Module 13 Nyquist Stability Criterion (4 hours)
Module 13 Nyquist Stability Criterion (4 hours)
13.1 Conformal Mapping: Cauchy's Theorem (保角映射:柯西定理) Recall Stability Problem: To determine the relative stability of a closed-loop system we must investigate the characteristic equation of the system 1+GH(s)=0 Where GH(S)or 1+ GH(s)is a complex function of s, and the difference between GH(s) and 1+ GH(s) is only 1. So (1)We can investigate 1+ GH(s) through GH(S) (2)How to investigate GH(S)?----If s has a variation, then GH(s) has a variation certainly. We can suppose the variation of s, to see the change of GH(S)
13.1 Conformal Mapping: Cauchy’s Theorem ( 保角映射: 柯西定理 ) Recall Stability Problem: To determine the relative stability of a closed-loop system, we must investigate the characteristic equation of the system: 1+ GH(s) = 0 Where GH(s) or 1+ GH(s) is a complex function of s , and the difference between GH(s) and 1+ GH(s) is only 1. So (1) We can investigate 1+ GH(s) through GH(s) ; (2) How to investigate GH(s) ? ---- If s has a variation , then GH(s) has a variation certainly. We can suppose the variation of s, to see the change of GH(s)
Mapping F(S) S-2 M -2 F(s)= M∠φ P (s-P2) =∑∠-=,-∑么s We are concern with the mapping of contours in the s-plane by a function F(s). A contour map is a contour or trajectory in one plane mapped or translated another plane by a relation F(S) Since s is a complex variable: s=o +jo, the function F(s) is itself complex; it can be defined as F(s)=u+jv and can be represented on a complex F(s)-plane with coordinates u and v S1=-1+1 F(S)=23sF1=2+j2 [F8 Mapping
= − − = M s p s z F s i j ( ) ( ) ( ) s F(s) ⎯Mapping ⎯ ⎯→ We are concern with the mapping of contours in the s – plane by a function F(s) . A contour map is a contour or trajectory in one plane mapped or translated another plane by a relation F(s) . Since s is a complex variable: s = σ +jω, the function F(s) is itself complex; it can be defined as F(s) = u + jv and can be represented on a complex F(s) – plane with coordinates u and v. jω σ [s] u [F(s)] jv S1=-1+j1 F1=-2+j2 Mapping F(S)=2s − − = i j s p s z M = − j − − pi s z s
As an example, let us consider a function F(s=25 I and a contour in the s-plane. The mapping of the s- plane unit square contour to the F(s)-plane is accomplished through the relation F(s), and so 20+ l1+jy=F(s)=2+1=2(a+jO)+1 v=20 F(s-plane j2H S-plane 0 2 B
s-plane F(s)-plane As an example, let us consider a function F(s) = 2s + 1 and a contour in the s – plane. The mapping of the s – plane unit square contour to the F(s) – plane is accomplished through the relation F(s) , and so u + j v = F(s) = 2s +1= 2( + j ) +1 2 2 1 = = + v u
Example 2 F(s) s+2 1+jI D →F() 1+l+2 jI A 0 少B
Example 2. 2 ( ) + = s s F s 1 1 1 2 1 1 : 1 1 ( ) j j j D s j F s D = − + + − + = − + =