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## Copyright (C) 2009, 2010, 2011 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 -*-
## @deftypefn{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u})
## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u}, @var{t})
## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u}, @var{t}, @var{x0})
## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u}, @var{t}, @var{[]}, @var{method})
## @deftypefnx{Function File} {[@var{y}, @var{t}, @var{x}] =} lsim (@var{sys}, @var{u}, @var{t}, @var{x0}, @var{method})
## Simulate LTI model response to arbitrary inputs. If no output arguments are given,
## the system response is plotted on the screen.
##
## @strong{Inputs}
## @table @var
## @item sys
## LTI model. System must be proper, i.e. it must not have more zeros than poles.
## @item u
## Vector or array of input signal. Needs @code{length(t)} rows and as many columns
## as there are inputs. If @var{sys} is a single-input system, row vectors @var{u}
## of length @code{length(t)} are accepted as well.
## @item t
## Time vector. Should be evenly spaced. If @var{sys} is a continuous-time system
## and @var{t} is a real scalar, @var{sys} is discretized with sampling time
## @code{tsam = t/(rows(u)-1)}. If @var{sys} is a discrete-time system and @var{t}
## is not specified, vector @var{t} is assumed to be @code{0 : tsam : tsam*(rows(u)-1)}.
## @item x0
## Vector of initial conditions for each state. If not specified, a zero vector is assumed.
## @item method
## Discretization method for continuous-time models. Default value is zoh
## (zero-order hold). All methods from @code{c2d} are supported.
## @end table
##
## @strong{Outputs}
## @table @var
## @item y
## Output response array. Has as many rows as time samples (length of t)
## and as many columns as outputs.
## @item t
## Time row vector. It is always evenly spaced.
## @item x
## State trajectories array. Has @code{length (t)} rows and as many columns as states.
## @end table
##
## @seealso{impulse, initial, step}
## @end deftypefn
## Author: Lukas Reichlin <lukas.reichlin@gmail.com>
## Created: October 2009
## Version: 0.3
function [y_r, t_r, x_r] = lsim (sys, u, t = [], x0 = [], method = "zoh")
## TODO: multiplot feature: lsim (sys1, "b", sys2, "r", ..., u, t)
if (nargin < 2 || nargin > 5)
print_usage ();
endif
if (! isa (sys, "ss"))
sys = ss (sys); # sys must be proper
endif
if (is_real_vector (u))
u = reshape (u, [], 1); # allow row vectors for single-input systems
elseif (! is_real_matrix (u))
error ("lsim: input signal u must be an array of real numbers");
endif
if (! is_real_vector (t) && ! isempty (t))
error ("lsim: time vector t must be real or empty");
endif
discrete = ! isct (sys); # static gains are treated as continuous-time systems
tsam = abs (get (sys, "tsam")); # use 1 second as default if tsam is unspecified (-1)
urows = rows (u);
ucols = columns (u);
if (discrete) # discrete system
if (isempty (t)) # lsim (sys, u)
dt = tsam;
tinitial = 0;
tfinal = tsam * (urows - 1);
elseif (length (t) == 1) # lsim (sys, u, tfinal)
dt = tsam;
tinitial = 0;
tfinal = t;
else # lsim (sys, u, t, ...)
warning ("lsim: spacing of time vector has no effect on sampling time of discrete system");
dt = tsam;
tinitial = t(1);
tfinal = t(end);
endif
else # continuous system
if (isempty (t)) # lsim (sys, u, [], ...)
error ("lsim: invalid time vector");
elseif (length (t) == 1) # lsim (sys, u, tfinal, ...)
dt = t / (urows - 1);
tinitial = 0;
tfinal = t;
else # lsim (sys, u, t, ...)
dt = t(2) - t(1); # assume that t is regularly spaced
tinitial = t(1);
tfinal = t(end);
endif
sys = c2d (sys, dt, method); # convert to discrete-time model
endif
[A, B, C, D] = ssdata (sys);
n = rows (A); # number of states
m = columns (B); # number of inputs
p = rows (C); # number of outputs
t = reshape (tinitial : dt : tfinal, [], 1); # time vector
trows = length (t);
if (urows != trows)
error ("lsim: input vector u must have %d rows", trows);
endif
if (ucols != m)
error ("lsim: input vector u must have %d columns", m);
endif
## preallocate memory
y = zeros (trows, p);
x_arr = zeros (trows, n);
## initial conditions
if (isempty (x0))
x0 = zeros (n, 1);
elseif (n != length (x0) || ! is_real_vector (x0))
error ("initial: x0 must be a vector with %d elements", n);
endif
x = reshape (x0, [], 1); # make sure that x is a column vector
## simulation
for k = 1 : trows
y(k, :) = C * x + D * u(k, :).';
x_arr(k, :) = x;
x = A * x + B * u(k, :).';
endfor
if (nargout == 0) # plot information
str = ["Linear Simulation Results of ", inputname(1)];
outname = get (sys, "outname");
outname = __labels__ (outname, "y_");
if (discrete) # discrete system
for k = 1 : p
subplot (p, 1, k);
stairs (t, y(:, k));
grid ("on");
if (k == 1)
title (str);
endif
ylabel (sprintf ("Amplitude %s", outname{k}));
endfor
xlabel ("Time [s]");
else # continuous system
for k = 1 : p
subplot (p, 1, k);
plot (t, y(:, k));
grid ("on");
if (k == 1)
title (str);
endif
ylabel (sprintf ("Amplitude %s", outname{k}));
endfor
xlabel ("Time [s]");
endif
else # return values
y_r = y;
t_r = t;
x_r = x_arr;
endif
endfunction
## TODO: add test cases
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