I'm trying to change this filter transfer function to state space representation
$ y_t=\frac{1+b_1 z^{-1}}{1+a_1 z^{-1} +a_2 z^{-2}}u_t $
I tried writing it as time series
$ y_t+a_1 y_{t-1}+a_2 y_{t-2}=u_t+b_1 u_{t-1} $
Here is where I am not sure how to continue, I wrote an expression
$ x_t=\frac{1}{1+a_1 z^{-1}+a_2 z^{-2}}u_{t-1} $
such that
$ y_t=(1+b_1z^{-1})x_{t+1}=x_{t+1}+b_1x_t $
and
$ (1+a_1 z^{-1} +a_2 z^{-2})x_t=u_{t-1}$
The idea was to get:
$ x_t=u_{t-1}-a_1x_{t-1}-a_2x_{t-2} $ $ (1)$
$ x_{t+1}=u_{t}-a_1x_{t}-a_2x_{t-1} $ $(2)$
renaming
$ x_{t+1}=x_{a,t+1} $
$ x_{t}=x_{b,t+1} $
and write my state equations as
$ \begin{bmatrix} x_{a,t+1}\\ x_{b,t+1}\end{bmatrix}=\begin{bmatrix} -a_1 && -a_2\\ 1&&0\end{bmatrix} \begin{bmatrix} x_{a,t}\\ x_{b,t}\end{bmatrix} +\begin{bmatrix} 1\\ 0\end{bmatrix}u_t$
from (2) my states are
$ x_{a,t}=x_t $
$ x_{b,t}=x_{t-1} $
But my problem comes when I want to state my output equation, which is
$ y_t=x_{t+1}+b_1x_t $
but $ x_{t+1} $ is not one of my states so I'm not being able to express my output equation in terms of $x_{a,t}$ and $x_{b,t}$
I would appreciate any hint on where I am making the mistake, thanks for your help.
