Compute the left normalized coprime factorization of a SISO system.

sys = zpk([1 -1+2i -1-2i],[-1 2+1i 2-1i],1);
[fact,Ml,Nl] = lncf(sys);

Examine the original system and its factors.

sys

sys =
 
  (s-1) (s^2 + 2s + 5)
  --------------------
  (s+1) (s^2 - 4s + 5)
 
Continuous-time zero/pole/gain model.

zpk(Ml)

ans =
 
  0.70711 (s+1) (s^2 - 4s + 5)
  ----------------------------
    (s+1) (s^2 + 3.162s + 5)
 
Continuous-time zero/pole/gain model.

zpk(Nl)

ans =
 
  0.70711 (s-1) (s^2 + 2s + 5)
  ----------------------------
    (s+1) (s^2 + 3.162s + 5)
 
Continuous-time zero/pole/gain model.

The numerators of the factors Ml and Nl are the denominator and numerator of sys, respectively. Thus, sys = Ml\Nl. lncf chooses the denominators of the factors such that the system $\left[{\mathit{M}}_{\mathit{l}}\left(\mathit{j}\omega \right),{\mathit{N}}_{\mathit{l}}\left(\mathit{j}\omega \right)\right]$ is a unit vector at all frequencies. To confirm that property of the factorization, examine the singular values of fact, which is a stable minimal realization of $\left[{\mathit{M}}_{\mathit{l}}\left(\mathit{j}\omega \right),{\mathit{N}}_{\mathit{l}}\left(\mathit{j}\omega \right)\right]$.

sigma(fact)

Within a small numerical error, the singular value of fact is 1 (0 dB) at all frequencies.

Compute the left normalized coprime factorization of a state-space model that has two outputs, two inputs, and three states.

rng(0); % for reproducibility
sys = rss(3,2,2);
[fact,Ml,Nl] = lncf(sys);

fact is a stable minimal realization of the factorization given by [Ml,Nl].

isstable(fact)

ans = logical
   1

Another property of fact is that its frequency response F(jω) is an orthogonal matrix at all frequencies (F(jω)’F(jω) = I). Confirm this property by examining the singular values of fact. Within a small numerical error, the singular values are 1 (0 dB) at all frequencies.

sigma(fact)

Confirm that the factors satisfy sys = Ml\Nl by examining the singular values of both.

sigma(sys,'b-',Ml\Nl,'r--')