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.TH flts 1 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
flts - time response (discrete time, sampled system)
.SH CALLING SEQUENCE
.nf
[y [,x]]=flts(u,sl [,x0])
[y]=flts(u,sl [,past])
.fi
.SH PARAMETERS
.TP 10
u
: matrix (input vector)
.TP 10
sl
: list (linear system \fVsyslin\fR)
.TP 10
x0
: vector (initial state ; default value=\fV0\fR)
.TP 10
past
: matrix (of the past ; default value=\fV0\fR)
.TP 10
x,y
: matrices (state and output)
.SH DESCRIPTION
.LP
State-space form:
.LP
\fVsl\fR is a \fVsyslin\fR list containing the matrices of the
following linear system
.LP
\fVsl=syslin('d',A,B,C,D)\fR (see \fVsyslin\fR):
.nf
x[t+1] = A x[t] + B u[t]
y[t] = C x[t] + D u[t]
.fi
or, more generally, if \fVD\fR is a polynomial matrix (\fVp = degree(D(z))\fR) :
.IG
.nf
D(z)=D_0 + z D_1 + z^2 D_2 +..+ z^p D_p
y[t] = C x[t] + D_0 u[t] + D_1 u[t+1] +..+ D_[p] u[t+p]
.fi
.FI
.LA $$ D(z)=D_0 + z D_1 + z^2 D_2 +..+ z^p D_p $$
.LA $$ y_{t} = C x_{t} + D_0 u_{t} + D_1 u_{t+1} +..+ D_{p}u_{t+p} $$
.IG
.nf
u=[u0,u1,... un] (input)
y=[y0,y1,... yn-p] (output)
x=x[n-p+1] (final state, used as \fVx0\fR at next call to flts)
.fi
.FI
.LA $$ u=[u_0,u_1,... u_n] (input) $$
.LA $$ y=[y_0,y_1,... y_{n-p}] (output) $$
.LA $$ x=x_{n-p+1} $$ (final state, used as x0 at next call to flts)
.PP
Transfer form:
.LP
\fV y=flts(u,sl[,past])\fR. Here \fVsl\fR is a linear system in
transfer matrix representation i.e
.LP
\fVsl=syslin('d',transfer_matrix)\fR (see \fVsyslin\fR).
.IG
.nf
past = [u ,..., u ]
[ -nd -1]
[y ,..., y ]
[ -nd -1]
.fi
.FI
.LA
.LA $$ past = \left[ \matrix{
.LA u_{-nd} & \ldots & u_{-1} \cr
.LA y_{-nd} & \ldots & u_{-1}
.LA } \right] $$
.LA
is the matrix of past values of u and y.
.LP
nd is the maximum of degrees of lcm's of each row of the denominator
matrix of sl.
.nf
u=[u0 u1 ... un] (input)
y=[y0 y1 ... yn] (output)
.fi
p is the difference between maximum degree of numerator and
maximum degree of denominator
.SH EXAMPLE
.nf
sl=syslin('d',1,1,1);u=1:10;
y=flts(u,sl);
plot2d2("onn",(1:size(u,'c'))',y')
[y1,x1]=flts(u(1:5),sl);y2=flts(u(6:10),sl,x1);
y-[y1,y2]
//With polynomial D:
z=poly(0,'z');
D=1+z+z^2; p =degree(D);
sl=syslin('d',1,1,1,D);
y=flts(u,sl);[y1,x1]=flts(u(1:5),sl);
y2=flts(u(5-p+1:10),sl,x1); // (update)
y-[y1,y2]
//Delay (transfer form): flts(u,1/z)
// Usual responses
z=poly(0,'z');
h=(1-2*z)/(z^2+0.3*z+1)
u=zeros(1,20);u(1)=1;
imprep=flts(u,tf2ss(h)); //Impulse response
plot2d2("onn",(1:size(u,'c'))',imprep')
u=ones(1,20);
stprep=flts(u,tf2ss(h)); //Step response
plot2d2("onn",(1:size(u,'c'))',stprep')
//
// Other examples
A=[1 2 3;0 2 4;0 0 1];B=[1 0;0 0;0 1];C=eye(3,3);Sys=syslin('d',A,B,C);
H=ss2tf(Sys); u=[1;-1]*(1:10);
//
yh=flts(u,H); ys=flts(u,Sys);
norm(yh-ys,1)
//hot restart
[ys1,x]=flts(u(:,1:4),Sys);ys2=flts(u(:,5:10),Sys,x);
norm([ys1,ys2]-ys,1)
//
yh1=flts(u(:,1:4),H);yh2=flts(u(:,5:10),H,[u(:,2:4);yh(:,2:4)]);
norm([yh1,yh2]-yh,1)
//with D<>0
D=[-3 8;4 -0.5;2.2 0.9];
Sys=syslin('d',A,B,C,D);
H=ss2tf(Sys); u=[1;-1]*(1:10);
rh=flts(u,H); rs=flts(u,Sys);
norm(rh-rs,1)
//hot restart
[ys1,x]=flts(u(:,1:4),Sys);ys2=flts(u(:,5:10),Sys,x);
norm([ys1,ys2]-rs,1)
//With H:
yh1=flts(u(:,1:4),H);yh2=flts(u(:,5:10),H,[u(:,2:4); yh1(:,2:4)]);
norm([yh1,yh2]-rh)
.fi
.SH SEE ALSO
ltitr, dsimul, rtitr
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