MATH319 Slides 162 Internal stability of

S F L

163 Youla’s stabilising controllers

We now allow $G(s)$ to be an arbitrary rational function, and write it in the form $G=P/Q$ with coprime $P,Q\in{\cal S}$ as in Frame 159.

Theorem

Suppose that $G=P/Q$ has a coprime factorisation $PX+QY=1$ where $P,Q,X,Y\in{\cal S}$ . Then $K=X/Y$ internally stabilises $S F L$ .

Actually Youla found all of the stabilising controllers, but we give the simplest case. By previous results, there exists a $S I S O$ such that $K$ is the transfer function, so in this sense we have shown that any $S I S O$ with rational transfer function can be stabilized by a rational (possibly unstable) controller.