Clamped Beam

Description: Clamped Beam Model

This benchmark models a cantilever beam, which is a beam clamped on one end. More details can be found in and , .

For larger beam-type benchmarks see the linear 1d beam and electrostatic beam benchmarks.

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: beam.zip and are stored as MATLAB .mat file.

Here is Python code for loading the matrices ($A$ is stored as a sparse matrix that is mostly full and $C$ is stored as an array of 8-bit unsigned integers):

import numpy as np
from scipy.io import loadmat

mat = loadmat('beam.mat')
A = mat['A'].toarray()
B = mat['B']
C = mat['C'].astype(np.float_)

The $(A,B,C)$ represents a second-order system

$\begin{array}{rl}\ddot{q}(t)+E_{so}\dot{q}(t)+a{K}_{so}q(t)& =a{B}_{so}u(t),\\ y(t)& =C_{so}q(t),\end{array}$

as

$\begin{array}{rl}A& =\left(\begin{array}{cc}0& aI\\ -K_{so}& -E_{so}\end{array}\right),\\ B& =\left(\begin{array}{c}0\\ B_{so}\end{array}\right),\\ C& =\left(\begin{array}{cc}{C}_{so}& 0\end{array}\right),\end{array}$

where $a\approx 21.3896$.

Here is Python code for checking the structure and extracting the second-order matrices:

n = 348
n2 = n // 2

assert np.all(A[:n2, :n2] == 0)
assert np.all(A[:n2, n2:] == A[0, n2] * np.eye(n2))
assert np.all(B[:n2] == 0)
assert np.all(C[:, :n2] == 0)

a = A[0, n2]
Eso = -A[n2:, n2:]
Kso = -a * A[n2:, :n2]
Bso = a * B[n2:]
Cso = C[:, :n2]

Dimensions

First differential order

System structure:

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

System dimensions:

$A\in {\mathbb{ℝ}}^{348\times 348}$, $B\in {\mathbb{ℝ}}^{348\times 1}$, $C\in {\mathbb{ℝ}}^{1\times 348}$.

Second differential order

System structure:

$\begin{array}{rl}\ddot{x}(t)+E\dot{x}(t)+Kx(t)& =Bu(t)\\ y(t)& =C_p\dot{x}(t)\end{array}$

System dimensions:

$E,K\in {\mathbb{ℝ}}^{174\times 174}$, $B\in {\mathbb{ℝ}}^{174\times 1}$, $C_p\in {\mathbb{ℝ}}^{1\times 174}$.

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_beam,
 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{morAntSG01,
 author =       {A.C. Antoulas, D.C. Sorensen and S. Gugercin},
 title =        {A survey of model reduction methods for large-scale systems},
 journal =      {Contemporary Mathematics},
 volume =       {280},
 pages =        {193--219},
 year =         {2001},
 doi =          {10.1090/conm/280}
}