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

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

where , , and .

We derive as:

We then have the following states:

where

we can see that

On to fill out the rest of the matrices as:

where .