Linear System

Definition

$$\begin{align*} \dot{x}(t)=Ax(t)+Bu(t)+c \\ y(t)=Cx(t)+Du(t)+k \end{align*}$$

where $x(t) \in \mathbb{R}^{n}$ is the system state, $u(t) \in \mathbb{R}^{m}$ is the system input, $y(t) \in \mathbb{R}^{p}$ is the system output, and $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n}$, $c \in \mathbb{R}^{n}$, $C \in \mathbb{R}^{p \times n}$, $D \in \mathbb{R}^{p \times m}$, $k \in \mathbb{R}^{p}$.

Example

$$\begin{align*} \begin{bmatrix} \dot{x_{0}} \\ \dot{x_{1}} \end{bmatrix} &= \begin{bmatrix} -2 & 0 \\ 1 & -3 \end{bmatrix} \begin{bmatrix} x_{0} \\ x_{1} \end{bmatrix} + \begin{bmatrix} 1 \\ 1 \end{bmatrix} u \\ y &= \begin{bmatrix} 1 \ 0 \end{bmatrix} \begin{bmatrix} x_{0} \\ x_{1} \end{bmatrix} \end{align*}$$