Module 10 Rules for plotting the root locus (4 hours)
Module 10 Rules for Plotting the Root Locus (4 hours)
(P173) In this section the concepts outlined previously will be developed further into some straightforward guidelines for plotting more complex root loci, which will be illustrated by focusing on a specific example Root locus:(review of previous lecture) 1+G(s)H(s)=0 G(s)H(s)=-1 GH=1 magnitude equation ZGH=-180 phase equation
(P173) In this section the concepts outlined previously will be developed further into some straightforward guidelines for plotting more complex root loci, which will be illustrated by focusing on a specific example
Rule #1 The Starting Points and the End Points of the root locus(根轨迹的起点和终点 The locus starts at the open-loop poles( the closed-loop poles for K=0), and finishes at the open-loop zeros( the closed-loop zeros for K=oo The number of segments going to infinity is n-m 根轨迹始于开环极点,终于开环零点。趋于无穷大的 线段条数为m-m。若n>m,则有n-m条根轨迹终止于无穷 远处;若m>n,则有m-n条根轨迹起始于无穷远处。)
• The locus starts at the open-loop poles ( the closed-loop poles for K = 0 ), and finishes at the open-loop zeros (the closed-loop zeros for K= ). The number of segments going to infinity is n-m. (根轨迹始于开环极点，终于开环零点。趋于无穷大的 线段条数为n-m。若n>m, 则有n-m条根轨迹终止于无穷 远处；若m>n，则有 m-n条根轨迹起始于无穷远处。) Rule #1 The Starting Points and the End Points of the Root Locus (根轨迹的起点和终点）
[ 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)
Rule #2 The Segments of the Root Locus on the Real Axis(实轴上的根轨迹) Segment of the real axis to the left of on an odd number of poles or zeros are segments of the root locus, remembering that complex poles or zeros have no effect (实轴上的根轨迹,是其右侧的开环零、极点数之和为奇数的 所在线段。或者说,实轴上,对应零、极点数之和为奇数的 左边线段为根轨迹。复数零、极点对该线段没有影响。) [Proving I p3 180° l80° P1 p2
• Segment of the real axis to the left of on an odd number of poles or zeros are segments of the root locus, remembering that complex poles or zeros have no effect. (实轴上的根轨迹，是其右侧的开环零、极点数之和为奇数的 所在线段。或者说，实轴上，对应零、极点数之和为奇数的 左边线段为根轨迹。复数零、极点对该线段没有影响。) Rule #2 The Segments of the Root Locus on the Real Axis(实轴上的根轨迹） [ Proving ] j 1 p 3 p2 p j 1 p p2 p3 1 p 1 p1 s − 1 p2 s − 1 p3 s − 1 s 180 1 s 180