State Space Representation

a_n \frac{d^n y}{dt^n} + a_{n-1} \frac{d^{n-1} y}{dt^{n-1}} + \dots + a_0 y(t) = b_q \frac{d^q u}{dt^q} + \dots + b_0 u(t)

\begin{align} \dot{x}(t) &= A x(t) + B u(t), \quad x(0) = x_0 \\ y(t) &= C x(t) + D u(t) \end{align}

$x(t) \in \mathbb{R}^n$ : State vector (order = number of states = order of the differential equation)
$u(t) \in \mathbb{R}^m$ : Input vector
$y(t) \in \mathbb{R}^r$ : Output vector
$A \in \mathbb{R}^{n \times n}$ : System matrix (describes the dynamics)
$B \in \mathbb{R}^{n \times m}$ : Input matrix (how inputs affect the states)
$C \in \mathbb{R}^{r \times n}$ : Output matrix (how outputs are calculated from states)
$D \in \mathbb{R}^{r \times m}$ : Feedthrough matrix (direct influence of inputs on outputs)

with:

System Matrix $A$ :

A = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ -a_0 & -a_1 & -a_2 & \cdots & -a_{n-1} \end{bmatrix}

Input Matrix $B$ :

B = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{bmatrix}

Output Matrix $C$ (for output equation coefficients $b_0, b_1, \dots, b_q$ ):

C = \begin{bmatrix} b_0 - b_n a_0 & b_1 - b_n a_1 & \dots & b_{n-1} - b_n a_{n-1} \end{bmatrix}

Feedthrough Matrix $D$ :

D = b_0

Special case for $q \lt n$ :

C = \begin{bmatrix} b_0 & b_1 & \dots & b_{q} & 0 & \dots & 0 \end{bmatrix}

D = 0