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## Copyright (C) 2009, 2010, 2012 Lukas F. Reichlin
##
## This file is part of LTI Syncope.
##
## LTI Syncope is free software: you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## LTI Syncope 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 General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with LTI Syncope. If not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## Common code for the time response functions step, impulse and initial.
## Author: Lukas Reichlin <lukas.reichlin@gmail.com>
## Created: October 2009
## Version: 0.4
function [y, t, x] = __time_response__ (response, args, sysname, plotflag)
sys_idx = find (cellfun (@isa, args, {"lti"})); # look for LTI models, 'find' needed for plot styles
sys_cell = cellfun (@ss, args(sys_idx), "uniformoutput", false); # convert to state-space
if (! size_equal (sys_cell{:}))
error ("%s: models must have equal sizes", response);
endif
vec_idx = find (cellfun (@is_real_matrix, args)); # indices of vector arguments
n_vec = length (vec_idx); # number of vector arguments
n_sys = length (sys_cell); # number of LTI systems
tfinal = [];
dt = [];
x0 = [];
## extract tfinal/t, dt, x0 from args
if (strcmpi (response, "initial"))
if (n_vec < 1)
error ("initial: require initial state vector 'x0'");
else # initial state vector x0 specified
arg = args{vec_idx(1)};
if (is_real_vector (arg))
x0 = arg;
else
error ("initial: initial state vector 'x0' must be a vector of real values");
endif
if (n_vec > 1) # tfinal or time vector t specified
arg = args{vec_idx(2)};
if (issample (arg))
tfinal = arg;
elseif (isempty (arg))
## tfinal = []; # nothing to do here
elseif (is_real_vector (arg))
dt = abs (arg(2) - arg(1)); # assume that t is regularly spaced
tfinal = arg(end);
else
warning ("initial: argument number %d ignored", vec_idx(2));
endif
if (n_vec > 2) # sampling time dt specified
arg = args{vec_idx(3)};
if (issample (arg))
dt = arg;
else
warning ("initial: argument number %d ignored", vec_idx(3));
endif
if (n_vec > 3)
warning ("initial: ignored");
endif
endif
endif
endif
else # step or impulse response
if (n_vec > 0) # tfinal or time vector t specified
arg = args{vec_idx(1)};
if (issample (arg))
tfinal = arg;
elseif (isempty (arg))
## tfinal = []; # nothing to do here
elseif (is_real_vector (arg))
dt = abs (arg(2) - arg(1)); # assume that t is regularly spaced
tfinal = arg(end);
else
warning ("%s: argument number %d ignored", response, vec_idx(1));
endif
if (n_vec > 1) # sampling time dt specified
arg = args{vec_idx(2)};
if (issample (arg))
dt = arg;
else
warning ("%s: argument number %d ignored", response, vec_idx(2));
endif
if (n_vec > 2)
warning ("%s: ignored", response);
endif
endif
endif
endif
## TODO: share common code between initial and step/impulse
[tfinal, dt] = cellfun (@__sim_horizon__, sys_cell, {tfinal}, {dt}, "uniformoutput", false);
tfinal = max ([tfinal{:}]);
ct_idx = cellfun (@isct, sys_cell);
sys_dt_cell = sys_cell;
tmp = cellfun (@c2d, sys_cell(ct_idx), dt(ct_idx), {"zoh"}, "uniformoutput", false);
sys_dt_cell(ct_idx) = tmp;
## time vector
t = @cellfun (@(dt) reshape (0 : dt : tfinal, [], 1), dt, "uniformoutput", false);
## function [y, x_arr] = __initial_response__ (sys, sys_dt, t, x0)
## function [y, x_arr] = __step_response__ (sys_dt, t)
## function [y, x_arr] = __impulse_response__ (sys, sys_dt, t)
## function [y, x_arr] = __ramp_response__ (sys_dt, t)
switch (response)
case "initial"
[y, x] = cellfun (@__initial_response__, sys_dt_cell, t, {x0}, "uniformoutput", false);
case "step"
[y, x] = cellfun (@__step_response__, sys_dt_cell, t, "uniformoutput", false);
case "impulse"
[y, x] = cellfun (@__impulse_response__, sys_cell, sys_dt_cell, t, "uniformoutput", false);
case "ramp"
[y, x] = cellfun (@__ramp_response__, sys_dt_cell, t, "uniformoutput", false);
otherwise
error ("time_response: invalid response type");
endswitch
if (plotflag) # display plot
[p, m] = size (sys_cell{1});
switch (response)
case "initial"
str = "Response to Initial Conditions";
cols = 1;
## yfinal = zeros (p, 1);
case "step"
str = "Step Response";
cols = m;
## yfinal = dcgain (sys_cell{1});
case "impulse"
str = "Impulse Response";
cols = m;
## yfinal = zeros (p, m);
case "ramp"
str = "Ramp Response";
cols = m;
otherwise
error ("time_response: invalid response type");
endswitch
style_idx = find (cellfun (@ischar, args));
outname = get (sys_cell{end}, "outname");
outname = __labels__ (outname, "y");
colororder = get (gca, "colororder");
rc = rows (colororder);
for k = 1 : n_sys # for every system
if (k == n_sys)
lim = numel (args);
else
lim = sys_idx(k+1);
endif
style = args(style_idx(style_idx > sys_idx(k) & style_idx <= lim));
if (isempty (style))
color = colororder(1+rem (k-1, rc), :);
style = {"color", color};
endif
if (ct_idx(k)) # continuous-time system
for i = 1 : p # for every output
for j = 1 : cols # for every input (except for initial where cols=1)
subplot (p, cols, (i-1)*cols+j);
plot (t{k}, y{k}(:, i, j), style{:});
hold on;
grid on;
if (k == n_sys)
axis tight
ylim (__axis_margin__ (ylim))
if (j == 1)
ylabel (outname{i});
if (i == 1)
title (str);
endif
endif
endif
endfor
endfor
else # discrete-time system
for i = 1 : p # for every output
for j = 1 : cols # for every input (except for initial where cols=1)
subplot (p, cols, (i-1)*cols+j);
stairs (t{k}, y{k}(:, i, j), style{:});
hold on;
grid on;
if (k == n_sys)
axis tight;
ylim (__axis_margin__ (ylim))
if (j == 1)
ylabel (outname{i});
if (i == 1)
title (str);
endif
endif
endif
endfor
endfor
endif
endfor
xlabel ("Time [s]");
if (p == 1 && m == 1)
legend (sysname)
endif
hold off;
endif
endfunction
function [y, x_arr] = __initial_response__ (sys_dt, t, x0)
[F, G, C, D] = ssdata (sys_dt); # system must be proper
n = rows (F); # number of states
m = columns (G); # number of inputs
p = rows (C); # number of outputs
l_t = length (t);
## preallocate memory
y = zeros (l_t, p);
x_arr = zeros (l_t, n);
## initial conditions
x = reshape (x0, [], 1); # make sure that x is a column vector
if (n != length (x0) || ! is_real_vector (x0))
error ("initial: x0 must be a real vector with %d elements", n);
endif
## simulation
for k = 1 : l_t
y(k, :) = C * x;
x_arr(k, :) = x;
x = F * x;
endfor
endfunction
function [y, x_arr] = __step_response__ (sys_dt, t)
[F, G, C, D] = ssdata (sys_dt); # system must be proper
n = rows (F); # number of states
m = columns (G); # number of inputs
p = rows (C); # number of outputs
l_t = length (t);
## preallocate memory
y = zeros (l_t, p, m);
x_arr = zeros (l_t, n, m);
for j = 1 : m # for every input channel
## initial conditions
x = zeros (n, 1);
u = zeros (m, 1);
u(j) = 1;
## simulation
for k = 1 : l_t
y(k, :, j) = C * x + D * u;
x_arr(k, :, j) = x;
x = F * x + G * u;
endfor
endfor
endfunction
function [y, x_arr] = __impulse_response__ (sys, sys_dt, t)
[~, B] = ssdata (sys);
[F, G, C, D, dt] = ssdata (sys_dt); # system must be proper
dt = abs (dt); # use 1 second if tsam is unspecified (-1)
discrete = ! isct (sys);
n = rows (F); # number of states
m = columns (G); # number of inputs
p = rows (C); # number of outputs
l_t = length (t);
## preallocate memory
y = zeros (l_t, p, m);
x_arr = zeros (l_t, n, m);
for j = 1 : m # for every input channel
## initial conditions
u = zeros (m, 1);
u(j) = 1;
if (discrete)
x = zeros (n, 1); # zero by definition
y(1, :, j) = D * u / dt;
x_arr(1, :, j) = x;
x = G * u / dt;
else
x = B * u; # B, not G!
y(1, :, j) = C * x;
x_arr(1, :, j) = x;
x = F * x;
endif
## simulation
for k = 2 : l_t
y (k, :, j) = C * x;
x_arr(k, :, j) = x;
x = F * x;
endfor
endfor
if (discrete)
y *= dt;
x_arr *= dt;
endif
endfunction
function [y, x_arr] = __ramp_response__ (sys_dt, t)
[F, G, C, D] = ssdata (sys_dt); # system must be proper
n = rows (F); # number of states
m = columns (G); # number of inputs
p = rows (C); # number of outputs
l_t = length (t);
## preallocate memory
y = zeros (l_t, p, m);
x_arr = zeros (l_t, n, m);
for j = 1 : m # for every input channel
## initial conditions
x = zeros (n, 1);
u = zeros (m, l_t);
u(j, :) = t;
## simulation
for k = 1 : l_t
y(k, :, j) = C * x + D * u(:, k);
x_arr(k, :, j) = x;
x = F * x + G * u(:, k);
endfor
endfor
endfunction
function [tfinal, dt] = __sim_horizon__ (sys, tfinal, Ts)
## code based on __stepimp__.m of Kai P. Mueller and A. Scottedward Hodel
TOL = 1.0e-10; # values below TOL are assumed to be zero
N_MIN = 50; # min number of points
N_MAX = 2000; # max number of points
N_DEF = 1000; # default number of points
T_DEF = 10; # default simulation time
ev = pole (sys);
n = length (ev); # number of states/poles
continuous = isct (sys);
discrete = ! continuous;
if (discrete)
dt = Ts = abs (get (sys, "tsam"));
## perform bilinear transformation on poles in z
for k = 1 : n
pol = ev(k);
if (abs (pol + 1) < TOL)
ev(k) = 0;
else
ev(k) = 2 / Ts * (pol - 1) / (pol + 1);
endif
endfor
endif
## remove poles near zero from eigenvalue array ev
nk = n;
for k = 1 : n
if (abs (real (ev(k))) < TOL)
ev(k) = 0;
nk -= 1;
endif
endfor
if (nk == 0)
if (isempty (tfinal))
tfinal = T_DEF;
endif
if (continuous)
dt = tfinal / N_DEF;
endif
else
ev = ev(find (ev));
ev_max = max (abs (ev));
if (continuous)
dt = 0.2 * pi / ev_max;
endif
if (isempty (tfinal))
ev_min = min (abs (real (ev)));
tfinal = 5.0 / ev_min;
## round up
yy = 10^(ceil (log10 (tfinal)) - 1);
tfinal = yy * ceil (tfinal / yy);
endif
if (continuous)
N = tfinal / dt;
if (N < N_MIN)
dt = tfinal / N_MIN;
endif
if (N > N_MAX)
dt = tfinal / N_MAX;
endif
endif
endif
if (continuous && ! isempty (Ts)) # catch case cont. system with dt specified
dt = Ts;
endif
endfunction