nyquist stability criterian
TRANSCRIPT
Given open loop frequency response , determine closed loop system stability
s - plane q(s) plane
take... )p - (s )p - (s
... )z - (s )z- (s (s) q
21
21
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.
.
.P1
P2
P3
P4
s
<
Taking a closed contour in s plane, find its mapping in q(s) plane.
. .
..
P1P2
P3P4
q (s)
<
Contour is q(s) plane will also be a closed path
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.
.
.P1
P2
P3
P4
s
<
..
Let q (s) = (s + 1)
q (s)
.
.
.
.P1
P2
P3
P4
<
.
.
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.P1
P2
P3
P4
s
<
.
let q (s) = (s - 1)
q (s)
.
.
.
.P1
P2
P3
P4
<
The difference between both the figures:first one : origin not encircledsecond one : origin encircled
1 - s
1 qlet
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.
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.P1
P2
P3
P4
s
<
.1
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.P1
P4
P3
P2
>
Ex.
<
S- plane
>
q(s)-plane
Origin encircled in CCW direction.
Hence : when the singularity is inside the circle in s plane, Origin is encircled in C W ( if zero)CCW ( if pole )
Ex.
<
.
S-plane
<
q(s)-plane
q (s) : 1 + GH
... )p (s )p (s
... )z (s )z (s K HG
where
21
21
..... )p (s )p (s
... )p (s )p (s ... )z (s )z (s K
HG 1
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The zeros of this function are closed loop poles
The poles of this function are open loop poles.
For stability, zeros of this function (1+GH) must not lie in RHP
Choose Nyquist contour in s plane in such a way that it includes entire RHP
.
.
.0
>>
<
j
j
R
eR j
if there are Z zeros and P poles of 1 + GH in R H P then number of encirclements about origin in CCW direction are:
Plot the mapping of s plane contour to 1+GH
N = P - Z
For closed loop system to be stable Z = 0 - Nyquist stability criterion
N = P
G H = (1 + GH) - 1
<1 + GH - 1 1 + GH<1 + GH encircling origin means GH encircling -1 + j 0
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A
OB
>>
<
C
Nyquist contour
O A : s = 0 to + j polar plot
C O : complex conjugate of polar plot
ABC : put s = R e j
where R
varies from +90 to -90
polar plot
. . 0
<
1
EX:1 s
1 G(s)H(s)
.
.
.O
>>
<
A
B
C
.ABC
O
>
<
90 - 0 - 90
R , eR s j
For A B C,
0 )H(s) s (G
P = 0
N = 0
0 Z System stable