MATLAB Examples

Contents

BRBT for descriptor systems

This demo script contains the application of the generalized bounded-real balanced truncation method (ml_brbt_dss) on a test bounded-real descriptor system with finite stable and infinite eigenvalues of the form

$$
\setlength\arraycolsep{2pt}
\begin{array}{rl}
E\dot{x}(t) & = Ax(t) + Bu(t),\\
y(t) & = Cx(t) + Du(t).
\end{array}
$$

After loading the demo data, the optional parameters are assigned here explicitly and the ml_brbt_dss function is called with its different input interfaces. The dss object version is only called if the System Control Toolbox (Matlab) or the Control Package (Octave) is installed/loaded.

To show the performance of the model reduction method, the sigma error of the full-order and reduced-order model is plotted. In case the System Control Toolbox is installed, also a bode magnitude plot of the error system is shown.

%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU Affero General Public License as published
% by the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
% GNU Affero General Public License for more details.
%
% You should have received a copy of the GNU Affero General Public License
% along with this program.  If not, see <http://www.gnu.org/licenses/>.
%
% Copyright (C) 2006-2017 Peter Benner, Steffen W. R. Werner
%

Initialization

For demonstration, a random bounded-real descriptor system example was generated by the script morlab_data_desc_br.m and saved in morlab_data_desc_br.mat. The number of finite stable, finite unstable and infinite eigenvalues as well as the complete size of the system is saved in the datainfo structure.

if exist('OCTAVE_VERSION', 'builtin')
    warning('off', 'Octave:data-file-in-path');
    load morlab_data_desc_br.mat;
    warning('on', 'Octave:data-file-in-path');
else
    load morlab_data_desc_br.mat;
end

% Get information about installed/loaded toolboxes.
hasControlPkg = size(license('inuse', 'control'), 2);
hasControlTbx = license('test', 'control_toolbox');

Construction of the descriptor system structure

To test the different input-output formats, the struct and state-space object shapes of the descriptor system are formulated here.

sys_struct = struct('A', A, 'B', B, 'C', C, 'D', D, 'E', E);

if hasControlPkg || hasControlTbx
    sys_ss = dss(A, B, C, D, E);
end

Set of optional parameters

The default values are mainly taken here, which can be modified. Alternative values depending on the system are commented out. Usually for using default values the corresponding parameters are not set or empty. Also, the function call "opts = ml_morlabopts('ml_brbt_dss')" generates an empty option struct of the following form.

% Option struct for generalized discrete-time Lyapunov equation solver.
gdlyapopts = struct(...
    'AbsTol' , 0, ...
    'Index'  , Inf, ... % Index = 3
    'Info'   , 0, ... % Info = 1
    'MaxIter', 100, ...
    'RelTol' , 1.0e+01 * datainfo.ni * eps);

% Option struct for decomposition of infinite part.
infdecopts = struct(...
    'AbsTol'   , 0, ...
    'Dimension', -1, ... % Dimension = datainfo.ni
    'Info'     , 0, ... % Info = 1
    'MaxIter'  , 100, ...
    'RelTol'   , 1.0e+02 * datainfo.n * eps, ...
    'RankTol'  , log(datainfo.n) * eps);

% Option struct for Lyapunov solver inside the Riccati solver.
lyapopts = struct(...
    'AbsTol' , 0, ...
    'CompTol', log(datainfo.np) * eps, ...
    'Info'   , 0, ... % Info = 1
    'MaxIter', 100, ...
    'RelTol' , 1.0e+01 * datainfo.np * eps);

% Option struct for Riccati equation solver.
pcareopts = struct(...
    'AbsTol'  , 0, ...
    'CompTol' , log(datainfo.np) * eps, ...
    'Info'    , 0, ... % Info = 1
    'lyapopts', lyapopts, ...
    'MaxIter' , 100, ...
    'RelTol'  , 1.0e+02 * datainfo.np * eps);

% Option struct for the complete function.
opts = struct(...
    'DecompTol'       , log(datainfo.n) * eps, ...
    'gdlyapopts'      , gdlyapopts, ...
    'ImproperTrunc'   , log(datainfo.n) * eps, ...
    'Index'           , Inf, ... % Index = 3
    'infdecopts'      , infdecopts, ...
    'Method'          , 'sr', ... % Method = 'bfsr'
    'Order'           , 10, ...
    'OrderComputation', 'tolerance', ... % OrderComputation = 'order'
    'pcareopts'       , pcareopts, ...
    'Tolerance'       , 1.0e-02);

Application of the function

Here the application of the ml_brbt_dss function is shown for different interfaces and input-data. The default calls are commented out.

% Application with single matrices.
% [Ar, Br, Cr, Dr, Er, info] = ml_brbt_dss(A, B, C, D, E);
[Ar, Br, Cr, Dr, Er, info] = ml_brbt_dss(A, B, C, D, E, opts);

% Application with structure.
% [rom_struct, info_struct] = ml_brbt_dss(sys_struct);
[rom_struct, info_struct] = ml_brbt_dss(sys_struct, opts);

% Application with state-space object.
if hasControlPkg || hasControlTbx
    % [rom_ss, info_ss] = ml_brbt_dss(sys_ss);
    [rom_ss, info_ss] = ml_brbt_dss(sys_ss, opts);
end

Report

As visualization, a sigmaplot of the error system is made for the descriptor system structures and a bode magnitude plot for the state- space objects.

% Sigmaplot of the error system.
figure;
ml_sigmaplot(sys_struct, rom_struct, -4, 4, 100, info.AbsErrBound, 'b.');
legend('Error bound', 'Sigma error');
title({'BRBT\_DSS (sigmaplot, error system)'; ...
    ['Full order = ' int2str(size(A, 1)) '; ' ...
    'Reduced-order = ' int2str(size(Ar, 1))]});

if hasControlTbx
    % Bode magnitude plot of the error system.
    bodeopts          = bodeoptions('cstprefs');
    bodeopts.MagUnits = 'abs';

    figure;
    bodemag(sys_ss - rom_ss, bodeopts);
    title({'BRBT\_DSS (Bode magnitude plot, error system)'; ...
        ['Full order = ' int2str(size(A, 1)) '; ' ...
        'Reduced-order = ' int2str(size(Ar, 1))]});
end