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//
// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
// Copyright (C) ????-2008 - INRIA
//
// This file is distributed under the same license as the Scilab package.
//
function demo_pendule()
mode(-1)
exec("SCI/modules/cacsd/demos/pendulum/simulation.sci", -1);
exec("SCI/modules/cacsd/demos/pendulum/graphics.sci", -1);
//disable displayed lines control
display_props=lines(); lines(0)
my_handle = scf(100001);
clf(my_handle,"reset");
show_window();
wdim=xget('wdim')
//mode(1)
dpnd()
//
// equations
//----------
//state =[x x' theta theta']
//
mb=0.1;mc=1;l=0.3;m=4*mc+mb; //constants
//
messagebox(['Open loop simulation'
' '
' y0 = [0;0;0.1;0]; //initial state [x x'' theta theta'']'
' t = 0.03*(1:180); //observation dates'
' y = ode(y0,0,0.03*(1:180),ivpd); //differential equation integration'
' //Display'
' P = initialize_display(y0(1),y0(3))'
' for k=1:size(y,2), set_pendulum(P,y(1,k),y(3,k));end'],"modal");
//
y0=[0;0;0.1;0];
y=ode(y0,0,0.03*(1:180),ivpd);
P=initialize_display(y0(1),y0(3));
for k=1:size(y,2), set_pendulum(P,y(1,k),y(3,k));end
//
messagebox(['Linearization'
' '
' x0=[0;0;0;0];u0=0;'
' [f,g,h,j]=lin(pendu,x0,u0);'
' pe=syslin(''c'',f,g,h,j);'
' // display of the linear system'
' ssprint(pe)'],"modal")
//
mode(1)
x0=[0;0;0;0];u0=0;
[f,g,h,j]=lin(pendu,x0,u0);
pe=syslin('c',f,g,h,j);
ssprint(pe)
mode(-1)
//
messagebox(['Checking the result'
' //comparison with linear model computed by hand';
''
' f1=[0 1 0 0'
' 0 0 -3*mb*9.81/m 0'
' 0 0 0 1'
' 0 0 6*(mb+mc)*9.81/(m*l) 0];'
' '
' g1=[0 ; 4/m ; 0 ; -6/(m*l)];'
' '
' h1=[1 0 0 0'
' 0 0 1 0];'
' '
' err=norm(f-f1,1)+norm(g-g1,1)+norm(h-h1,1)+norm(j,1)'],"modal")
//
mode(1)
f1=[0 1 0 0
0 0 -3*mb*9.81/m 0
0 0 0 1
0 0 6*(mb+mc)*9.81/(m*l) 0];
g1=[0 ; 4/m ; 0 ; -6/(m*l)];
h1=[1 0 0 0
0 0 1 0];
err=norm(f-f1,1)+norm(g-g1,1)+norm(h-h1,1)+norm(j,1)
mode(-1)
messagebox(['Linear system properties analysis'
' spec(f) //stability (unstable system !)'
' n=contr(f,g) //controlability'
' '
' //observability'
' m1=contr(f'',h(1,:)'') '
' [m2,t]=contr(f'',h(2,:)'')'],"modal");
//---------
mode(1)
spec(f) //stability (unstable system !)
n=contr(f,g) //controlability
//observability
m1=contr(f',h(1,:)')
[m2,t]=contr(f',h(2,:)')
mode(-1)
//
messagebox(['Synthesis of a stabilizing controller using poles placement'
' '
'// only x and theta are observed : contruction of an observer'
'// to estimate the state : z''=(f-k*h)*z+k*y+g*u'
' to=0.1; '
' k=ppol(f'',h'',-ones(4,1)/to)'' //observer gain'
'// compute the compensator gain'
' kr=ppol(f,g,-ones(4,1)/to)'],"modal");
//-------------------------------------------------
//
//pole placement technique
//only x and theta are observed : contruction of an observer
//to estimate the state : z'=(f-k*h)*z+k*y+g*u
//
to=0.1; //
k=ppol(f',h',-ones(4,1)/to)' //observer gain
//
//verification
//
// norm( poly(f-k*h,'z')-poly(-ones(4,1)/to,'z'))
//
kr=ppol(f,g,-ones(4,1)/to) //compensator gain
//
messagebox(['Build full linear system pendulum-observer-compensator'
' '
' ft=[f-g*kr -g*kr'
' 0*f f-k*h]'
' '
' gt=[g;0*g];'
' ht=[h,0*h];'
''
' pr=syslin(''c'',ft,gt,ht);'
''
'//Check the closed loop dynamic'
' spec(pr.A)'
' '
'//transfer matrix representation'
' hr=clean(ss2tf(pr),1.d-10)'
' '
'//frequency analysis'
' black(pr,0.01,100,[''position'',''theta''])'
' g_margin(pr(1,1))'],"modal");
//---------------------------------------------
//
//state: [x x-z]
//
mode(1)
ft=[f-g*kr -g*kr
0*f f-k*h];
gt=[g;0*g];
ht=[h,0*h];
pr=syslin('c',ft,gt,ht);
// closed loop dynamics:
spec(pr(2))
//transfer matrix representation
hr=clean(ss2tf(pr),1.d-10)
//frequency analysis
clf()
black(pr,0.01,100,['position','theta'])
g_margin(pr(1,1))
mode(-1)
//
messagebox(['Sampled system'
' '
' t=to/5; //sampling period'
' prd=dscr(pr,t); //discrete model'
' spec(prd.A) //poles of the discrete model'],"modal");
//---------------
//
mode(1)
t=to/5;
prd=dscr(pr,t);
spec(prd(2))
mode(-1)
//
messagebox(['Impulse response'
' '
' x0=[0;0;0;0;0;0;0;0]; //initial state'
' u(1,180)=0;u(1,1)=1; //impulse'
' y=flts(u,prd,x0); //siscrete system simulation'
' draw1()'],"modal");
//-----------------
//
mode(1)
x0=[0;0;0;0;0;0;0;0];
u(1,180)=0;u(1,1)=1;
y=flts(u,prd,x0);
draw1()
mode(-1)
//
messagebox(['Compensation of the non linear system with'
'linear regulator'
''
' t0=0;t1=t*(1:125);'
' x0=[0 0 0.4 0 0 0 0 0]''; // initial state'
' yd=ode(x0,t0,t1,regu); //integration of differential equation'
' draw2()'],"modal");
;
//--------------------------------------
//
//simulation
//
mode(1)
t0=0;t1=t*(1:125);
x0=[0 0 0.4 0 0 0 0 0]'; //
yd=ode(x0,t0,t1,regu);
draw2()
mode(-1)
lines(display_props(2))
messagebox('The end',"modal");
endfunction
demo_pendule();
clear demo_pendule;
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