11 the principle of the argument complex function

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Unformatted text preview: Stability The area to the right of the path Γ is the area enclosed by Γ . 11 The Principle of the Argument Complex Function Mapping • S-plane → F(s) complex plane mapping. jω s2 Γ s3 Im{F} unique s1 F(s1) σ Re{F} non unique F(s2) F(s3) • If F(s) is analytic along the path Γ (no poles of F(s) on Γ) and s starts at s = s1 and traces a closed path terminating at s1 , then F(s) will trace a closed path in the F plane starting at F(s1) and terminating at F(s1) . 12/13/2008 HSK - EPM 381 - Stability 12 The Principle of the Argument • The principle of the argument states that for an The arbitrary closed path Γ in the s-plane, the corresponding closed path in the F plane will encircle the origin as many times as the difference between the number of zeroes of F(s) and poles of F(s) located in the area enclosed by the path Γ . • The direction of the encirclements of origin is the The direction same as the path Γ if the number of zeroes of F(s) is same greater than the poles. 12/13/2008 HSK - EPM 381 - Stability 13 The Nyquist Path • Define the Nyquist Define path Γ such that it encloses the right hand side of the splane, but does not go through any poles of F(s). 12/13/2008 jω j∞ × Poles of F(s) HSK - EPM 381 - Stability ∞ → R Γ σ × × -j∞ 14 The Nyquist Criterion • Map F(s) for the Nyquist path enclosing the RHS. • Then the number of clockwise encirclements of the origin of the F(s) plane is N = Z – P , where Z is the number of zeroes of F(s) in the RHS, P is the number of poles of F(s) in the RHS. Z, corresponds to the number of pol...
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