State Space Representation

SSR takes the following form

$$\begin{array}{r}\dot{x}(t)=Ax(t)+Bu(t)\\ y(t)=Cx(t)+Du(t)\end{array}$$

The general solution in the time domain is

$$\begin{array}{r}y(t)=C{e}^{At}x(0)+C{\int }_0^t{e}^{A(t-\tau )}Bu(\tau )d\tau +Du(t)\end{array}$$

The transfer function (or impulse response) is given by

$$G(s)=\frac{Y(s)}{U(s)}=C(sI-A{)}^{-1}B+D$$