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## Copyright (C) 1996, 1998, 2000, 2004, 2005, 2007 Kai P. Mueller
##
##
## This program 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.
##
## 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
## General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program; see the file COPYING. If not, see
## <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {} dhinfdemo ()
## Demonstrate the functions available to design a discrete
## @iftex
## @tex
## $ { \cal H }_\infty $
## @end tex
## @end iftex
## @ifinfo
## H-infinity
## @end ifinfo
## controller. This is not a true discrete design. The
## design is carried out in continuous time while the effect of sampling
## is described by a bilinear transformation of the sampled system.
## This method works quite well if the sampling period is "small"
## compared to the plant time constants.
##
## Continuous plant:
## @iftex
## @tex
## $$ G(s) = { 1 \over (s+2) (s+1) } $$
## @end tex
## @end iftex
## @ifinfo
## @example
## @group
## 1
## G(s) = --------------
## (s + 2)(s + 1)
## @end group
## @end example
## @end ifinfo
##
## Discretised plant with @acronym{ZOH} (Sampling period = @var{Ts} = 1 second):
## @iftex
## @tex
## $$ G(z) = { 0.39958z + 0.14700 \over (z - 0.36788) (z - 0.13533) } $$
## @end tex
## @end iftex
## @ifinfo
## @example
## @group
## 0.39958z + 0.14700
## G(z) = --------------------------
## (z - 0.36788)(z - 0.13533)
## @end group
## @end example
## @end ifinfo
##
## @example
## @group
## +----+
## -------------------->| W1 |---> v1
## z | +----+
## ----|-------------+ || T || => min.
## | | vz infty
## | +---+ v +----+
## *--->| G |--->O--*-->| W2 |---> v2
## | +---+ | +----+
## | |
## | +---+ |
## -----| K |<-------
## +---+
## @end group
## @end example
##
## @noindent
## W1 and W2 are the robustness and performancs weighting functions.
## @end deftypefn
## K. Mueller, <mueller@ifr.ing.tu-bs.de>
## Technical University of Braunschweig, IfR
echo off
disp(" ");
disp(" --------------------------------------------------");
disp(" Discrete H_infinity optimal control for the plant:");
disp(" ");
disp(" 0.39958z + 0.14700");
disp(" G(s) = --------------------------");
disp(" (z - 0.36788)(z - 0.13533)");
disp(" --------------------------------------------------");
disp(" ");
disp("sampling time:")
cmd = "Ts = 1.0;";
disp(cmd);
eval(cmd);
disp("weighting on actuator value u");
cmd = "W1 = wgt1o(0.1, 200.0, 50.0);";
disp(cmd);
eval(cmd);
disp("weighting on controlled variable y");
cmd = "W2 = wgt1o(350.0, 0.05, 0.0002);";
disp(cmd);
eval(cmd);
## omega axis (column vector)
ww = vec(logspace(-4.99, 3.99, 100));
disp("Create ZOH equivalent model of a continuous plant");
cmd = "G = tf(2,[1 3 2]); Gd = c2d(G, Ts);";
run_cmd
## w-plane (continuous representation of the sampled system)
disp("W-plane transform of discrete time system:");
cmd = "Gw = d2c(Gd, \"bi\");";
run_cmd
disp(" ");
disp(" o building P...");
## need One as the pseudo transfer function One = 1
cmd = "One = ugain(1);";
disp(cmd);
eval(cmd);
cmd = " psys = buildssic([1 4;2 4;3 1],[3],[2 3 5],[3 4],Gw,W1,W2,One);";
run_cmd;
disp(" o controller design...");
cmd = "[K, gfin, GWC] = hinfsyn(psys, 1, 1, 0.1, 10.0, 0.02);";
run_cmd
disp(" ");
fig_n = 1;
yn = input(" * Plot magnitudes of W1KS and W2S? [n]: ","S");
if (length(yn) >= 1)
if ((yn(1) == "y") || (yn(1) == 'Y'))
disp(" o magnitudes of W1KS and W2S...");
gwx = sysprune(GWC, 1, 1);
mag1 = bode(gwx, ww);
if (columns(mag1) > 1); mag1 = mag1'; endif
gwx = sysprune(GWC, 2, 1);
mag2 = bode(gwx, ww);
if (columns(mag2) > 1); mag2 = mag2'; endif
figure(fig_n)
fig_n = fig_n + 1;
loglog(ww, [mag1 mag2]);
grid ("on");
endif
endif
Kd = c2d(K, "bi", Ts);
GG = buildssic([1 2; 2 1], [], [1 2], [-2], Gd, Kd);
disp(" o closed loop poles...");
damp(GG);
disp(" ");
yn = input(" * Plot closed loop step responses? [n]: ","S");
if (length(yn) >= 1)
if ((yn(1) == "y") || (yn(1) == 'Y'))
disp(" o step responses of T and KS...");
figure(fig_n)
step(GG, 1, 10);
endif
endif
## --------- End of dhinfdemo/kpm
|
