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
• Splane → 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 splane, 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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This note was uploaded on 03/03/2010 for the course AUTOMATIC 335 taught by Professor ? during the Winter '10 term at Ain Shams University.
 Winter '10
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