Image Processing Reference
In-Depth Information
V
go
=
U
go
U
g
=
u
g
V
h
= V
ho
+
V
h
b
11
+
b
21
b
22
+
+
U
l
=
u
l
V
l
= V
lo
+
V
l
X
lo
=
U
lo
FIGURE 9.6
Block diagram representation of linear electrostatic system (Equation 9.22)
shown with nominal actuator values.
after the integrator are {u
1
, u
2
} and {u
g
, u
l
}, respectively. The control system is
modeled in state-space form (Equation 9.21) by introducing the integrator, where
A
¼
I is the 2
2 identity matrix, B is the 2
2 Jacobian matrix given by
b
11
0
B
¼
(
9
:
23
)
b
21
b
22
The state vector x(k), control vector v(k), and the output vector y(k) are given by
;
;
V
V
l
u
g
¼ D
U
g
u
l
¼ D
U
l
x
(
k
) ¼
v
(
k
) ¼
y
(
k
) ¼
x
(
k
)
(
9
:
24
)
It is clear from Figure 9.6 that if the slope, b
22
, is equal to zero, the exposed voltage,
V
l
, is not affected by the laser intensity. In other words, when b
22
¼
0 the states x(k)
are not fully controllable using only the two electrostatic actuators. As the photo-
conductor charge is reduced in magnitude or increased from more negative to
positive value (see Figure 9.7 at nominal X(Ul)¼
l
)
cm
2
), the slope, b
22
,
decreases rapidly. If the slope b
22
is too small, it can result in loss of controllability
as described above.
¼
8 ergs
=
Example 9.1
Test the following electrostatic control system for controllability. The system is at
the nominal operating point {V
go
¼
cm
2
}.
600V, X
lo
¼
8 ergs
=
x(k)
v(k)
10
01
0
:
9798
0
x(k þ
1)
¼
þ
0
:
3315 11
:
1026
x(k)
10
01
y(k)
¼
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