I proposed using an improper transfer function to model a simple tuned-mass damper (read more here) with the transfer function:

$\frac{y}{u} = \frac{mcs^{3}+mks^{2}}{ms^{2}+cs+k}$

To convert this to state-space representation we need to use the more generalized descriptor state-space notation which introduces the “E” term:

$E\dot{x}(t)=Ax(t)+Bu(t)$

$y(t)=Cx(t)+Du(t)$

where $E,A \in \mathbb{R}^{n\times n}$$B \in \mathbb{R}^{n\times m}$$C \in \mathbb{R}^{p\times n}$ and $D \in \mathbb{R}^{p\times m}$.

We derive $C$ as:

$y = C \times \left[\begin{matrix}x_1\\x_2\\x_3\\x_4\end{matrix}\right]$

$y = \left[\begin{matrix}mc,mk,0,0\end{matrix}\right]\times \left[\begin{matrix}x_1\\x_2\\x_3\\x_4\end{matrix}\right]$

We then have the following states:

$\left[\begin{matrix}x_1\\x_2\\x_3\\x_4\end{matrix}\right]=\left[\begin{matrix}us^3/d\\us^2/d\\us/d\\u/d\end{matrix}\right]$

$\left[\begin{matrix}\dot{x}_1\\\dot{x}_2\\\dot{x}_3\\\dot{x}_4\end{matrix}\right]=\left[\begin{matrix}us^4/d\\us^3/d\\us^2/d\\us/d\end{matrix}\right]$

where $d = ms^2+cs+k$

we can see that

$\begin{matrix}u&=&d\times x_4&\\u&=&ms^2x_4+csx_4+kx_4&\\u&=&m\dot{x}_3+cx_3+kx_4&\\\end{matrix}$

On to fill out the rest of the matrices as:

$\begin{matrix}E&\dot{X}&=&A&X&+&B&u\\ \left[\begin{matrix}0&1&0&0\\0&0&1&0\\0&0&m&0\\0&0&0&1\end{matrix}\right]&\left[\begin{matrix}\dot{x}_1\\\dot{x}_2\\\dot{x}_3\\\dot{x}_4\end{matrix}\right]&=&\left[\begin{matrix} 1&0&0&0\\0&1&0&0\\0&0&-c&-k\\0&0&0&1 \end{matrix}\right]&\left[\begin{matrix}x_1\\x_2\\x_3\\x_4\end{matrix}\right]&+&\left[\begin{matrix}0\\0\\1\\0\end{matrix}\right]&u\end{matrix}$

$\begin{matrix}Y&=&C&X&+&D&u&\\Y&=&\begin{matrix}\left[mc,mk,0,0\right]\end{matrix}&\left[\begin{matrix}x_1\\x_2\\x_3\\x_4\end{matrix}\right]&&\end{matrix}$

where $D=0$.