I need help to convert this dynamic system in a space state model I need some help with this dynamic model, which I got from a electrical system. The values a,b, c and d are non-zero reals. If someone has an idea of how to convert this into state space model I will be grateful.
PS: The most difficult from my analysis is tha have a second-order input
au''(t) = by''(t) + cy'(t) + dy(t)
 A: This is not a full-fledged solution, but I remember there is a formula that you can directly convert the Ordinary Differential Equation (ODE)
$$b \ddot{y} + c \dot{y} + d y = a \ddot{u}$$
to the State Space form
$$\mathbf{\dot{x}} = \mathbf{A} \mathbf{x} + \mathbf{B} u$$
$$y = \mathbf{C} \mathbf{x} + \mathbf{D} u.$$
However, without relying too much on the rote memorization of the formula, you can try applying what was previously learned by critically thinking about integrating both sides:
$$\int \int \left\{\ddot{y} + \frac{c}{b} \dot{y} + \frac{d}{b} y\right\} = \frac{a}{b} \int \int \ddot{u}$$
to obtain a function of $y$ on the left-hand side, and $u$ on the right-hand side. If you compare with the output $y$ in the State-space model, then you can deduce that the direct transmission matrix $\mathbf{D}$ is no longer equal to zero. This implies that
$$y        = \alpha        x_{1} + \beta u $$
$$\dot{y}  = \alpha  \dot{x}_{1} + \beta \dot{u} $$
$$\ddot{y} = \alpha \ddot{x}_{1} + \beta \ddot{u} $$
If the ODE is rearranged to become
$$\ddot{y} = - \frac{c}{b} \dot{y} - \frac{d}{b} y + \frac{a}{b} \ddot{u}$$
I think you will get the idea and can carry on from this point.
A: For SISO systems, build the transfer function
$$\dfrac{Y(s)}{U(s)}=H(s)=\dfrac{N(s)}{D(s)}$$
where $N(s)=\sum_{i=0}^mb_is^i$ and $D(s)=\sum_{i=0}^na_is^i$, where $m\le n$ and $a_n=1$.
Then, the idea is to decompose $H(s)$ as
$$\dfrac{Y(s)}{U(s)}=\dfrac{Y(s)}{X(s)}\dfrac{X(s)}{U(s)},$$
where $X(s)$ is the state. Then, the idea is to "integrate the input to the state" and "differentiate the state to the output". Mathematically speaking, we define
$$\dfrac{X(s)}{U(s)}=\dfrac{1}{D(s)}\ \mathrm{and}\ \dfrac{Y(s)}{X(s)}=N(s).$$
This yields
$$\sum_{i=0}^na_ix^{(i)}(t)=u(t),$$
which can be written in state-space form as
$$\mathbf{\dot{x}} = \mathbf{A} \mathbf{x} + \mathbf{B} u$$ where $\mathbf x=(x,\ldots,x^{(n-1)})$ and
$$\mathbf A=\begin{bmatrix} 0 & 1 & 0 & \ldots & 0\\
0 & 0 & 1 & \ldots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \ldots & 1\\
-a_0 & -a_1 & -a_2 & \ldots & -a_{n-1}
\end{bmatrix},\ \mathbf B=\begin{bmatrix}0\\ 0 \\\vdots\\0 \\1\end{bmatrix}.$$
Similarly, we have that
$$\sum_{i=0}^mb_ix^{(i)}(t)=y(t),$$
which can be written in state-space form as
$$\mathbf{y} = \mathbf{C} \mathbf{x} + \mathbf{D} u,$$
where
$$\mathbf C=\begin{bmatrix} 
b_0-a_0b_n & b_1-a_1b_{n} & b_2-a_2b_{n} & \ldots & b_{n-2}-a_{n-1}b_n
\end{bmatrix},\ \mathbf  D=b_n.$$
