Summary of rules for plotting root locus Rule #l The Starting points and the End points of the root locus(根轨迹的起点和终点) Rule #2 The Segments of the Root Locus on the Real axis(实轴上的根轨迹) Rule #3 The Symmetry and the Asymptotes of the root locus(根轨迹的对称性和渐近线) Rule #4 The Real Axis intercept of the △ asymptotes(渐近线和实轴的交点)
Summary of Rules for Plotting Root Locus Rule #1 The Starting Points and the End Points of the Root Locus (根轨迹的起点和终点） Rule #2 The Segments of the Root Locus on the Real Axis (实轴上的根轨迹） Rule #3 The Symmetry and the Asymptotes of the Root Locus (根轨迹的对称性和渐近线） Rule #4 The Real Axis intercept of the Asymptotes (渐近线和实轴的交点）
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) Rule #o The root Locus Crossing with the Imaginary Axis(根轨迹与虚轴的交点) Rule #7 The Breakaway Point of the Root Locus(+#th 迹的分离点) Rule #8 The angle between the direction of emergence (or entry) of q coincident poles(or zeros) on the real axis 根轨迹离开或进入实轴上q重极点(或零点)方向之间的夹角)
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） Rule #6 The Root Locus Crossing with the Imaginary Axis (根轨迹与虚轴的交点） Rule #7 The Breakaway Point of the Root Locus (根轨 迹的分离点） Rule #8 The angle between the direction of emergence (or entry) of q coincident poles (or zeros) on the real axis (根轨迹离开或进入实轴上q重极点（或零点）方向之间的夹角）
Rule #9 The gain at a selected point SI on the locus (在某特定点上的根轨迹增益K) Rule#10 The sum of the closed- loop poles(闭环极点 之和) Rule #l The number of branches of the root locus (根轨迹的分支数)
Rule #9 The gain at a selected point s1 on the locus (在某特定点上的根轨迹增益K） Rule #10 The sum of the closed-loop poles (闭环极点 之和） Rule #11 The number of branches of the root locus (根轨迹的分支数）
Instructional objectives. At the end of this lecture students should be able to Find the directions of the pole asymptotes Find the values of k for which the poled cross the imaginary aXiS Determine the root locus and select“good” values for K
Instructional objectives: At the end of this lecture students should be able to •Find the directions of the pole asymptotes • Find the values of K for which the poled cross the imaginary axis • Determine the root locus and select “good” values for K
Module 11 System Design Using the Root Locus (2 hours)
Module 11 System Design Using the Root Locus (2 hours)
System design in the complex plane 1-21.8 (1)T=-|-tan 4.6 4.6 (2)7 O≥ Stable but sluggish Stable but sluggish (3)PO%=100 10d1 0≤5≤0.6 0.6 →≥0.6(1-PO%/100 (4) or—etc 1+K.K (2) K orK≥
Example 3 Satellite attitude control 4 S(S 4 (s-1) 4 → nstable 1+G 4 1+ s2-s+4 S(S Let's try to use the following controller K(s+a) (S+b) s(S-1 Controller
s+a) 4 (S+b) S(S Controller Lets assume that a=5: 6=1 4K(s+5) (s+1)(s-0.5+1.94j)(s-0.5-1.94j
Lets assume that a=1: 6=5 K=2 Find the intersection with the imaginary axIs 4K(s+1) (s+52-s+4) s3+4s2+(4K-1)s+20+4K=0; 4)2+20+4K=0 3+(4K-1)0=0→0=0;02=4K-1 .=-1.5 -16K+4+20+4K=0 12K=24→K=2