Consider the following system.

G0 = ss(zpk([-1 -5 1+2i 1-2i],[-100 1+2i 1-2i -10],1e3));
H = G0'*G0;

G0 has a mix of stable and unstable dynamics. H is a self-conjugate system whose dynamics consist of the poles and zeros of G0 and their reflections across the imaginary axis. Use spectral factorization to separate the stable poles and zeros into G and the unstable poles and zeros into G'.

[G,S] = spectralfact(H);

Confirm that G is stable and minimum phase, by checking that all its poles and zeros fall in the left half-plane (Re(s) < 0).

p = pole(G)

p = 4×1 complex
102 ×

  -0.0100 + 0.0200i
  -0.0100 - 0.0200i
  -0.1000 + 0.0000i
  -1.0000 + 0.0000i

z = zero(G)

z = 4×1 complex

  -1.0000 + 2.0000i
  -1.0000 - 2.0000i
  -1.0000 + 0.0000i
  -5.0000 + 0.0000i

G also has unit feedthrough.

G.D

ans = 1

Because H is SISO, S is a scalar. If H were MIMO, the dimensions of S would match the I/O dimensions of H.

S

S = 1000000

Confirm that G and S satisfy H = G'*S*G by comparing the original system to the difference between the original and factored systems. sigmaplot throws a warning because the difference is very small.

Hf = G'*S*G;
sigmaplot(H,H-Hf)

Warning: The frequency response has poor relative accuracy. This may be because the response is nearly zero or infinite at all frequencies, or because the state-space realization is ill conditioned. Use the "prescale" command to investigate further.

Suppose that you have the following 2-output, 2-input state-space model, F.

A = [-1.1  0.37;
     0.37 -0.95];
B = [0.72 0.71;
     0    -0.20];
C =  [0.12 1.40
      1.49 1.41];
D = [0.67 0.7172;
    -1.2  0];
F = ss(A,B,C,D);

Suppose further that you have a symmetric 2-by-2 matrix, R.

R =   [0.65    0.61
       0.61   -3.42];

Compute the spectral factorization of the system given by H = F'*R*F, without explicitly computing H.

[G,S] = spectralfact(F,R);

G is a minimum-phase system with identity feedthrough.

G.D

ans = 2×2

     1     0
     0     1

Because F is has two inputs and two outputs, both R and S are 2-by-2 matrices.

Confirm that G'*S*G = F'*R*F by comparing the original factorization to the difference between the two factorizations. The singular values of the difference are far below those of the original system.

Ff = F'*R*F;
Gf = G'*S*G;
sigmaplot(Ff,Ff-Gf)

Consider the following discrete-time system.

F = zpk(-1.76,[-1+i -1-i],-4,0.002);

F has poles and zeros outside the unit circle. Use spectralfact to compute a system G with stable poles and zeros, such that G'*G = F'*F.

G = spectralfact(F,[])

G =
 
  -3.52 z (z+0.5682)
  ------------------
   (z^2 + z + 0.5)
 
Sample time: 0.002 seconds
Discrete-time zero/pole/gain model.

Unlike F, G has no poles or zeroes outside the unit circle. G does have an additional zero at z = 0, which is a reflection of the unstable zero at z = Inf in F.

pzplot(G)

Confirm that G'*G = F'*F by comparing the original factorization to the difference between the two factorizations. The singular values of the difference are far below those of the original factorization.

Ff = F'*F;
Gf = G'*G;
sigmaplot(Ff,Ff-Gf)