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a) F(s) -s - zi
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b) F(s) = 1/(s - pi)
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c) F(s) : s - zi
Fig.5.17 nlustration ofcontour mapprng.
corresponding points in the F-plane and obtain the contour B. Then the contour A
in the s-plane is said to be mapped to the contour B in the F -plane. We assume
that the mapping is one to one, i.e., for every point in the s-plane, there is one and
only one corresponding point in the F -plane and vice versa.
To understand the concept ofmapping further,let F(s) - s - Zi and let the point
s - zi, which is the zero of F (s), lie outside the contour A as shown in Fig. 5.17a.
Instead of using the coordinates of point P,let us use the vector approach. Every
point P on the contour A is associated with a vector V. Let V' be the image
vector in the F-plane. For this case, IV'I = IVI and Z V' - / V. Now as we move
clockwise along the contour A, the magnitude and phase of the vector V varjr.
The phase oscillates betwcen.the two limiting values ~i and @z. In this case, a
clockwise movement along the contour A corresponds to a clockwise movement
along the image contour B in the F-plane.
Now let F(s) = l/(s - pi) and let the pole s = Pi lie outside the contour A as
shown in Fig. 5.17b. For this case, IV'I = 1/I VI and L V' = -Z V. As a result, the
contour A in the first quadrant maps to contour B in the fourth quadrant. Observe
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW 475
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d) F(s) - l/(s - pi)
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PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL3(52)
