Root Locus
Table of Contents
$$\text{System} = {s^2 + s + 1 \over s^3 + 4s^2 + Ks + 1}$$
$$1 + KG(s) = 0$$
rlocus(G(s))
$$1+KG(s) = 0 \implies KG(s) = -1 = 1 \angle (2k+1) \pi, \quad k = 0, \pm1, \pm 2, \cdots$$
Poles of $G(s)$ are when $P(s) = 0$, $K = 0$
Zeros of $G(s)$ are when $Q(s) = 0$, as $K \rightarrow \infty, P(s) + \infty Q(s) = 0$
So closed loop poles travel from poles of $G(s)$ to zeros of $G(s)$
$$KG(s) = \frac{K}{s(s+1)(s+2)}$$
$$KG(s) \approx \frac{K}{s^3}$$
Given the system transfer function
$$G(s) = \beta\frac{(s-z_1)(s-z_2)\cdots}{(s-p_1)(s-p_2)\cdots} = \beta \frac{s^m - \left(\sum z_i \right)s^{m-1} + \cdots}{s^n - \left(\sum p_i \right)s^{n-1} + \cdots}, \quad \text{assume }\; n>m$$
For a large value of $s$, $G(s)$ approximately looks like having $n-m$ repeated poles at $\sigma_A$ on the real axis
Break-away is the point where loci leave the real axis.
Break-in is the point where loci enter the real axis.
The method is to maximize and minimizes the gain, $K$, using differential calculus.