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.TH odedc G "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
odedc - discrete/continuous ode solver
.SH CALLING SEQUENCE
.nf
yt=odedc(y0,nd,stdel,t0,t,f)
.fi
.SH PARAMETERS
.TP 10
y0
: real column vector (initial conditions),\fVy0=[y0c;y0d]\fR where
\fVy0d\fR has \fVnd\fR components.
.TP
nd
: integer, dimension of \fVy0d\fR
.TP
stdel
: real vector with one or two entries, \fVstdel=[h, delta]\fR (with
\fVdelta=0\fR as default value).
.TP
t0
: real scalar (initial time).
.TP
t
: real (row) vector, instants where \fVyt\fR is calculated
.TP
f
: Scilab "external" i.e. function or character string or list with
calling sequence: \fVyp=f(t,yc,yd,flag)\fR
.SH DESCRIPTION
.LP
\fVy=odedc([y0c;y0d],nd,[h,delta],t0,t,f)\fR
computes the solution of a mixed discrete/continuous system.
The discrete system state \fVyd_k\fR is embedded into a piecewise constant \fVyd(t)\fR time function as follows: \fVyd(t)= yd_k\fV for \fVt\fR in \fV[t_k=delay+k*h,t_(k+1)=delay+(k+1)*h[\fR (with \fVdelay=h*delta\fR).
The simulated equations are now:
.LP
dyc/dt=f(t,yc(t),yd(t),0) , for t in [t_k,t_(k+1)[
yc(t0)=y0c
.LP
and at instants \fVt_k\fR the discrete variable \fVyd\fR is updated by:
.LP
yd(t_k+)=f(yc(t_k-),yd(t_k-),1)
.LP
Note that, using the definition of \fVyd(t)\fR the last equation gives
.LP
yd_k = f (t_k,yc(t_k-),yd(t_(k-1)),1) (yc is time-continuous: yc(t_k-)=yc(tk))
.LP
The calling parameters of \fVf\fR are fixed: \fVycd=f(t,yc,yd,flag)\fR;
this function must return either the derivative of the vector \fVyc\fR if
\fVflag=0\fR or the update of \fVyd\fR if \fVflag=1\fR.
.LP
\fVycd=dot(yc)\fR must be a vector with same dimension as \fVyc\fR
if \fVflag=0\fR and \fVycd=update(yd)\fR must be a vector with same
dimension as \fVyd\fR if \fVflag=1\fR
.LP
\fVt\fR is a vector of instants where the solution \fVy\fR is computed.
.LP
\fVy\fR is the vector \fVy=[y(t(1)),y(t(2)),...]\fR.
This function can be called with the same optional parameters as the
\fVode\fR function (provided \fVnd\fR and \fVstdel\fR are given
in the calling sequence as second and third parameters).
It particular integration flags, tolerances can be set. Optional
parameters can be set by the \fVodeoptions\fR function.
.LP
An example for calling an external routine is given in directory
\fVdefault/fydot2.f\fR
External routines can be dynamically linked (see \fVlink\fR).
.SH EXAMPLE
.nf
//Linear system with switching input
deff('xdu=phis(t,x,u,flag)','if flag==0 then xdu=A*x+B*u; else xdu=1-u;end');
x0=[1;1];A=[-1,2;-2,-1];B=[1;2];u=0;nu=1;stdel=[1,0];u0=0;t=0:0.05:10;
xu=odedc([x0;u0],nu,stdel,0,t,phis);x=xu(1:2,:);u=xu(3,:);
nx=2;
plot2d1('onn',t',x',[1:nx],'161');
plot2d2('onn',t',u',[nx+1:nx+nu],'000');
//Fortran external( see fydot2.f):
norm(xu-odedc([x0;u0],nu,stdel,0,t,'phis'),1)
//Sampled feedback
//
// | xcdot=fc(t,xc,u)
// (system) |
// | y=hc(t,xc)
//
//
// | xd+=fd(xd,y)
// (feedback) |
// | u=hd(t,xd)
//
deff('xcd=f(t,xc,xd,iflag)',...
['if iflag==0 then '
' xcd=fc(t,xc,e(t)-hd(t,xd));'
'else '
' xcd=fd(xd,hc(t,xc));'
'end']);
A=[-10,2,3;4,-10,6;7,8,-10];B=[1;1;1];C=[1,1,1];
Ad=[1/2,1;0,1/20];Bd=[1;1];Cd=[1,1];
deff('st=e(t)','st=sin(3*t)')
deff('xdot=fc(t,x,u)','xdot=A*x+B*u')
deff('y=hc(t,x)','y=C*x')
deff('xp=fd(x,y)','xp=Ad*x + Bd*y')
deff('u=hd(t,x)','u=Cd*x')
h=0.1;t0=0;t=0:0.1:2;
x0c=[0;0;0];x0d=[0;0];nd=2;
xcd=odedc([x0c;x0d],nd,h,t0,t,f);
norm(xcd-odedc([x0c;x0d],nd,h,t0,t,'fcd1')) // Fast calculation (see fydot2.f)
plot2d([t',t',t'],xcd(1:3,:)');
xset("window",2);plot2d2("gnn",[t',t'],xcd(4:5,:)');
xset("window",0);
.fi
.SH SEE ALSO
ode, odeoptions, csim, external
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