Penzl's FOM

Penzl's FOM

Background
Benchmark ID	penzlFOM_n1006m1q1
Category	slicot
System-Class	LTI-FOS
Parameters
nstates	1006
ninputs	1
noutputs	1
nparameters	0
components	A, B, C
Copyright
License	NA
Creator	User:Himpe
Editor	User:Himpe User:Yue
Location
NA

Description

This benchmark is an artificial example system of order $1006$ from also listed in . It has long been regarded as a standard "full order model" (FOM) for testing new methods.

The benchmark system consists of the following system components:

$\begin{array}{rcl}A& =& \left(\begin{array}{c}{A}_1\\ & A_2\\ & & A_3\\ & & & A_4\end{array}\right),\\ A_1& =& \left(\begin{array}{cc}-1& 100\\ -100& -1\end{array}\right),A_2=\left(\begin{array}{cc}-1& 200\\ -200& -1\end{array}\right),A_3=\left(\begin{array}{cc}-1& 400\\ -400& -1\end{array}\right),A_4=\left(\begin{array}{c}-1\\ & -2\\ & & \ddots \\ & & & -1000\end{array}\right),\\ B& =& {\left(\begin{array}{ccccccccc}10& 10& 10& 10& 10& 10& 1& \dots & 1\end{array}\right)}^T,\\ C& =& B^T.\end{array}$

This system is a theoretical construct, but features a non-smooth Bode plot with three spikes.

MIMO Variant

In a MIMO variant of this benchmark is utilized by adding random vectors to $B$ and $C$.

Parametric Variant

In , a parametric variant of this benchmark is formulated by redefining $A_1=\left(\begin{array}{cc}-1& p\\ -p& -1\end{array}\right).$

Origin

This benchmark is part of the SLICOT Benchmark Examples for Model Reduction.

Data

The system matrices $A$, $B$, $C$ are available from the SLICOT benchmarks page: fom.zip and are stored as MATLAB .mat file.

Dimensions

System structure:

$\begin{array}{rcl}\dot{x}(t)& =& Ax(t)+Bu(t)\\ y(t)& =& Cx(t)\end{array}$

System dimensions:

$A\in {\mathbb{ℝ}}^{1006\times 1006}$, $B\in {\mathbb{ℝ}}^{1006\times 1}$, $C\in {\mathbb{ℝ}}^{1\times 1006}$,

Citation

To cite this benchmark, use the following references:

• For the benchmark itself and its data:

Niconet e.V., SLICOT - Subroutine Library in Systems and Control Theory, http://www.slicot.org

@MANUAL{slicot_fom,
 title =        {{SLICOT} - Subroutine Library in Systems and Control Theory},
 organization = {Niconet e.V.}
 address =      {\url{http://www.slicot.org}},
 key =          {SLICOT}
}

• For the background on the benchmark:

@ARTICLE{morPen06,
 author =       {T. Penzl},
 title =        {Algorithms for Model Reduction of Large Dynamical Systems},
 journal =      {Linear Algebra and its Application},
 volume =       {415},
 number =       {2--3},
 pages =        {322--343},
 year =         {2006},
 doi =          {10.1016/j.laa.2006.01.007}
}