Multiplication of state space transfer function (state-space form)

Question

If I know the following transfer function (ss-form)

How to obtain the following efficiently:

Thanks

Answer 1

First note that the input of $\hat{U}^\sim (s)$ is the output of $\hat{U}(s)$. Therefore,

$$\begin{align} \dot{x}_1 &= A x_1 + B u \\ y &= C x_1 + D u \\ &\\ \dot{x}_2 &= -A^* x_2 - C^* y \\ z &= B^* x_2 + D^* y \end{align}$$

Now, put $y$ in the second system to obtain $$\begin{align} \dot{x}_2 &= -A^* x_2 - C^* C x_1 - C^* D u \\ \dot{x}_1 &= A x_1 + B u \\ z &= B^* x_2 + D^* C x_1 + D^* D u \end{align}$$