Introduction to the Nyquist criterion The Nyquist criterion relates the stability of a closed system to the open-loop frequency response and open loop pole location. Mapping . If we take a complex number on the s-plane and substitute into a function F (s), an- other complex number results. e.g. substitut- ing s =4+ j 3 into F (s)= s 2 +2s + 1 yields 16 + j 30. Contour . Consider a collection of points, called a contour A. Contour A can be mapped into Contour B, as shown in the next Figure. Figure above; Mapping contour A through F (s) to contour B. 1
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Introduction to the Nyquist criterion
The Nyquist criterion relates the stability of
a closed system to the open-loop frequency
response and open loop pole location.
Mapping. If we take a complex number on the
s-plane and substitute into a function F(s), an-
other complex number results. e.g. substitut-
ing s = 4 + j3 into F(s) = s2 + 2s + 1 yields
16 + j30.
Contour. Consider a collection of points, called
a contour A. Contour A can be mapped into
Contour B, as shown in the next Figure.
Figure above; Mapping contour A through F(s)
to contour B.
1
Assuming
F(s) =(s − z1)(s − z2) · · ·
(s − p1)(s − p2) · · ·
If we assume a clockwise direction for mapping
the points on contour A, the contour B maps
in a clockwise direction if F(s) has just one
zero. If the zero is enclosed by contour A,
then contour B enclose origin.
Alternatively, the mapping is in a counterclock-
wise direction if F(s) has just one pole, and if
the pole is enclosed by contour A, then contour
B enclose origin.
If there is the one pole and one zero is enclosed
by contour A, then contour B does not enclose
origin.
2
Figure above; Examples of contour mapping.
3
Consider the system in the Figure below.
Figure above; closed loop control system
Letting
G(s) =NG
DG
, H(s) =NH
DH
,
We found
T(s) =G(s)
1 + G(s)H(s)=
NGDH
DGDH + NGNH
Note that
1 + G(s)H(s) =DGDH + NGNH
DGDH
4
The poles of 1+G(s)H(s) are the same as the
poles of G(s)H(s), the open-looped system,
that are known. The zeros of 1 + G(s)H(s)
are the same as the poles of T(s), the closed-
looped system, that are unknown.
Because stable systems have T(s) with poles
only in the left half-plane, we apply the concept
of contour to use the entire right half-plane as
contour A, as shown in the Figure below.
Figure above; Contour enclosing right half-
plane to determine stability.
5
We try to construct contour B via
F(s) = G(s)H(s)
which is the same as that of 1 + G(s)H(s),
except that it is shifted to the right by (1, j0).