Penzl's FOM

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:

$\begin{array}{rcl} A &=& \begin{pmatrix} A_1 \\ & A_2 \\ & & A_3 \\ & & & A_4 \end{pmatrix}, \\ A_1 &=& \begin{pmatrix} -1 & 100 \\ -100 & -1 \end{pmatrix}, \; A_2 = \begin{pmatrix} -1 & 200 \\ -200 & -1 \end{pmatrix}, \; A_3 = \begin{pmatrix} -1 & 400 \\ -400 & -1 \end{pmatrix}, \; A_4 = \begin{pmatrix} -1 \\ & -2 \\ & & \ddots \\ & & & -1000 \end{pmatrix}, \\ B &=& \begin{pmatrix} 10 & 10 & 10 & 10 & 10 & 10 & 1 & \dots & 1 \end{pmatrix}^T, \\ C &=& B^T. \end{array}$

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 $A_1 = \begin{pmatrix} -1 & p \\ -p & -1 \end{pmatrix}.$

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}
}