Definition (Transfer function)

Consider the LTIC system $$\begin{cases} \dot{x}(t)=Ax(t)+Bu(t) \\ y(t)=Cx(t)+Du(t) \end{cases}$$where $x(0)=0$. We have the Laplace Transform version of this system: $$\begin{cases} s\hat{x}(s)&=A\hat{x}(s)+B\hat{u}(s) \\ \hat{y}(s)&=C\hat{x}(s)+D\hat{u}(s) \end{cases}$$as a result, the input-output behaviour is given by $$\hat{y}(s)=G(s)\hat{u}(s)$$where $$G(s)=C(sI-A)^{-1}+D$$ is called the transfer function of the above LTIC system. We adopt the following notation $$\begin{pmatrix}\begin{array}{c|c} A&B \\ \hline C &D \end{array}\end{pmatrix}(s):=C(sI-A)^{-1}B+D=G(s)$$