[ Proving I K∏I( s open-loop zero G(S ∏(S-p) pi--open- loop pole 1+G(s)=01(s-n1)+k(s-2)=0 At the starting point of the root locus: K=0 V(s-p2)=0,S=P1;(=1,2,…,n) At the end point of the root locus:K→>∞ and the characteristic equation can be written as (s-p)+∏(s-z,)=0 When K→>S=2 K[ Proving ] ( ) ( ) ( ) 1 1 i n i j m j s p K s z G s  −  − = = =  p open loop pole z open loop zero i j − − − − − − 1 ( ) 0 ( ) ( ) 0 1 1 + =  − +  − = = = j m j i n i G s s p K s z At the starting point of the root locus: K=0 (s p ) 0, s p ; (i 1, 2, , n)  − i = = i =  At the end point of the root locus: K → and the characteristic equation can be written as ( ) ( ) 0 1 1 1  − +  − = = = j m j i n i s p s z K when K → j s = z ( j =1, 2,  , m)