How can I estimate the model
$$A(q)y(t) = B(q)u(t) + C(q)e(t)$$
Where $e(t)$ is the noise and $u(t)$ is input and $y(t)$ is output of the model? I don't know how to find the $e(t)$ noise.
Notice that the kalman gain matix $K$ can be found from:
$$A = \begin{bmatrix} -a_1 & 1 & 0 & \cdots & 0\\ \vdots & & \ddots & & \\ -a_{n-1}& 0 & &\cdots & 1 \\ -a_n &0 & &\cdots & 1 \end{bmatrix} $$
$$ B = \begin{bmatrix} 0 & \dots & 0 &b_0 & \dots & b_m \end{bmatrix}^T $$
$$ \bar C = \begin{bmatrix} 1 &0 &\dots &0 \end{bmatrix} $$
$$ K = \begin{bmatrix} c_1 -a_1 & \dots & c_n-a_n \end{bmatrix}^T$$
Because $det(qI - A + K \bar C) = C(q)$
Source:
Adaptive Control by Karl Johan Åström - Second edition. Page 166.