morlab_demo_time_evaluation

MORLAB Demo: Time Evaluation and Visualization
General Handling
Multiple Systems and Error Plots
Remarks

MORLAB Demo: Time Evaluation and Visualization

This demo script contains the handling of the time evaluation and visualization functions (simulations).

See also ml_ct_dss_simulate_ie, ml_ct_dss_simulate_ss11, ml_ct_dss_simulate_ss21, ml_ct_soss_simulate_ie, ml_ct_soss_simulate_ss22, ml_ct_ss_simulate_ie, ml_ct_ss_simulate_ss11, ml_ct_ss_simulate_ss21, ml_dt_dss_simulate, ml_dt_ss_simulate.

%
% This file is part of the MORLAB toolbox
% (https://www.mpi-magdeburg.mpg.de/projects/morlab).
% Copyright (C) 2006-2023 Peter Benner, Jens Saak, and Steffen W. R. Werner
% All rights reserved.
% License: BSD 2-Clause License (see COPYING)
%

General Handling

For each supported system type there are time simulation routines in MORLAB:

continuous-time standard systems: implicit Euler ml_ct_ss_simulate_ie, SS11 ml_ct_ss_simulate_ss11, SS21 ml_ct_ss_simulate_ss21
continuous-time descriptor systems: implicit Euler ml_ct_dss_simulate_ie, SS11 ml_ct_dss_simulate_ss11, SS21 ml_ct_dss_simulate_ss21
continous-time second-order systems: implicit Euler ml_ct_soss_simulate_ie, SS22 ml_ct_soss_simulate_ss22
discrete-time standard systems: ml_dt_ss_simulate
discrete-time descriptor systems: ml_dt_dss_simulate

For a better demonstration, we load a prepared datafile containing matrices of a standard contractive state-space system:

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

For using the plot functions, we need to put the system matrices either into a struct or, if the System Control Toolbox (control package) is installed, an ss object. For simplicity we decide here for the struct version.

sys = struct( ...
    'A', A, ...
    'B', B, ...
    'C', C, ...
    'D', D);

We start with the most simplest way of using the visualization by just calling the implicit Euler function.

ml_ct_ss_simulate_ie(sys);

This produces by default a unit step response plot of the system over the time range [0, 10]. The time range, number of time points and further things can be changed by optional parameters. In general, the simulation functions can be used to simulate the system for arbitrary input signals. In the standard case, we have a system of the form

x'(t) = A*x(t) + B*u(t),
 y(t) = C*x(t) + D*u(t).

To change the input signal u(t) of the system, we need to give a function handle as optional parameter describing a function depending on time that is giving the appropriate number of input signals. We decide on the following signal

u = @(t) sin(t) * ones(3, 1) - 0.5;

Due to using subplots in case of several system outputs, generated plots cannot be overwritten easily. Therefore, we close the previous created plot.

close;

Now we can simulate the system with the new input signal over the time range [0, 30]:

opts = struct('InputFcn', u, 'TimeRange', 30);
ml_ct_ss_simulate_ie(sys, opts);

An important feature of the time evaluation functions are the output arguments. For the moment, we suppress the creation of the simulation plot and consider just the output arguments:

[t, y, info] = ml_sigmaplot(sys, struct('ShowPlot', 0));

We shortly explain the meaning of those outputs:

t contains the time points over which the system was simulated
y contains the outputs of the system generated by the simulated states,
info is a struct with the number of computed time points, the internal states x(t) and possibel error values between given systems, if this option was used.

Another sepcial feature to consider is the input range of the simulation. By this it is possible to determine a time range in which the input signal is applied and outside it is set to 0, e.g., we apply the new input signal only on the interval [5, 15] and simulate the system again until the time 30.

close;
opts.InputRange = [5 15];
ml_ct_ss_simulate_ie(sys, opts);

It shall be mentioned that the initial value of the internal states is by default 0 but can be changed also by the optional parameters.

Multiple Systems and Error Plots

Another feature of the MORLAB time evaluation functions is the ability to accept multiple systems. The given systems need to be of the same type and have the same number of inputs and outputs. We construct a second system

sys2 = struct( ...
    'A', -1, ...
    'B', ones(1, 3), ...
    'C', ones(2, 1));

Now the function will create to graphs per plot, for each system one.

close;
ml_ct_ss_simulate_ie(sys, sys2);

If multiple systems are given, the evaluation functions support difference modi. Those allow the computation of absolute and relative error outputs based on the given systems. In this case, the first given system is taken as the primary one and all following systems will be substracted from the first one to create the results. Let's compute as an example the relative error plot between sys and sys2.

close;
opts = struct('DiffMode', 'rel');
ml_ct_ss_simulate_ie(sys, sys2, opts);

The magnitude values of the error systems are then given in the info output argument.

Multiple systems can be given either as demonstrated above or in an one-dimensional cell array, where each entry is a system.

close;
ml_ct_ss_simulate_ie({sys, sys2});

Remarks

All frequency evaluation functions are basically equivalent in use but differ in the computed values and results. Therefore, all examples here work for all the functions in the same way.
The functions support sparse system matrices by using structs as input arguments.
The simulator functions in general store only the outputs of the system in the argument y. By setting the optional parameter StoreStates, also the internal states can be stored during the simulation and afterwards be found in the info struct.

Contents