sfact

discrete time spectral factorization

Calling Sequence

F=sfact(P)

Arguments

:P real polynomial matrix :

Description

Finds F, a spectral factor of P. P is a polynomial matrix such that each root of P has a mirror image w.r.t the unit circle. Problem is singular if a root is on the unit circle.

sfact(P) returns a polynomial matrix F(z) which is antistable and such that

P = F(z)* F(1/z) *z^n

For scalar polynomials a specific algorithm is implemented. Algorithms are adapted from Kucera’s book.

Examples

//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)

See Also

• gtild tilde operation
• fspecg stable factorization of continuous time dynamical systems