Note zeroes of fs roots of the characteristic equation

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) . GH • poles of F(s) ≡ poles of GH(s) (loop transfer function) GH • 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 loop becomes the task of finding if there are any zeroes of F(s) in the right hand side (RHS) of the s-plane. of • This is achieved through the application of the This principle of the argument, a result from general principle result complex number theory. complex • This involves a function mapping from the complex This s-plane to the complex F(s) or GH(s) plane. plane or GH 12/13/2008 HSK - EPM 381 - Stability 10 The Principle of the Argument: Some Definitions • Encirclements in the complex plane. Im Im Γ Path Γ is a Re clockwise encirclement of 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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