Let's say that a got an approximation state space mode:
$$\dot{x}(t) = Ax(t) + Bu(t)$$ $$y(t) = Cx(t) + Du(t)$$
All a know is that this state space model is just an approximation for the real process. But I have measured all the states over time $x(t)$
So I can add a vector to this state space mode:
$$\dot{x}(t) = A\theta x(t) + B\theta u(t)$$ $$y(t) = C\theta x(t) + D\theta u(t)$$
Where $$ \theta = \begin{bmatrix} a_1\\ a_2\\ \vdots \\ a_n\\ \end{bmatrix} $$
Then I need to choose the parameters $a_1, a_2, ... , a_n$ to make sure that the model is correct.
How do I choose the parameters $a_1, a_2, ... , a_n$ ?
Notice that $\theta$ is not a vector who going to change matrix $A$, only add some scalar to the system.
I might have wrong about this method, but my cource in system identification only teach out the theory.
Edit: Is there a better way to estimate a state space model?