Unformatted text preview: eroes of F(s) and -pi are the
poles of F(s) .
Note: zeroes of F(s) ≡ roots of the characteristic
equation ≡ poles of the closed-loop system.
12/13/2008 HSK - EPM 381 - Stability 8 The Nyquist Criterion:
Notes on Zeroes and Poles
• The zeroes of F(s) are the values of s that make F(s) = 0
and the poles are the values of s that make F(s) = ∞ .
• If GH ( s) = N L ( s) , then F (s) = 1 + N L ( s) = DL ( s) + N L ( s)
DL ( s) DL ( s ) DL ( s) • The denominator and numerator order of F(s) are equal
to the order of the loop transfer function GH(s) .
• poles of F(s) ≡ poles of GH(s) (loop transfer function)
• zeroes of F(s) ≡ roots of the characteristic equation,
DL ( s) + N L ( s) = 0
12/13/2008 HSK - EPM 381 - Stability 9 The Nyquist Criterion
The Principle of the Argument
• The stability analysis of the closed-loop system now
becomes the task of finding if there are any zeroes
of F(s) in the right hand side (RHS) of the s-plane.
• This is achieved through the application of the
principle of the argument, a result from general
complex number theory.
• This involves a function mapping from the complex
s-plane to the complex F(s) or GH(s) plane.
12/13/2008 HSK - EPM 381 - Stability 10 The Principle of the Argument:
• Encirclements in the complex plane.
Im Im Γ Path Γ is a
point A A Γ Re A counterclockwise
encirclement • Enclosements in the complex plane.
Im Im Γ 12/13/2008 Γ
Re Re HSK - EPM 381 -...
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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