someone can explain me how to identify a model of an unknown system of this general form:
$y(k)=y(k-1)\theta_1 +u(k-1)\theta_2$
I've to figure out the the $\theta$ value in order to reconstruct the model, with these data:
$u=[1.5,-0.3, 1, 0.5]^T$
$y=[-1, -4.2, -4.44, -7.328 ]^T$
If I understand your indexing scheme correctly, one of your equations (with $k=2$) reads:
$$-4.2 = -1 \theta_1 + 1.5 \theta_2.$$
And you have two more equations just like this, with $k=3,k=4$.
So you have three linear equations in the two unknowns $\theta_1,\theta_2$. If an exact solution exists, you can find it by Gaussian elimination, but this probably is not the case (typically if you have more linear equations than variables then no solution exists). As an alternative, you can find a least squares solution, which will always exist and will typically be unique when you have at least as many equations as variables.
Notably your setup is probably not exactly right, because the last element of $u$ never enters anywhere.