"
0 7149
0 0055
0 1180 #" 1 #
0 75
1
:
;
:
:
0 0055
0 0074
0 0447
0 75
;
:
;
:
:
;
:
0 1180
0 0447
0 2493
:
:
:
:
;
:
The controller stability plot in gure 11.22 was produced by nding points where the
transfer function ( ) 2 vanished for some frequency . Since = 1 (1 + std
0
)
S
j
!
=!
!
S
=
P
K
vanishes wherever std
0 or
has a
axis pole, the
axis poles of are exactly the
P
K
j
!
j
!
K
axis zeros of ( ) 2 the factor of 2 cancels the two zeros at = 0 that inherits
j
!
S
j
!
=!
!
s
S
from the two = 0 poles of std
0 . At each frequency , the linear equations in and
s
P
!
1
(a)
(b)
(c)
2
13 ( ) +
13 ( ) + (1
) 13 ( ) . = 0
<
;
H
j
!
H
j
!
;
;
H
j
!
!
1
(a)
(b)
(c)
2
13 ( ) +
13 ( ) + (1
) 13 ( ) . = 0
=
;
H
j
!
H
j
!
;
;
H
j
!
!
may be dependent, independent, or inconsistent their solution in the rst two cases gives
either a line or point in the (
) plane. When these lines and points are plotted over
all frequencies they determine subsets of slice over which the controller has a constant
H
K
number of unstable (right half-plane) poles. By checking any one controller inside each
subset of slice for open-loop stability, each subset of slice can be labeled as being achieved
H
H
by stable or unstable controllers.
