Building Model

Description: Motion Problem in a Building

This benchmark models the displacement of a multi-story building for example during an Earthquake. More details can be found in and , .

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: build.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('build.mat')
A = mat['A'].toarray()
B = mat['B']
C = mat['C'].astype(np.float_)

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

$\begin{array}{rl}\ddot{q}(t)+E_{so}\dot{q}(t)+K_{so}q(t)& =B_{so}u(t),\\ y(t)& =C_{so}\dot{q}(t),\end{array}$

as

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

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

n = 48
n2 = n // 2

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

Eso = -A[n2:, n2:]
Kso = -A[n2:, :n2]
Bso = 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:

$A\in {\mathbb{ℝ}}^{48\times 48}$, $B\in {\mathbb{ℝ}}^{48\times 1}$, $C\in {\mathbb{ℝ}}^{1\times 48}$.

Second differential order

System structure:

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

System dimensions:

$E,K\in {\mathbb{ℝ}}^{24\times 24}$, $B\in {\mathbb{ℝ}}^{24\times 1}$, $C_v\in {\mathbb{ℝ}}^{1\times 24}$.

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_build,
 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}
}