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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} {} hinfdemo ()
##
## @iftex
## @tex
## $ { \cal H }_\infty $
## @end tex
## @end iftex
## @ifinfo
## H-infinity
## @end ifinfo
## design demos for continuous @acronym{SISO} and @acronym{MIMO} systems and a
## discrete system. The @acronym{SISO} system is difficult to control because
## it is non-minimum-phase and unstable. The second design example
## controls the @command{jet707} plant, the linearized state space model of a
## Boeing 707-321 aircraft at @var{v}=80 m/s
## @iftex
## @tex
## ($M = 0.26$, $G_{a0} = -3^{\circ}$, ${\alpha}_0 = 4^{\circ}$, ${\kappa}= 50^{\circ}$).
## @end tex
## @end iftex
## @ifinfo
## (@var{M} = 0.26, @var{Ga0} = -3 deg, @var{alpha0} = 4 deg, @var{kappa} = 50 deg).
## @end ifinfo
## Inputs: (1) thrust and (2) elevator angle
## Outputs: (1) airspeed and (2) pitch angle. The discrete system is a
## stable and second order.
##
## @table @asis
## @item @acronym{SISO} plant:
##
## @iftex
## @tex
## $$ G(s) = { s-2 \over (s+2) (s-1) } $$
## @end tex
## @end iftex
## @ifinfo
## @example
## @group
## s - 2
## G(s) = --------------
## (s + 2)(s - 1)
## @end group
## @end example
## @end ifinfo
##
## @smallexample
## @group
##
## +----+
## -------------------->| W1 |---> v1
## z | +----+
## ----|-------------+
## | |
## | +---+ v y +----+
## u *--->| G |--->O--*-->| W2 |---> v2
## | +---+ | +----+
## | |
## | +---+ |
## -----| K |<-------
## +---+
## @end group
## @end smallexample
##
## @iftex
## @tex
## $$ { \rm min } \Vert T_{vz} \Vert _\infty $$
## @end tex
## @end iftex
## @ifinfo
## @example
## min || T ||
## vz infty
## @end example
## @end ifinfo
##
## @var{W1} und @var{W2} are the robustness and performance weighting
## functions.
##
## @item @acronym{MIMO} plant:
## The optimal controller minimizes the
## @iftex
## @tex
## $ { \cal H }_\infty $
## @end tex
## @end iftex
## @ifinfo
## H-infinity
## @end ifinfo
## norm of the
## augmented plant @var{P} (mixed-sensitivity problem):
## @smallexample
## @group
## w
## 1 -----------+
## | +----+
## +---------------------->| W1 |----> z1
## w | | +----+
## 2 ------------------------+
## | | |
## | v +----+ v +----+
## +--*-->o-->| G |-->o--*-->| W2 |---> z2
## | +----+ | +----+
## | |
## ^ v
## u y (to K)
## (from controller K)
## @end group
## @end smallexample
##
## @iftex
## @tex
## $$ \left [ \matrix{ z_1 \cr
## z_2 \cr
## y } \right ] =
## P \left [ \matrix{ w_1 \cr
## w_2 \cr
## u } \right ] $$
## @end tex
## @end iftex
## @ifinfo
## @smallexample
## @group
## + + + +
## | z | | w |
## | 1 | | 1 |
## | z | = [ P ] * | w |
## | 2 | | 2 |
## | y | | u |
## + + + +
## @end group
## @end smallexample
## @end ifinfo
##
## @item Discrete system:
## 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.
##
## @item The continuous plant:
## @iftex
## @tex
## $$ G(s) = { 1 \over (s+2)(s+1) } $$
## @end tex
## @end iftex
##
## @ifinfo
## @example
## @group
## 1
## G (s) = --------------
## k (s + 2)(s + 1)
##
## @end group
## @end example
## @end ifinfo
##
## is discretised with a @acronym{ZOH} (Sampling period = @var{Ts} = 1 second):
## @iftex
## @tex
## $$ G(z) = { 0.199788z + 0.073498 \over (z - 0.36788) (z - 0.13534) } $$
## @end tex
## @end iftex
## @ifinfo
## @example
## @group
##
## 0.199788z + 0.073498
## G(z) = --------------------------
## (z - 0.36788)(z - 0.13534)
## @end group
## @end example
## @end ifinfo
##
## @smallexample
## @group
##
## +----+
## -------------------->| W1 |---> v1
## z | +----+
## ----|-------------+
## | |
## | +---+ v +----+
## *--->| G |--->O--*-->| W2 |---> v2
## | +---+ | +----+
## | |
## | +---+ |
## -----| K |<-------
## +---+
## @end group
## @end smallexample
## @iftex
## @tex
## $$ { \rm min } \Vert T_{vz} \Vert _\infty $$
## @end tex
## @end iftex
## @ifinfo
## @example
## min || T ||
## vz infty
## @end example
## @end ifinfo
## @var{W1} and @var{W2} are the robustness and performance weighting
## functions.
## @end table
## @end deftypefn
## Author: Kai P. Mueller <mueller@ifr.ing.tu-bs.de>
## Created: April 30, 1998
yn = [];
while (length(yn) < 1)
yn = input(" * [s]iso, [m]imo, or [d]iscrete design? [no default]: ","S");
endwhile
if ((yn(1) == "s") | (yn(1) == 'S'))
sys_type = 1;
elseif ((yn(1) == "m") | (yn(1) == 'M'))
sys_type = 2;
elseif ((yn(1) == "d") | (yn(1) == 'D'))
sys_type = 3;
else
disp(" *** no system type specified, hinfdemo terminated.");
return;
endif
echo off
switch (sys_type)
case (1)
## siso
disp(" ");
disp(" ----------------------------------------------");
disp(" H_infinity optimal control for the SISO plant:");
disp(" ");
disp(" s - 2");
disp(" G(s) = --------------");
disp(" (s + 2)(s - 1)");
disp(" ");
disp(" ----------------------------------------------");
disp(" ");
## weighting on actuator u
W1 = wgt1o(0.05, 100.0, 425.0);
## weighting on controlled variable y
W2 = wgt1o(10.0, 0.05, 0.001);
## plant
G = tf2sys([1 -2],[1 1 -2]);
## need One as the pseudo transfer function One = 1
One = ugain(1);
disp(" o forming P...");
psys = buildssic([1 4;2 4;3 1],[3],[2 3 5],[3 4],G,W1,W2,One);
disp(" ");
disp(" o controller design...");
[K, gfin, GW]=hinfsyn(psys, 1, 1, 0.1, 10.0, 0.02);
disp(" ");
disp("-- OK ----------------------------------------------");
disp(" Closed loop poles:");
damp(GW);
## disp(" o Testing H_infinity norm: (hinfnorm does not work)");
## hinfnorm(GW);
disp(" ");
yn = input(" * Plot closed loop step response? [n]: ","S");
if (length(yn) >= 1)
if ((yn(1) == "y") || (yn(1) == 'Y'))
disp(" o step responses of T and KS...");
GW = buildssic([1 2; 2 1], [], [1 2], [-2], G, K);
figure(1);
step(GW, 1, 10);
endif
endif
case (2)
## mimo
disp(" ");
disp(" -----------------------------------------------");
disp(" H_inf optimal control for the jet707 plant");
disp(" -----------------------------------------------");
disp(" ");
## Weighting function on u (robustness weight)
ww1 = wgt1o(0.01,5,0.9);
ww2 = wgt1o(0.01,5,2.2);
W1 = buildssic([1 0;2 0],[],[1 2],[1 2],ww1,ww2);
## Weighting function on y (performance weight)
ww1 = wgt1o(250,0.1,0.0001);
ww2 = wgt1o(250,0.1,0.0002);
W2 = buildssic([1 0;2 0],[],[1 2],[1 2],ww1,ww2);
## plant (2 x 2 system)
G = jet707;
disp(" o forming P...");
One = ugain(2);
Clst = [1 7; 2 8; 3 7; 4 8; 5 1; 6 2];
P = buildssic(Clst,[5 6],[3:6 9 10],[1 2 5:8],G,W1,W2,One);
disp(" ");
disp(" o controller design...");
K = hinfsyn(P, 2, 2, 0.25, 10.0, 0.005);
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...");
GW = buildssic([1 3;2 4;3 1;4 2],[],[1 2 3 4],[-3 -4],G,K);
disp(" ");
disp(" FIGURE 1: speed refence => 1, pitch angle ref. => 0");
disp(" ===================================================");
disp(" y1: speed (should be 1)");
disp(" y2: pitch angle (should remain 0)");
disp(" y3: thrust (should be a slow transient)");
disp(" y6: elevator (should be a faster transient)");
disp(" ");
disp(" FIGURE 2: speed refence => 0, pitch angle ref. => 1");
disp(" ===================================================");
disp(" y1: speed (should remain 0)");
disp(" y2: pitch angle (should be 1)");
disp(" y3: thrust (should be a slow transient)");
disp(" y6: elevator (should be a faster transient)");
disp(" ");
figure(1)
step(GW);
figure(2)
step(GW,2);
endif
endif
case (3)
## discrete
disp(" ");
disp(" --------------------------------------------------");
disp(" Discrete H_infinity optimal control for the plant:");
disp(" ");
disp(" 0.199788z + 0.073498");
disp(" G(s) = --------------------------");
disp(" (z - 0.36788)(z - 0.13533)");
disp(" --------------------------------------------------");
disp(" ");
## sampling time
Ts = 1.0;
## weighting on actuator value u
W1 = wgt1o(0.1, 200.0, 50.0);
## weighting on controlled variable y
W2 = wgt1o(350.0, 0.05, 0.0002);
## omega axis
ww = logspace(-4.99, 3.99, 100);
if (columns(ww) > 1); ww = ww'; endif
## continuous plant
G = tf2sys(2,[1 3 2]);
## discrete plant with zoh
Gd = c2d(G, Ts);
## w-plane (continuous representation of the sampled system)
Gw = d2c(Gd, "bi");
disp(" ");
disp(" o building P...");
## need One as the pseudo transfer function One = 1
One = ugain(1);
psys = buildssic([1 4;2 4;3 1],[3],[2 3 5],[3 4],Gw,W1,W2,One);
disp(" o controller design...");
[K, gfin, GWC] = hinfsyn(psys, 1, 1, 0.1, 10.0, 0.02);
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
endswitch
disp(" o hinfdemo terminated successfully.");
## KPM-hinfdemo/End
|
