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.TH ARL2 4 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
arl2 - SISO model realization by L2 transfer approximation
.SH CALLING SEQUENCE
.nf
h=arl2(y,den0,n [,imp])
h=arl2(y,den0,n [,imp],'all')
[den,num,err]=arl2(y,den0,n [,imp])
[den,num,err]=arl2(y,den0,n [,imp],'all')
.fi
.SH PARAMETERS
.TP 10
y
: real vector or polynomial in \fVz^-1\fR, it contains the
coefficients of the Fourier's series of the rational system to
approximate (the impulse response)
.TP
den0
: a polynomial which gives an initial guess of the solution, it may be
\fVpoly(1,'z','c')\fR
.TP
n
: integer, the degree of approximating transfer function (degree of
den)
.TP
imp
: integer in \fV(0,1,2)\fR (verbose mode)
.TP
h
: transfer function \fVnum/den\fR or transfer matrix (column vector)
when flag \fV'all'\fR is given.
.TP
den
: polynomial or vector of polynomials, contains the denominator(s) of
the solution(s)
.TP
num
: polynomial or vector of polynomials, contains the numerator(s) of
the solution(s)
.TP
err
: real constant or vector , the l2-error achieved for each solutions
.SH DESCRIPTION
\fV[den,num,err]=arl2(y,den0,n [,imp]) \fR finds a pair of polynomials
\fVnum\fR and \fVden\fR such that the transfer function \fVnum/den\fR
is stable and its impulse response approximates (with a minimal l2
norm) the vector \fVy\fR assumed to be completed by an infinite number
of zeros.
.IG
If y(z) = y(1)(1/z)+y(2)(1/z^2)+ ...+ y(ny)(1/z^ny)
.FI
.LA If~:$$ y(z)= y(1)({ 1 \over z})+y(2)({ 1 \over z})^2+\dots
.LA + y(ny)({ 1 \over z})^ny $$
then l2-norm of \fVnum/den - y(z)\fR is \fVerr\fR.
.LP
\fVn\fR is the degree of the polynomial \fVden\fR.
.LP
The \fVnum/den\fR transfer function is a L2 approximant of the
Fourier's series of the rational system.
.LP
Various intermediate results are printed according to \fVimp\fR.
.LP
\fV[den,num,err]=arl2(y,den0,n [,imp],'all') \fR returns in the
vectors of polynomials \fVnum\fR and \fVden\fR a set of local
optimums for the problem. The solutions are sorted with increasing
errors \fVerr\fR. In this case \fVden0\fR is already assumed to be
\fVpoly(1,'z','c')\fR
.SH EXAMPLE
.nf
v=ones(1,20);
xbasc();
plot2d1('enn',0,[v';zeros(80,1)],2,'051',' ',[1,-0.5,100,1.5])
[d,n,e]=arl2(v,poly(1,'z','c'),1)
plot2d1('enn',0,ldiv(n,d,100),2,'000')
[d,n,e]=arl2(v,d,3)
plot2d1('enn',0,ldiv(n,d,100),3,'000')
[d,n,e]=arl2(v,d,8)
plot2d1('enn',0,ldiv(n,d,100),5,'000')
[d,n,e]=arl2(v,poly(1,'z','c'),4,'all')
plot2d1('enn',0,ldiv(n(1),d(1),100),10,'000')
.fi
.SH SEE ALSO
ldiv, imrep2ss, time_id, armax, frep2tf
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