13. 2 The application of Cauchy's Theorem Nyquist Stability Criterion Cauchys theorem: If a closed curve T encircles P poles and Z zeros in a clockwise direction, then the image curve TF-F(T)encircles the origin N=Z-P times in the clockwise direction 13.2.1 F(s)- plane G(s) H(sI The stability is determined by the zeros of the characteristic equation: F(s)≡1+G(s)H(S)=0
13.2 The application of Cauchy’s Theorem: Nyquist Stability Criterion 13.2.1 F(s) — plane
m▲ Suppose:F(S)=1+GH(s) Investigate F(s by GH(S)) Re F(S) GH(S) K∏( GH(S) II(s-p) F(s)=1+ KII(s-=)∏(s-p)+K( ∏I(s-p) (S-P1) (S-S;) (S-p) Pi-open-loop pole, and the pole of gh(s)and F(s open-loop zero S,- the zero of f(s), that is the closed-loop pole
Suppose: F(s) =1+ GH(s) − − = ( ) ( ) ( ) i j s p K s z GH s − − = − − + − = − − = + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 i i i i j i j s p s s s p s p K s z s p K s z F s F(s) GH(s) ( Investigate F(s) by GH(s) ) i p — open–loop pole, and the pole of GH(s) and F(s). j z — open–loop zero. i s — the zero of F(s), that is the closed–loop pole