154 Example

Let

$$G(s)=\frac{s^2+5}{s^2-s+1}.$$

Then with $s=(1-\lambda )/\lambda$, we have

$$G(s)=\frac{{(1-\lambda )}^2+5{\lambda }^2}{{(1-\lambda )}^2-\lambda (1-\lambda )+{\lambda }^2}=\frac{6{\lambda }^2-2\lambda +1}{3{\lambda }^2-3\lambda +1}.$$

Then by the Euclidean algorithm

$$\left(\frac{24\lambda -2}{7}\right)\left(3{\lambda }^2-3\lambda +1\right)-\left(\frac{12\lambda -9}{7}\right)\left(6{\lambda }^2-2\lambda +1\right)=1$$

so letting $\lambda =1/(1+s)$, we have $PX+QY=1$ with $P,Q,X,Y\in 𝒮$

$$\left(\frac{22-2s}{7(1+s)}\right)\left(\frac{3}{{(1+s)}^2}-\frac{3}{1+s}+1\right)-\left(\frac{3-9s}{7(1+s)}\right)\left(\frac{6}{{(1+s)}^2}-\frac{2}{1+s}+1\right)=1.$$