146 Simple feedback loop

(i) Suppose that $G$ has $k$ inputs and $m$ outputs. We choose $K$ to have $m$ inputs and $k$ outputs. Then the simple feedback loop (SFL) has transfer function

$$T={(I+GK)}^{-1}G.$$

(ii) Suppose that $G$ and $K$ are rational matrix functions. The poles of $G$ are called open loop poles; the poles of $T$ are called closed loop poles. (iii) PID (proportional-integral-differentiator) controllers $K$ have the form

$$K(s)=a+\frac{b}{s}+cs.$$