I saw these two properties about root locus, but I was unable to demonstrate why this happen. Can someone help me?

Typical Pole-Zero Configurations and Corresponding Root Loci.
In summarizing, we show several open-loop pole-zero configurations and their corresponding root loci in Table 6–1. The pattern of the root loci depends only on the relative separation of the open-loop poles and zeros. If the number of open-loop poles exceeds the number of finite zeros by three or more, there is a value of the gain $K$ beyond which root loci enter the right-half $s$ plane, and thus the system can become unstable. A stable system must have all its closed-loop poles in the left-half $s$ plane.

Comments on the Root-Locus Plots.
It is noted that the characteristic equation of the negative feedback control system whose open-loop transfer function is $$G(s)H(s)=\frac{K\left(s^m+b_1s^{m-1}+\cdots+b_m\right)}{s^n+a_1s^{n-1}+\cdots+a_n}\quad(n\ge m)$$ is an $n$th-degree algebraic equation in $s$. If the order of the numerator of $G(s)H(s)$ is lower than that of the denominator by two or more (which means that there are two or more zeros at infinity), then the coefficient $a_1$ is the negative sum of the roots of the equation and is independent of $K$. In such a case, if some of the roots move on the locus toward the left as $K$ is increased, then the other roots must move toward the right as $K$ is increased. This information is helpful in finding the general shape of the root loci.


Let $H$ be the loop transfer function and suppose the system is controlled using inverting feedback with gain $K$. The closed loop TF is $C = H/(1+KH)$ (I assume a SISO system). Clearly as $K$ becomes large, $1+KH \approx KH$ and thus $C \approx 1/K$. That is "closed loop poles go to open loop zeros as gain is increased" or any equivalent statement you find in every undergrad textbook. Since you suppose $n \geq m +2$ then there are at least two poles which have nowhere to go--these must go to infinity. Since $K$ can only extend the bandwidth and not alter the asymptotic behavior of the overall transfer function (waterbed principle), the overall order of poles at plus and minus infinity must match.


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