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
Rule #3 The Symmetry and the Asymptotes of the root locus(根轨迹的对称性和渐近线) The loci are symmetrical about the real axis since complex roots are always in conjugate pairs.(根轨迹关 于实轴对称,因为复数根总是成对出现的。) The angle between adjacent asymptotes is 360%/(n-m) and to obey the symmetry rule, the negative real axis is one asymptote when n- m is odo.(相邻的渐近线之间的夹角是 360%(m-m)并同样服从对称规律。当n-m是奇数时,负实轴也是一个渐 近线。) The Angle of the asymptotes and real axis is:(渐近线与实 轴正向的夹角是) 2k+1 丌(k=0,1,…,n-m-1)
• The loci are symmetrical about the real axis since complex roots are always in conjugate pairs.（根轨迹关 于实轴对称，因为复数根总是成对出现的。） • The angle between adjacent asymptotes is 360º/(n-m), and to obey the symmetry rule, the negative real axis is one asymptote when n-m is odd.（相邻的渐近线之间的夹角是 360º/(n-m), 并同样服从对称规律。当n-m 是奇数时，负实轴也是一个渐 近线。） • The Angle of the asymptotes and real axis is: (渐近线与实 轴正向的夹角是） Rule #3 The Symmetry and the Asymptotes of the Root Locus (根轨迹的对称性和渐近线） ( 0,1, , 1) 2 1 = − − − + = k n m n m k a
Rule#4 The Real Axis intercept of the Asymptotes(断近线和实轴的交点) The asymptotes intersect the real axis at oa ∑p1-∑ 1-m where pi is the sum of the real parts of the open-loop poles(including complex roots)and 2 is the sum of the real parts of the open- loop zeros(also including complex zeros) ∑P1-∑ (渐近线与实轴的交点是可=三nm,式中∑P 是开环极点的实部的和(包括复数极点);∑ 是开环零点的实部的和(包括复数零点)
• The asymptotes intersect the real axis at , where is the sum of the real parts of the open-loop poles (including complex roots) and is the sum of the real parts of the open-loop zeros (also including complex zeros). （渐近线与实轴的交点是 ，式中 是开环极点的实部的和（包括复数极点）； 是开环零点的实部的和（包括复数零点）。） Rule #4 The Real Axis intercept of the Asymptotes (渐近线和实轴的交点） a n m p z i j a − − = pi j z j z i p n m p z i j a − − =
SP ∑p1-∑ 1-m 0+(-1)+(-2) 60° 3-0 2k+1 丌(k=0,1,…,n-m-1) 丌/3:k=0 〈元 丌/5=-丌/3:k=2
SP. j − 2 −1 0 1 3 0 0 ( 1) ( 2) = − − + − + − = − − = n m p z i j a = − = = = = = − − − + = / 5 / 3; 2 ; 1 / 3; 0 ( 0,1, , 1) 2 1 k k k k n m n m k a 60
Rule #5 The Angle of Emergence from Complex Poles and The Angle of Entry into Complex Zeros(根轨迹 的出射角和入射角) The angle of emergence from complex poles is given by 180 X(angles of the vectors from all other open-loop poles to the poles in question)+ 2(angles of the vectors from the open-loop zeros to the complex pole in question 0n=1809-∑∠(P-p/)+∑(P-=) The angle of entry into a complex zero may be found from the same rule and then the sign changed to produce the final result 0:=180°-∑∠(=1-1)+∑(=1-P)
• The angle of emergence from complex poles is given by 180º – Σ(angles of the vectors from all other open-loop poles to the poles in question) + Σ(angles of the vectors from the open-loop zeros to the complex pole in question). Rule #5 The Angle of Emergence from Complex Poles and The Angle of Entry into Complex Zeros (根轨迹 的出射角和入射角） = = = − − + − n j i j m j p i j i j p p p z i 1 1 180 ( ) ( ) • The angle of entry into a complex zero may be found from the same rule and then the sign changed to produce the final result. = = = − − + − m j i j n j z i j i pj z z z i 1 1 180 ( ) ( )
SP K(S+5) GH(s)=-2 K(S+5) s(s2+4s+8)s(s+2-j2)(s+2+j2 6.2+12=180°(∠(-2+2-0)+∠(-2+j12)-(-2-2) +∠(-2+j2)-(-5)) 2 4 180°-tg-2 tg ttg 632 1K=32 180°-135°-90°+33°=-12° 6.32 Fig. 10.4 Imaginary-axis crossing point
SP. ( 2 2)( 2 2) ( 5) ( 4 8) ( 5) ( ) 2 s s j s j K s s s s K s G H s + − + + + = + + + = = − − + = − − + − = − + − + − − = − − + − + − + − − − − − − − + 180 135 90 33 12 3 2 0 4 2 2 180 (( 2 2) ( 5)) 180 ( ( 2 2 0) (( 2 2) ( 2 2)) 1 1 1 2 2 t g t g t g j j j j j