For every controller, there is a representative gain $K$.

From Block Diagrams (WIP), Laplace Transform, and PID Control, we know that for a typical closed-loop system, the characteristic equation is equal to:

$$ 1+KG_c(s)G_p(s)H(s)=0 $$

Where

$K$ is the representative gain (for the controller,

$G_c(s)$ is the transfer function of the controller,

$G_p(s)$ is the transfer function of the plant,

$H(s)$ is the transfer function of the sensor.

Root locus is the locus of the closed-loop poles (the solutions of the characteristic equation) when the representative gain $K$ varies from 0 to infinity.

• In other words, it plots the “trajectory” (the locus) of the closed-loop poles as $K$ changes, on the S-plane.

The open loop transfer function:

Note that for the closed loop above, the transfer function is $\frac{G(s)}{1+G(s)H(s)}$.

The open-loop transfer function for above is $G(s)H(s)$.

Representative gain K가 바뀜에 따라 Closed-loop Pole들이 어떻게 되는지 알고 싶음. 즉 Characteristic Equation (N차 방정식)을 풀어서, 그 솔루션이 어떻게 되는지를 보고 싶다는 말.

• Why? Closed loop pole의 값 (a.k.a. S-plane 상 위치) 에 따라 stability가 정해지니까 보고싶음
• 4차 방정식부터는 근의 공식이라는게 존재하지 않음. Oh no~
• 예전에는 컴퓨터가 없어서 그림으로 이걸 풀었음 (Evans? 1910’s)

$1+KG_c(s)G_p(s)H(s)=0$ → $G_c(s)G_p(s)H(s)=-\frac{1}{K}$

where (0<K<inf)

The solutions to the above equations are complex numbers. So, they will have magnitude and “angle” on the S-plane. Thus, they need to satisfy the follwing two criteria:

1. Magnitude Criterion
   1. $\left|G_c(s)G_p(s)H(s)\right|=\left|-\frac{1}{K}\right|=\frac{1}{K}$