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 = − + + − + = − + =
Example 3.F(S (b)
Example 3. 2 1 ( ) + = s s F s
S-2 FC(s-2 M P II(s-p) =M∠p ∠(S-=) (S-p1) △S-z △M= A=以A=∑A(s==)∑A(s-P) 中 contour F Contour 中r P2 P1 中 (b)
= − − = M s p s z F s i j ( ) ( ) ( ) − − = i j s p s z M =( − ) −( − ) j pi s z s − − = i j s p s z M =( − ) −( − ) j pi s z s
Cauchy's Theorem The encirclement of the poles and zeros of F(s) can be related to the encirclement of the origin in the F(s)-plane by Cauchy's theorem, commonly known as the principle of the argument, which state If a contour Is in the s-plane encircles Z zeros and P poles of f(s) and does not pass through any poles or zeros of f(s)and the traversal is in the clockwise direction along the contour, the corresponding contour TF in the f(s-plane encircles the origin of the F(S)- plane n=z-p times in the clockwise direction
• The encirclement of the poles and zeros of F(s) can be related to the encirclement of the origin in the F(s)-plane by Cauchy’s theorem, commonly known as the principle of the argument, which state: Cauchy’s Theorem • If a contour Γs in the s-plane encircles Z zeros and P poles of F(s) and does not pass through any poles or zeros of F(s) and the traversal is in the clockwise direction along the contour, the corresponding contour ΓF in the F(s)-plane encircles the origin of the F(s)- plane N = Z – P times in the clockwise direction
Ex. 4 Ts (a) J J Ex 5 (b)
Ex. 4 Ex. 5
Im F(s) 3|-2-1 Re f(s) Plot of F(sy .The image of the path encircling the zero encircles the origin once in the clockwise direction .The image of the path encircling the pole encircles the origin once in the counter-clockwise direction .The image of the path encircling neither does not encircle the origin