149 Example of proportional-integral feedback

Find a feedback controller of the form $-a-b/s$ that stabilises the plant with transfer function $G(s)=s/(s-1)(s+3)$.

The plant has poles at $s=1,-3$, hence is unstable. We consider

$$\frac{G}{1+GK}=\frac{1}{(1/G)+K}=\frac{s}{s^2+(2-a)s-b-3}.$$

For stability we require positive coefficients, so $2-a>0$ and $-b-3>0$. In particular, we choose $b=-4$ and $a=0$, and get

$$\frac{G}{1+GK}=\frac{s}{{(1+s)}^2},$$

which is stable.