How to deal with a negative pole (unstable) in the pre-filter of a control system?
In general, in order to stabilise a system, however complex, if you have the transfer function $G(s)$, you introduce a feedback loop with a new function $F(s)$.

Write the closed loop transfer function for the new system with the added $F(s)$, and then find $F(s)$ in order for the new system to be stable. This is like the first exercise in any control book in order to offer an example of stabilising a system via negative Feedback.

Check the book from Ogata on control engineering for reference.