Convection Reaction

Convection Reaction

Background
Benchmark ID	convectionReaction_n84m1q1
Category	slicot
System-Class	LTI-FOS
Parameters
nstates	84
ninputs	1
noutputs	1
nparameters	0
components	A, B, C
Copyright
License	NA
Creator	User:Himpe
Editor	User:Himpe User:Mlinaric
Location
NA

Description

This benchmark models a chemical reaction by a convection-reaction partial differential equation on the unit square, given by:

$\frac{\partial x}{\partial t}=\frac{{\partial }^{2}x}{\partial y^2}+\frac{{\partial }^{2}x}{\partial z^2}+20\frac{\partial x}{\partial z}-180x+f(y,z)x(t),$

with Dirichlet boundary conditions and discretized with centered difference approximation.

The input vector $B$ is composed of random elements and the output vector equals the input vector $C=B^T$.

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

Here is Python code for loading the matrices ($A$ is a sparse matrix of 16-bit integers and $B$ and $C$ are full matrices stored as sparse matrices):

import numpy as np
from scipy.io import loadmat

mat = loadmat('build.mat')
A = mat['A'].astype(np.float64)
B = mat['B'].toarray()
C = mat['C'].toarray()

Dimensions

System structure:

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

System dimensions:

$A\in {\mathbb{ℝ}}^{84\times 84}$, $B\in {\mathbb{ℝ}}^{84\times 1}$, $C\in {\mathbb{ℝ}}^{1\times 84}$.

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_pde,
 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{Saa88,
 author =       {Y. Saad},
 title =        {Projection and deflation method for partial pole assignment in linear state feedback},
 journal =      {IEEE Transactions on Automatic Control},
 volume =       {33},
 number =       {3},
 pages =        {290--297},
 year =         {1988},
 doi =          {10.1109/9.406}
}