Clamped Beam

Clamped Beam

Background
Benchmark ID	clampedBeam_n348m1q1
Category	slicot
System-Class	LTI-FOS
Parameters
nstates	348
ninputs	1
noutputs	1
nparameters	0
components	A, B, C
Copyright
License	NA
Creator	Christian Himpe
Editor	Christian Himpe Petar Mlinarić
Location
NA

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.float64)

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

\[ \begin{align} \ddot{q}(t) + E_{so} \dot{q}(t) + a K_{so} q(t) &= a B_{so} u(t), \\ y(t) &= C_{so} q(t), \end{align} \]

as

\[ \begin{align} A &= \begin{pmatrix} 0 & a I \\ -K_{so} & -E_{so} \end{pmatrix}, \\ B &= \begin{pmatrix} 0 \\ B_{so} \end{pmatrix}, \\ C &= \begin{pmatrix} C_{so} & 0 \end{pmatrix}, \end{align} \]

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:

\[ \begin{align} \dot{x}(t) &= A x(t) + B u(t) \\ y(t) &= C x(t) \end{align} \]

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:

\[ \begin{align} \ddot{x}(t) + E \dot{x}(t) + K x(t) &= B u(t) \\ y(t) &= C_p x(t) \end{align} \]

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}
}