148 Example of proportional feedback

Consider a SISO system with transfer function $G(s)=p(s)/q(s)$ with $p,q$ complex polynomials. The zeros of $p$ are called open loop zeros, while the zeros of $q$ are called open loop poles. Then the controller $-K=-k$ with constant $k>0$ gives a proportional negative feedback loop with

$$T={(I+KG)}^{-1}G=\frac{p}{kp+q}.$$

Example. Let $p(s)=s^2$ and $q(s)=bs+c$ where $b,c>0$. Then $G(s)$ is not proper; whereas

$$T(s)=\frac{s^2}{k{s}^2+bs+c}$$

is proper and has poles in the open left half plane at

$$s=\frac{-b\pm \sqrt{b^2-4ck}}{2k},$$

so the controlled system with $T$ is stable.