Improper Transfer Function to State Space

by ⋅ Aside

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.