Spectral factorization of polynomial x(s)

As a part of the algorithm that is used for Optimal robust stabilization I have to find stable polynomial such that:

Where and are numerator and denominator of a transfer function , and .

I have managed to find and by using:

s = tf('s');

% input of transfer function

transfer_function = input("Input the transfer function: ")

%ex: 1/s^3

[b,a] = tfdata(transfer_function);

arrayB = cell2mat(b);

arrayA = cell2mat(a);

syms s;

b = poly2sym(arrayB,s);

a = poly2sym(arrayA,s);

% b(-s) and a(-s)

minus_b = subs(b,s,-s);

minus_a = subs(a,s,-s);

expression = a*minus_a+b*minus_b;

But I am having trouble finding and .

I have tried using spectral factorization with the function spectralfact(), but it does not seem to work. Can you give me any tips on how to procede from here?

Answer 1

[Torsten]

Let

a(s)*a(-s)+b(s)*b(-s) = p0 + p1*s + ... + p2n*s^(2*n) 

Determine the roots of

a(s)*a(-s) + b(s)*b(-s)

The 2*n roots appear in pairs

s1,-s1,s2,-s2,...,sn,-sn

Take

s1,s2,...,sn

and build the polynomial

d(s)=a*(s-s1)*(s-s2)*...*(s-sn)    

with a constant "a" to be determined.

Then

d(s)*d(-s) = a^2*(-1)^n*(s+s1)*(s+s2)*...*(s+sn)*(s-s1)*(s-s2)*...*(s-sn)

This means that

a(s)*a(-s) + b(s)*b(-s) = d(s)*d(-s)

if

p2n = a^2*(-1)^n

thus

a = sqrt(p2n/(-1)^n)

I leave it to you to implement this in MATLAB.

[Omar Decevic]

Thank you very much!