158 Controlling rational systems

Proposition

Let $G=P/Q$ be a complex rational function, as above. Then the rational $K=X/Y$ is such that

{{G}\over{1+GK}},\quad{{K}\over{1+GK}},\quad{{1}\over{1+GK}},\quad{{GK}\over{1% +GK}}

are all stable rational functions.

Proof. We recall that $PX+QY=1$ , we we have

{{G}\over{1+GK}}={{P/Q}\over{1+PX/QY}}=PY;

{{K}\over{1+GK}}={{X/Y}\over{1+PX/QY}}=XQ;