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Nyquist examples - The Nyquist Stability Criterion...

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The Nyquist Stability Criterion Functions of a Complex Variable Suppose F ( s ) is a function of s = σ + A curve (contour) in the s -plane will map to a curve (contour) in the F ( s )-plane For a rational function F ( s ) = n ( s ) d ( s ) , the poles and zeros determine important qualitative properties of the contour c 2 as we should see. Example: a) Suppose F ( s ) = s - s 0 , where s 0 is a constant. Suppose c 1 is a circle centered a s 0 . c 2 will be a circle centered at the origin with the same radius. Moreover, if we start at a point s on c 1 and traverse c 1 once in the clockwise direction, F ( s ) will transverse c 2 once in the clockwise direction. b) Now suppose F ( s ) = 1 s - s 0 In this case, c 2 will be a circle centered at the origin with radius 1 radius of c 1 . Moreover, if we traverse c 1 once in the clockwise direction, F ( s ) will traverse c 2 once in the ‘counterclockwise’ direction. (!) Suppose the point s 0 is outside the contour c 1 . Notice that if we traverse c 1 once in the clockwise direction, the total change in the angle of s - s 0 will be less than 360 . In other words, s - s 0 will not 1
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