MATLAB Examples

Contents

LQGBT for standard systems

This demo script contains the application of the LQG balanced truncation method (ml_lqgbt) on a test standard system with stable and unstable eigenvalues of the form

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

After loading the demo data, the optional parameters are assigned here explicitly and the ml_lqgbt function is called with its different input interfaces. The ss 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. And if the System Control Toolbox and the Robust Control toolbox are installed, a sigma error and bode magnitude plot of the error system resulting of the coprime factorization is shown with the corresponding error bound.

%
% 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 standard system example was generated by the script morlab_data_std_unstab.m and saved in morlab_data_std_unstab.mat. The number of stable and unstable 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_std_unstab.mat;
    warning('on', 'Octave:data-file-in-path');
else
    load morlab_data_std_unstab.mat;
end

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

Construction of the standard system structure

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

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

if hasControlPkg || hasControlTbx
    sys_ss = ss(A, B, C, D);
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_lqgbt')" generates an empty option struct of the following form.

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

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

% Option struct for Lyapunov or Bernoulli equation solver.
stabmethodopts = struct(...
    'AbsTol' , 0, ...
    'Info'   , 0, ... % Info = 1
    'MaxIter', 100, ...
    'RelTol' , 1.0e+01 * datainfo.n * eps);

% Option structs for the decomposition of unstable part.
stabsignmopts = struct(...
    'AbsTol' , 0, ...
    'Info'   , 0, ... % Info = 1
    'MaxIter', 100, ...
    'RelTol' , 1.0e+02 * datainfo.n * eps);

% Option struct for the complete function.
opts = struct(...
    'Beta'            , 0.1, ...
    'careopts'        , careopts, ...
    'Method'          , 'sr', ... % Method = 'bfsr'
    'Order'           , 10, ...
    'OrderComputation', 'tolerance', ... % OrderComputation = 'order'
    'stabmethodopts'  , stabmethodopts, ...
    'stabsignmopts'   , stabsignmopts, ...
    'StabMethod'      , 'abe', ... % StabMethod = 'lyap'
    'Tolerance'       , 1.0e-02, ...
    'UnstabDim'       , -1); % UnstabDim = 8

Application of the function

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

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

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

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

Report

As visualization, a sigmaplot of the error system of the (right) coprime factors is made for the standard 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, 200);
legend('Sigma error');
title({'LQGBT (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({'LQGBT (Bode magnitude plot, error system)'; ...
        ['Full order = ' int2str(size(A, 1)) '; ' ...
        'Reduced-order = ' int2str(size(Ar, 1))]});

    if hasRobustTbx
        % Coprime factorization.
        [~, sysINFO] = ncfmr(sys_ss, size(sys_ss.a, 1));
        [~, romINFO] = ncfmr(rom_ss, size(rom_ss.a, 1));

        sys_GR = struct(...
            'A', sysINFO.GR.A, ...
            'B', sysINFO.GR.B, ...
            'C', sysINFO.GR.C, ...
            'D', sysINFO.GR.D);

        rom_GR = struct(...
            'A', romINFO.GR.A, ...
            'B', romINFO.GR.B, ...
            'C', romINFO.GR.C, ...
            'D', romINFO.GR.D);

        % Sigmaplot of the coprime error system.
        figure;
        ml_sigmaplot(sys_GR, rom_GR, -4, 4, 100, info.AbsErrBound, 'b.');
        legend('Error bound', 'Sigma error of coprime factors');
        title({'LQGBT (sigmaplot, coprime error system)'; ...
            ['Full order = ' int2str(size(A, 1)) '; ' ...
            'Reduced-order = ' int2str(size(Ar, 1))]});

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

        figure;
        bodemag(sysINFO.GR - romINFO.GR, bodeopts);
        title({'LQGBT (Bode magnitude plot, coprime error system)'; ...
            ['Full order = ' int2str(size(A, 1)) '; ' ...
            'Reduced-order = ' int2str(size(Ar, 1))]});
    end
end