MOESP and N4SID not always working about sippy

[CPCLAB-UNIPI]

Dear Jelmer44,
sorry for the delay in our reply.

To answer your issue please consider two different aspects which make your generated matrices different from the actual ones.

1. sinusoidal inputs are surely less informative than specific input signals as PRBS (pseudo-random binary signal) or GBN (generalized binary noise), as they only excite your system at the unique frequency of the input (0.3 in your case).

2. In addition, note that when you are identifying a input-output model (e.g., transfer functions TF), there is only one correct model which corresponds exactly to your actual dynamics.
   Otherwise, when managing with state-space (SS) models, there are infinite similar models which correspond to your actual dynamics.
   Note that transformation from SS to TF is unique, while the opposite in not, as there are infinite transformations called "realizations" and infinite SS systems similar by a base-matrix T.

Consider the following Input-output equivalence between two state-space models (with same number of states):
original system
xk+1 = Axk + Buk
yk = Cxk + Duk

similar system
with zk = T xk
zk+1 = TAT^{-1}zk + TBuk = At zk + Bt uk
yk = CT^{-1}zk + Duk = Ct zk + D uk

Invariant parameters:
D does not change
A and At have same eigenvalues (poles of G(z))
Gain matrix G(1) = C(I - A)^{-1}B + D is the same Gt (1) = Ct (I - At)^{-1}Bt + Dt

Therefore, your identified systems may be all similar to the actual one.

Hoping to have clarified all your doubts.
Best