{1}\\ B_{2}\end{bmatrix}, \check{C} = \begin{bmatrix} C_{1} & C_{2}\end{bmatrix},</math>
where <math>A_{22} \in \mathbb{R}^{k \times k}</math>, <math>B_{2} \in \mathbb{R}^{k \times m}</math> and <math>C_{2} \in \mathbb{R}^{p \times k}</math>.
5. Compute the transformation
<math>\begin{align}\tilde{A} & = \Gamma^{-1}(\varsigma_{r+1}^{2}A_{11}^{T} + \Sigma A_{11} \Sigma + \varsigma_{r+1}C_{1}^{T}UB_{1}^{T}),\\ \tilde{B} & = \Gamma^{-1}(\Sigma B_{1} - \varsigma_{r+1}C_{1}^{T}U),\\ \tilde{C} & = C_{1}\Sigma - \varsigma_{r+1}UB_{1}^{T},\\ \tilde{D} & = D + \varsigma_{r+1}U,\end{align}</math>
with <math>U = \left(C_{2}^{T}\right)^{\dagger}B_{2}</math> and <math>\Gamma = \Sigma^{2} - \varsigma_{r+1}^{2}I_{n-k}</math>.
6. Compute the additive decomposition
<math>\tilde{G}(s) = \tilde{C}(sI_{n-k} - \tilde{A})^{-1}\tilde{B} + \tilde{D} = G_{r}(s) + F(s),</math>
where <math>F</math> is anti-stable and <math>G_{r}</math> is the <math>r</math>-th order stable Hankel-norm approximation.

== References ==

<references>
<ref name="morBenQQ04">P. Benner, E. S. Quintana-Ortí, and G. Quintana-Ortí. Computing optimal Hankel norm approximations of large-scale systems. In 2004 43rd IEEE Conference on Decision and Control (CDC), volume 3, pages 3078-3083, Atlantis, Paradise Island, Bahamas, December 2004. Institute of Electrical and Electronics Engineers.</ref>
<ref name="morGlo84">K. Glover. All optimal Hankel-norm approximations of linear multivariable systems and their <math style="vertical-align: top;">L^{\infty}</math>-error norms. Internat. J. Control, 39(6):1115-1193, 1984.</ref>
</references>

2018-04-24T20:40:24Z

Werner: Added CFD formulation and linearization.

{{preliminary}} 

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:DAE order unspecified]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:MIMO]]
[[Category:Sparse]]

==Description: Active Control of a Supersonic Engine Inlet==

<figure id="fig1">[[File:Supersonic_Inlet1.png|750px|thumb|right|<caption>Steady-state Mach contours inside diffuser. Freestream Mach number
is 2.2.</caption>]]</figure>

This example considers unsteady flow through a supersonic diffuser as shown in <xr id="fig1"/>.
The diffuser operates at a nominal Mach number of 2.2, however it is subject to perturbations in the incoming flow, which may be due (for
example) to atmospheric variations.
In nominal operation, there is a strong shock downstream of the diffuser throat, as can be seen from the Mach contours
plotted in Figure <xr id="fig1"/>.
Incoming disturbances can cause the shock to move forward towards the throat. When the shock sits at the throat, the inlet
is unstable, since any disturbance that moves the shock slightly upstream will cause it to move forward rapidly, leading to unstart of the inlet. This is extremely undesirable, since unstart results in a large loss of thrust.
In order to prevent unstart from occurring, one option is to actively control the position of the shock.
This control may be effected through flow bleeding upstream of the diffuser throat.

A complete description of the benchmark can be downloaded as PDF file [https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Supersonic%20Engine%20Inlet%20%2838866%29/files/fileinnercontentproxy.2010-01-31.5794999947 here].

===Active Flow Control Setup===

<figure id="fig2">[[File:Supersonic_Inlet2.png|600px|thumb|right|<caption>Supersonic diffuser active flow control problem setup.</caption>]]</figure>

<xr id="fig2"/> presents the schematic of the actuation mechanism.
Incoming flow with possible disturbances enters the inlet and is sensed using pressure sensors.
The controller then adjusts the bleed upstream of the throat in order to control the position of the shock and to prevent it from moving upstream.
In simulations, it is difficult to automatically determine the shock location.
The average Mach number at the diffuser throat provides an appropriate surrogate that can be easily computed.
There are several transfer functions of interest in this problem.
The shock position will be controlled by monitoring the average Mach number at the diffuser throat.
The reduced-order model must capture the dynamics of this output in response to two inputs: the incoming flow disturbance and the bleed actuation.
In addition, total pressure measurements at the diffuser wall are used for sensing.

===CFD Formulation===

The unsteady, two-dimensional flow of an inviscid, compressible fluid is governed by the Euler equations.
The usual statements of mass, momentum, and energy can be written in integral form as

:<math>
\begin{align}
\frac{\partial}{\partial t}\iint\rho\mathrm{d}V + \oint\rho Q\cdot\mathrm{dA} & = 0,\\
\frac{\partial}{\partial t}\iint\rho Q\mathrm{d}V + \oint\rho Q (Q\cdot\mathrm{dA}) + \oint p \mathrm{dA} & = 0,\\
\frac{\partial}{\partial t}\iint\rho E\mathrm{d}V + \oint\rho H (Q\cdot\mathrm{dA}) + \oint p Q\cdot\mathrm{dA} & = 0,
\end{align}
</math>

where <math>\rho</math>, <math>Q</math>, <math>H</math>, <math>E</math>, and <math>p</math> denote density, flow velocity, total enthalpy, energy, and pressure, respectively.
The CFD formulation for this problem uses a finite volume method and is described fully in <ref name="lassaux2002"/> and <ref name="willcox2005"/>.
The unknown flow quantities used are the density, streamwise velocity component, normal velocity component, and enthalpy at each point in the computational grid.
Note that the local flow velocity components <math>q</math> and <math>q^{\perp}</math> are defined using a streamline computational grid that is computed for the steady-state solution.
<math>q</math> is the projection of the flow velocity on the meanline direction of the grid cell, and <math>q^{\perp}</math> is the normalto-meanline component.
To simplify the implementation of the integral energy equation, total enthalpy is also used in place of energy.
The vector of unknowns at each node <math>i</math> is therefore

:<math>
x_{i} = \begin{bmatrix} \rho_{i}, & q_{i}, & q^{\perp}_{i}, & H_{i} \end{bmatrix}^{T}.
</math>

Two physically different kinds of boundary conditions exist: inflow/outflow conditions, and conditions applied at a solid wall.
At a solid wall, the usual no-slip condition of zero normal flow velocity is easily applied as <math>q^{\perp} = 0</math>.
In addition, we will allow for mass addition or removal (bleed) at various positions along the wall.
The bleed condition is also easily specified.
We set

:<math>
q^{\perp} = \frac{\dot{m}}{\rho},
</math>

where <math>\dot{m}</math> is the specified mass flux per unit length along the bleed slot.
At inflow boundaries, Riemann boundary conditions are used.
For the diffuser problem considered here, all inflow boundaries are supersonic, and hence we impose inlet vorticity, entropy and Riemann’s invariants.
At the exit of the duct, we impose outlet pressure.

===Linearized CFD Matrices===

The two-dimensional integral Euler equations are linearized about the steadystate solution to obtain an unsteady system of the form

:<math>
\begin{align}
E\dot{x}(t) & = Ax(t) + Bu(t),\\
y(t) & = Cx(t),
\end{align}
</math>

The descriptor matrix <math>E</math> arises from the particular CFD formulation.
In addition, the matrix <math>E</math> contains some zero rows that are due to implementation of boundary conditions.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38866, see also <ref name="willcox2005"/>.

==Data==

The matrices are in [http://math.nist.gov/MatrixMarket/ Matrix Market] format [https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Supersonic%20Engine%20Inlet%20%2838866%29/files/fileinnercontentproxy.2010-01-31.5849953094 inlet.tar.gz].
The size of the file is 5.4 MB.
The matrix name is used as an extension of the matrix file.

==Dimensions==

System structure:

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

System dimensions:

<math>E \in \mathbb{R}^{11730 \times 11730}</math>,
<math>A \in \mathbb{R}^{11730 \times 11730}</math>,
<math>B \in \mathbb{R}^{11730 \times 2}</math>,
<math>C \in \mathbb{R}^{1 \times 11730}</math>

==References==

<references>

<ref name="korvink2005"> J.G. Korvink, E.B. Rudnyi, [https://doi.org/10.1007/3-540-27909-1_11 Oberwolfach Benchmark Collection], Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 311--315, 2005.</ref>

<ref name="lassaux2002">G. Lassaux. High-Fidelity Reduced-Order Aerodynamic Models: Application to Active Control of Engine Inlets. Master’s thesis, Dept. of Aeronautics and Astronautics, MIT, June 2002.</ref>

<ref name="willcox2005">K. Willcox , G. Lassaux, [https://doi.org/10.1007/3-540-27909-1_20 Model Reduction of an Actively Controlled Supersonic Diffuser]. In: Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 357--361, 2005.</ref>

</references>

Supersonic Engine Inlet

2018-04-24T20:09:31Z

Werner: Added description and setup.

Steel Profile

2018-04-24T15:21:06Z

Windscreen

2018-04-24T11:33:31Z

Werner: Updated BibTeX entry.

{{preliminary}} 

[[Category:benchmark]]
[[Category:Oberwolfach]]
[[Category:Linear]]
[[Category:Affine parameter representation]]
[[Category:Parametric]]
[[Category:Parametric 1 parameter]]
[[Category:SISO]]
[[Category:Sparse]]

==Description==
<figure id="fig1">[[File:Windscreen1.gif|490px|thumb|right|Figure 1]]</figure>
<figure id="fig2">[[File:Windscreen2.png|490px|thumb|right|Figure 2]]</figure>

This is an example for a model in the frequency domain of the form

:<math>
\begin{align}
K_d x - \omega^2 M x & = f \\
y & = f^* x
\end{align}
</math>

where <math>f</math> represents a unit point load in one unknown of the state vector.
<math>M</math> is a symmetric positive-definite matrix and <math>K_d = (1+i\gamma) K</math> where <math>K</math> is symmetric positive semi-definite.

The test problem is a structural model of a car windscreen. <ref name="meerbergen2007"/>
This is a 3D problem discretized with <math>7564</math> nodes and <math>5400</math> linear hexahedral elements (3 layers of <math>60 \times 30</math> elements).
The mesh is shown in <xr id="fig1"/>.
The material is glass with the following properties:
The Young modulus is <math>7\times10^{10}\mathrm{N}/\mathrm{m}^2</math>, the density is <math>2490 \mathrm{kg}/\mathrm{m}^3</math>, and the Poisson ratio is <math>0.23</math>. The natural damping is <math>10\%</math>, i.e. <math>\gamma=0.1</math>.
The structural boundaries are free (free-free boundary conditions).
The windscreen is subjected to a point force applied on a corner.
The goal of the model reduction is the fast evaluation of <math>y</math>.
Model reduction is used as a fast linear solver for a sequence of parametrized linear systems.

The discretized problem has dimension <math>n=22692</math>.
The goal is to estimate <math>x(\omega)</math> for <math>\omega\in[0.5,200]</math>.
In order to generate the plots the frequency range was discretized as <math>\{\omega_1,\ldots,\omega_m\} =
\{0.5j,j=1,\ldots,m\}</math> with <math>m=400</math>.

<xr id="fig1"/> shows the mesh of the car windscreen and <xr id="fig2"/> the frequency response <math>\vert \Re(y(\omega)) \vert</math>.

==Origin==

This benchmark is part of the '''Oberwolfach Benchmark Collection'''<ref name="korvink2005"/>; No. 38886.

==Data==

Download matrices in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format:

* [https://portal.uni-freiburg.de/imteksimulation/downloads/benchmark/Windscreen%20%2838886%29/files/fileinnercontentproxy.2010-02-26.3407583176 windscreen.tar.gz] (21.5 MB)

The archive contains files <tt>windscreen.K</tt>, <tt>windscreen.M</tt> and <tt>windscreen.B</tt> representing <math>K_d</math>, <math>-M</math> and <math>f</math> accordingly.

==Dimensions==

System structure:
:<math>
\begin{align}
(K + \omega^2 M) x & = B \\
y & = B^{\mathrm{T}} x
\end{align}
</math>
with <math>\omega \in [0.5, 200]</math>.

System dimensions:

<math>K \in \mathbb{C}^{22692 \times 22692}</math>,
<math>M \in \mathbb{R}^{22692 \times 22692}</math>,
<math>B \in \mathbb{R}^{22692 \times 1}</math>.

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

* For the benchmark itself and its data:
:: Oberwolfach Benchmark Collection, '''Windscreen'''. hosted at MORwiki - Model Order Reduction Wiki, 2004. http://modelreduction.org/index.php/Windscreen

@MISC{morwiki_windscreen,
author = <nowiki>{{Oberwolfach Benchmark Collection}}</nowiki>,
title = {Windscreen},
howpublished = {hosted at {MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Windscreen}</nowiki>,
year = 2004
}

* For the background on the benchmark:

@article{Mee,
author = {K. Meerbergen},
title = {Fast frequency response computation for Rayleigh damping},
journal = {International Journal for Numerical Methods in Engineering},
volume = 73,
number = 1,
pages = {96--106},
year = 2007,
doi = {10.1002/nme.2058},
}

==References==

<references>

<ref name="korvink2005"> J.G. Korvink, E.B. Rudnyi, [https://doi.org/10.1007/3-540-27909-1_11 Oberwolfach Benchmark Collection], In: Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol 45: 311--315, 2005.</ref>

<ref name="meerbergen2007"> K. Meerbergen, [https://doi.org/10.1002/nme.2058 Fast frequency response computation for Rayleigh damping], International Journal for Numerical Methods in Engineering, '''73'''(1): 96--106, 2007.</ref>

</references>

2018-04-24T08:59:12Z

Werner: /* Citation */

{{preliminary}} 

[[Category:benchmark]]
[[Category:linear]]
[[Category:second differential order]]
[[Category:SISO]]

==Description==

The '''flexible space structure''' benchmark <ref name="gawronski90"/>,<ref name="gawronski91"/>,<ref name="gawronski96"/> is a procedural modal model which represents structural dynamics with a selectable number actuators and sensors. This model is used for truss structures in space environments i.e. the COFS-1 (Control of Flexible Structures) mast flight experiment <ref name="horner86"/>,<ref name="horta86"/>.

===Model===

In modal form the '''flexible space structure''' model for <math>K</math> modes, <math>M</math> actuators and <math>Q</math> sensors is of second order and given by:

:<math>\ddot{\nu}(t) = (2 \xi \circ \omega) \circ \dot{\nu}(t) + (\omega \circ \omega) \circ \nu = Bu(t)</math>

:<math>y(t) = C_r\dot{\nu}(t) + C_d\nu(t)</math>

with the parameters <math>\xi \in \mathbb{R}_{>0}^K</math> (damping ratio), <math>\omega \in \mathbb{R}_{>0}^K</math> (natural frequency) and using the Hadamard product <math>\circ</math>.
The first order representation follows for <math>x(t) = (\dot{\nu}(t), \omega_1\nu_1, \dots, \omega_K\nu_K)</math> by:

:<math>\dot{x}(t) = Ax(t) + Bu(t) </math>

:<math>y(t) = Cx(t)</math>

with the matrices:

:<math>A := \begin{pmatrix} A_1 & & \\ & \ddots & \\ & & A_K \end{pmatrix}, \; B := \begin{pmatrix} B_1 \\ \vdots \\ B_K \end{pmatrix}, \; C := \begin{pmatrix} C_1 & \dots & C_K \end{pmatrix}, </math>

and their components:

:<math>A_k := \begin{pmatrix} -2\xi_k\omega_k & -\omega_k \\ \omega_k & 0 \end{pmatrix}, \; B_k := \begin{pmatrix} b_k \\ 0 \end{pmatrix}, \; C_k := \begin{pmatrix} c_{rk} & \frac{c_{dk}}{\omega_k} \end{pmatrix},</math>

where <math>b_k \in \mathbb{R}^{1 \times M}</math> and <math>c_{rk}, c_{dk} \in \mathbb{R}^{Q \times 1}</math>.

===Benchmark Specifics===

For this benchmark the system matrix is block diagonal and thus chosen to be sparse.
The parameters <math>\xi</math> and <math>\omega</math> are sampled from a uniform random distributions <math>\mathcal{U}_{[0,\frac{1}{1000}]}^K</math> and <math>\mathcal{U}_{[0,100]}^K</math> respectively.
The components of the input matrix <math>b_k</math> are sampled form a uniform random distribution <math>\mathcal{U}_{[0,1]}</math>,
while the output matrix <math>C</math> is sampled from a uniform random distribution <math>\mathcal{U}_{[0,10]}</math> completely w.l.o.g, since if the components of <math>C_d</math> are random their scaling can be ignored.

==Data==

The following Matlab code assembles the above described <math>A</math>, <math>B</math> and <math>C</math> matrix for a given number of modes <math>K</math>, actuators (inputs) <math>M</math> and sensors (outputs) <math>Q</math>.

<div class="thumbinner" style="width:540px;text-align:left;">
<source lang="matlab">
function [A,B,C] = fss(K,M,Q)

rand('seed',1009);
xi = rand(1,K)*0.001; % Sample damping ratio
omega = rand(1,K)*100.0; % Sample natural frequencies

A_k = cellfun(@(p) sparse([-2.0*p(1)*p(2),-p(2);p(2),0]), ...
num2cell([xi;omega],1),'UniformOutput',0);

A = blkdiag(A_k{:});

B = kron(rand(K,M),[1;0]);

C = 10.0*rand(Q,2*K);
end
</source>
</div>

==Dimensions==

System structure:
:<math>
\begin{align}
\dot{x}(t) &= Ax(t) + Bu(t) \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{2K \times 2K}</math>,
<math>B \in \mathbb{R}^{2K \times M}</math>,
<math>C \in \mathbb{R}^{Q \times 2K}</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Flexible Space Structures'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Flexible_Space_Structures

@MISC{morwiki-flexspacstruc,
author = <nowiki>{{The MORwiki Community}}</nowiki>,
title = {Flexible Space Structures},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Flexible_Space_Structures}</nowiki>,
year = 2018
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morGawW91 morGawW91] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morGawW91 BibTeX])

==Reference==

<references>

<ref name="gawronski90">W. Gawronski and J.N. Juang. "[https://doi.org/10.1016/B978-0-12-012736-8.50010-3 Model Reduction for Flexible Structures]", Control and Dynamic Systems, 36: 143--222, 1990.</ref>

<ref name="gawronski91">W. Gawronski and T. Williams, "[http://doi.org/10.2514/3.20606 Model Reduction for Flexible Space Structures]", Journal of Guidance 14(1): 68--76, 1991</ref>

<ref name="gawronski96">W. Gawronski. "[https://doi.org/10.1007/3540760172_4 Model reduction]". In: Balanced Control of Flexible Structures. Lecture Notes in Control and Information Sciences, vol 211: 45--106, 1996.</ref>

<ref name="horner86">G.C. Horner. "[https://ntrs.nasa.gov/search.jsp?R=19870006596 COFS-1 Research Overview]". NASA / DOD Control Structures Interaction Technology: 233--251, 1986</ref>

<ref name="horta86">L.G. Horta, J.L. Walsh, G.C. Horner and J.P. Bailey. "[https://ntrs.nasa.gov/search.jsp?R=19870006613 Analysis and simulation of the MAST (COFS-1 flight hardware)]". NASA / DOD Control Structures Interaction Technology: 515--532, 1986.</ref>

</references>

==Contact==

''[[User:Himpe|Christian Himpe]]''

Flexible Space Structures

2018-04-24T08:58:40Z

Werner: Updated BibTeX entry.

{{preliminary}} 

[[Category:benchmark]]
[[Category:linear]]
[[Category:second differential order]]
[[Category:SISO]]

==Description==

The '''flexible space structure''' benchmark <ref name="gawronski90"/>,<ref name="gawronski91"/>,<ref name="gawronski96"/> is a procedural modal model which represents structural dynamics with a selectable number actuators and sensors. This model is used for truss structures in space environments i.e. the COFS-1 (Control of Flexible Structures) mast flight experiment <ref name="horner86"/>,<ref name="horta86"/>.

===Model===

In modal form the '''flexible space structure''' model for <math>K</math> modes, <math>M</math> actuators and <math>Q</math> sensors is of second order and given by:

:<math>\ddot{\nu}(t) = (2 \xi \circ \omega) \circ \dot{\nu}(t) + (\omega \circ \omega) \circ \nu = Bu(t)</math>

:<math>y(t) = C_r\dot{\nu}(t) + C_d\nu(t)</math>

with the parameters <math>\xi \in \mathbb{R}_{>0}^K</math> (damping ratio), <math>\omega \in \mathbb{R}_{>0}^K</math> (natural frequency) and using the Hadamard product <math>\circ</math>.
The first order representation follows for <math>x(t) = (\dot{\nu}(t), \omega_1\nu_1, \dots, \omega_K\nu_K)</math> by:

:<math>\dot{x}(t) = Ax(t) + Bu(t) </math>

:<math>y(t) = Cx(t)</math>

with the matrices:

:<math>A := \begin{pmatrix} A_1 & & \\ & \ddots & \\ & & A_K \end{pmatrix}, \; B := \begin{pmatrix} B_1 \\ \vdots \\ B_K \end{pmatrix}, \; C := \begin{pmatrix} C_1 & \dots & C_K \end{pmatrix}, </math>

and their components:

:<math>A_k := \begin{pmatrix} -2\xi_k\omega_k & -\omega_k \\ \omega_k & 0 \end{pmatrix}, \; B_k := \begin{pmatrix} b_k \\ 0 \end{pmatrix}, \; C_k := \begin{pmatrix} c_{rk} & \frac{c_{dk}}{\omega_k} \end{pmatrix},</math>

where <math>b_k \in \mathbb{R}^{1 \times M}</math> and <math>c_{rk}, c_{dk} \in \mathbb{R}^{Q \times 1}</math>.

===Benchmark Specifics===

For this benchmark the system matrix is block diagonal and thus chosen to be sparse.
The parameters <math>\xi</math> and <math>\omega</math> are sampled from a uniform random distributions <math>\mathcal{U}_{[0,\frac{1}{1000}]}^K</math> and <math>\mathcal{U}_{[0,100]}^K</math> respectively.
The components of the input matrix <math>b_k</math> are sampled form a uniform random distribution <math>\mathcal{U}_{[0,1]}</math>,
while the output matrix <math>C</math> is sampled from a uniform random distribution <math>\mathcal{U}_{[0,10]}</math> completely w.l.o.g, since if the components of <math>C_d</math> are random their scaling can be ignored.

==Data==

The following Matlab code assembles the above described <math>A</math>, <math>B</math> and <math>C</math> matrix for a given number of modes <math>K</math>, actuators (inputs) <math>M</math> and sensors (outputs) <math>Q</math>.

<div class="thumbinner" style="width:540px;text-align:left;">
<source lang="matlab">
function [A,B,C] = fss(K,M,Q)

rand('seed',1009);
xi = rand(1,K)*0.001; % Sample damping ratio
omega = rand(1,K)*100.0; % Sample natural frequencies

A_k = cellfun(@(p) sparse([-2.0*p(1)*p(2),-p(2);p(2),0]), ...
num2cell([xi;omega],1),'UniformOutput',0);

A = blkdiag(A_k{:});

B = kron(rand(K,M),[1;0]);

C = 10.0*rand(Q,2*K);
end
</source>
</div>

==Dimensions==

System structure:
:<math>
\begin{align}
\dot{x}(t) &= Ax(t) + Bu(t) \\
y(t) &= Cx(t)
\end{align}
</math>

System dimensions:

<math>A \in \mathbb{R}^{2K \times 2K}</math>,
<math>B \in \mathbb{R}^{2K \times M}</math>,
<math>C \in \mathbb{R}^{Q \times 2K}</math>.

==Citation==

To cite this benchmark, use the following references:

* For the benchmark itself and its data:
::The MORwiki Community, '''Flexible Space Structures'''. MORwiki - Model Order Reduction Wiki, 2018. http://modelreduction.org/index.php/Flexible_Space_Structures

@MISC{morwiki-flexspacstruc,
author = <nowiki>{{The MORwiki} Community}}</nowiki>,
title = {Flexible Space Structures},
howpublished = {{MORwiki} -- Model Order Reduction Wiki},
url = <nowiki>{http://modelreduction.org/index.php/Flexible_Space_Structures}</nowiki>,
year = 2018
}

* For the background on the benchmark: [https://morwiki.mpi-magdeburg.mpg.de/BibTeX/#morGawW91 morGawW91] ([https://morwiki.mpi-magdeburg.mpg.de/BibTeX/html/mor_bib.html#morGawW91 BibTeX])

==Reference==

<references>

<ref name="gawronski90">W. Gawronski and J.N. Juang. "[https://doi.org/10.1016/B978-0-12-012736-8.50010-3 Model Reduction for Flexible Structures]", Control and Dynamic Systems, 36: 143--222, 1990.</ref>

<ref name="gawronski91">W. Gawronski and T. Williams, "[http://doi.org/10.2514/3.20606 Model Reduction for Flexible Space Structures]", Journal of Guidance 14(1): 68--76, 1991</ref>

<ref name="gawronski96">W. Gawronski. "[https://doi.org/10.1007/3540760172_4 Model reduction]". In: Balanced Control of Flexible Structures. Lecture Notes in Control and Information Sciences, vol 211: 45--106, 1996.</ref>

<ref name="horner86">G.C. Horner. "[https://ntrs.nasa.gov/search.jsp?R=19870006596 COFS-1 Research Overview]". NASA / DOD Control Structures Interaction Technology: 233--251, 1986</ref>

<ref name="horta86">L.G. Horta, J.L. Walsh, G.C. Horner and J.P. Bailey. "[https://ntrs.nasa.gov/search.jsp?R=19870006613 Analysis and simulation of the MAST (COFS-1 flight hardware)]". NASA / DOD Control Structures Interaction Technology: 515--532, 1986.</ref>

</references>

==Contact==

''[[User:Himpe|Christian Himpe]]''