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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 ┃ ┃
┗━━━━━━━━┻━━━┛
Zeros of G(s)
┏━━━┳━━━━━┓
┃ Im J ┃ . ┃
┃ D' ┃( ~A-(l0_ ┃
┃-1 B' ┃~j r' ┃
┣━━━╋━━━━━┫
┃ ┃ ┃
┗━━━┻━━━━━┛
b)
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