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by one unit to the left of the origin. Then, instead of counting the encirclement of
the origin, we can count the encirclement of the point -l. Everything else remains
the same, and we can now use the Nyquist plot to determine the system stability,
With this modification, the Nyquist criterion for the stability of a closed-loop
system can be restated as follows.
    If a contour A in the s-plane that covers the entire right half of the s-plane is
mapped to the F-plane with the mapping function F (s) -. G(s)H(s), then the
number of closed-loop poles Z that lie in the right half of the s-p.lane equals
the number of open-loop poles P that are in the right half of the s-plane minus the
number of counterclockwise rotations N of the Nyquist plot around the point -1
in the F-plane, i.e., Z  - P - N. For stability of the closed-loop system, Z must
be equal to zero.
     To understand the Nyquist criterion, let us study two case.s shown in Fig. 5.18.
Let us assume that somehow we know the zeros of I + G(s)H(s), which are
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW        477
~
┏━━━━━━━┳━━━━━━━┓
┃        jco , ┃~'-         ~ ┃
┣━━━━━━━╋━━━━━━━┫
┃  x           ┃-/"'          ┃
┃/-            ┃              ┃
┃~             ┃              ┃
┃   x          ┃              ┃
┗━━━━━━━┻━━━━━━━┛
s-plane
┏━━━┳━━━━━┓
┃   x  ┃"~-   .r  ┃
┣━━━╋━━━━━┫
┃   x  ┃          ┃
┗━━━┻━━━━━┛
Poles of G(8)H(s)
Zeros of G(s
a)
  ~n\s)     Im J    \        
┏━━━━━━━━┳━━━━┓
┃     D'         ┃..~' ~- ┃
┃   -1 B'        ┃~x yA-c ┃
┣━━━━━━━━╋━━━━┫
┃                ┃        ┃
┗━━━━━━━━┻━━━━┛
F-planc
┏━━┳━━━━┓
┃_/  ┃~est-l  ┃
┃~   ┃J       ┃
┣━━╋━━━━┫
┃    ┃        ┃
┗━━┻━━━━┛
F-plane
Ftg. 5.18     Nyquist plots for mapping function 1+ G(s)H(s).
Line
poles of the closed-loop transfer function T(s). The poles of 1  + G(s)H (s) are the
combined poles of theUPopen-loop transfer function G(s)H (s) and are known. Let
the open circles denote the zeros of 1+G(s)H (s) and cross the poles of G(s)H (s).
For Fig. 5.18a, there are no poles or zeros of 1 + G(s)H (s) in the right half of the
s-plane,i.e., P  - 0 and Z - O. Hence, the Nyquist plot will not encircle the point
-1 in the F-plane as shown in Fig. 5.18b. For this case, N - P - Z - 0 and the
system is stable. For Fig. 5.18b, we have one zero of 1 + G(s)H (s) located in the
right half ofthe s-plane (unstable system). Therefore, Z -  1. Furthermore, P  - 0
because there are no poles of 1  + G(s)H (s) located in right half of the s-plane.
Hence, according to the Nyquist criterion, N -- P - Z = -1,i.e., the Nyquist plot
in the F-plane will encircle the point -1  once in the clockwise direction as shown
in Fig. 5.19b.
   The number of encirclements can be converuently determined by drawing a
radialline from the point -1 and counting the number ofintersections with the
478            PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
x     Poles of G(s)
┏━━━━━━━━┳━━━┓
┃         jco J  ┃~~-   ┃
┣━━━━━━━━╋━━━┫
┃.r.  ~  ..      ┃Dy)C- ┃
┃3  -2 -1        ┃      ┃
　
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