自动控制理论-第七章

s e j （ 0, : 90 0 90 )
K K jv then, lim L( s) lim v jv lim v e 0 0 e 0 i.e. the contour encircles the origin v/2 times in the clockwise direction, and the radius is .
Im
j
=0-
r=∞
r=∞ 0
=-∞ =+∞
0 Re
=0+
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(b) The portion from = 0+ to =+
is the polar plot of L(s).
(c) The portion from = + to =
is mapped into the origin of the L(s)-plane.
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2. The plot of contour L for the range -< <0will be the complex conjugate of the plot for the range 0 + , and the polar plot of L( s ) will be symmetrical in the L( s)-plane about the real axis. So it is sufficient to construct the contour L for
(0+ ＋), and the conclusion changes to
that for the contour in the L( s)-plane, the number of counterclockwise encirclements of the ( 1, j0) point is equal to P/2.
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Example 7.1
K L( s ) G ( s ) H ( s ) (T1s 1)(T2 s 1)
Im
N=Z=0, so the system is stable.
-1
=∞
0
=0
K Re
=1/T
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Example 7.2 Assuming open loop transfer functien-loop transfer function.
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Nyquist stability criterion 1. A feedback system is stable if and only if the contour Γ L in the L(s)-plane does net encircle the (-1, 0) point when the number of poles of L(s) in the right-hand s-plane is zero (P=0). 2. A feedback system is stable if and only if, for the contour Γ L, the number of counterclockwise encirclements of the (-1, 0) point is equal to the number of poles of L(s) with positive real parts.
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7.1 Introduction
• •
Developed by H.Nyquist in 1932. Based on Cauchy’s theorem.
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• The frequency response can be obtained experimentally. • It can be utilized to investigate the relative stability.
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• The contour Γ F is known as the Nyquist diagram or ploar plot of F(s). • As L(s)=F(s)-1, the number of encirclements of the origin in F(s)-plane becomes the number of encirclements of -1 point in L(s)plane.
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3. The conclusion can be expanded to the system including delay unit.
4. If the contour L(j) overpass the (-1, j0) point, that is one close-loop pole on the jaxis, the system is critically stable.
＋
3. circle direction ： ( ) ， clockwise
We need to find the cross-over point and compare it with -1!
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4. The cross-over point K L( j ) j (4 3 12.2 ) 22.4 2 1 The cross-over point is at 4 3 12.2 0， ＝ 3.05, K K where L( j 3.05 )＝ － (1 22.4 3.05) 68.32
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5. System with v poles at the origin • The supplement curve must be draw. • The small semicircular detour around the pole at the origin can be represented by setting
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R(s) + -
Gc(s) H(s)
G(s)
Y(s)
Fig 7.1 A closed-loop system
F s 1 L( s) 1 Gc ( s )G ( s ) H ( s )
• Where L(s) is a rational function of s. • To ensure stability, it must be ascertained that all zeros of F(s) lie in the left-hand splane. • Propose a mapping of the right-hand splane in F(s)-plane.
(d) The portion from = to =0
is symmetrical to the polar plot.
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Note: 1.
s
lim L( s ) lim
j
k (s z j )
m
s
(s p )
i i 1
j 1 n
0
( m n)
Chapter 7 Stability in the Frequency Domain
7.1 7.2 7.3 7.4 7.5 Introduction Mapping Contours in the s-plane Nyquist Stability Criterion Stability Margin of System Dynamics performance of closed-loop from open-loop frequency characteristic 7.6 Summary
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7.2 Mapping Contours in the s-plane
• A contour map is a contour in one plane mapped into another plane by a relation F(s). Example: s F ( s) s2
D s -2 C -1 j A j 0 -j 1 B
Im
=0-
j
r=∞
r=∞ 0
=-∞ =+∞
0 Re
=0+
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(a) The origin of the s-plane
s e j , 0
: 90 0 90


K K j lim L( s) lim j lim e 0 0 e 0
n i 1 M
(s p )
k k 1
• The poles of F(s) are the poles of L(s). • The zeros of F(s) are the characteristic roots of the system.
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• For a system to be stable, all the zeros of F(s) must lie in the left-hand s-plane. • Choose a contour Γ s in the s-plane that encloses the entire right-hand s-plane, the number of encirclements of the origin of the j F(s)-plane is N=Z-P. Z: zeros in RHP P: poles in RHP r=∞ • So the number of unstable 0 poles of the system is Z=N+P
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Another example:
s F ( s) s 1 2
j A j 0 -j 1 B
D s -2 C b -1