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.TH SFACT G "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
sfact - discrete time spectral factorization
.SH CALLING SEQUENCE
.nf
F=sfact(P)
.fi
.SH PARAMETERS
.TP
P
: real polynomial matrix
.SH DESCRIPTION
Finds \fVF\fR, a spectral factor of \fVP\fR. \fVP\fR is a
polynomial matrix such that each root of \fVP\fR has a
mirror image w.r.t the unit circle. Problem is singular
if a root is on the unit circle.
.LP
.Vb sfact(P)
returns a polynomial matrix \fVF(z)\fR which is antistable and such that
.LP
\fVP = F(z)* F(1/z) *z^n\fR
.LP
For scalar polynomials a specific algorithm is implemented.
Algorithms are adapted from Kucera's book.
.SH EXAMPLE
.nf
//Simple polynomial example
z=poly(0,'z');
p=(z-1/2)*(2-z)
w=sfact(p);
w*numer(horner(w,1/z))
//matrix example
F1=[z-1/2,z+1/2,z^2+2;1,z,-z;z^3+2*z,z,1/2-z];
P=F1*gtild(F1,'d'); //P is symmetric
F=sfact(P)
roots(det(P))
roots(det(gtild(F,'d'))) //The stable roots
roots(det(F)) //The antistable roots
clean(P-F*gtild(F,'d'))
//Example of continuous time use
s=poly(0,'s');
p=-3*(s+(1+%i))*(s+(1-%i))*(s+0.5)*(s-0.5)*(s-(1+%i))*(s-(1-%i));p=real(p);
//p(s) = polynomial in s^2 , looks for stable f such that p=f(s)*f(-s)
w=horner(p,(1-s)/(1+s)); // bilinear transform w=p((1-s)/(1+s))
wn=numer(w); //take the numerator
fn=sfact(wn);f=numer(horner(fn,(1-s)/(s+1))); //Factor and back transform
f=f/sqrt(horner(f*gtild(f,'c'),0));f=f*sqrt(horner(p,0)); //normalization
roots(f) //f is stable
clean(f*gtild(f,'c')-p) //f(s)*f(-s) is p(s)
.fi
.SH SEE ALSO
gtild, fspecg
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