Image Processing Reference
In-Depth Information
The gain K
0
controls the rate at which the error signal between the true states and the
estimated states, e(k)
x(k) reaches zero steady state. A difference equation
can be obtained for the error signal by using Equations 9.100 and 9.102 as follows:
¼
x(k)
e
(
k
þ
1
) ¼
x
(
k
þ
1
)
x
(
k
þ
1
) ¼ (
A
K
0
C
)(
x
(
k
)
x
(
k
))
(
9
:
103
)
Since e(k)
¼
x(k)
x(k) Equation 9.103 can be written as
e
(
k
þ
1
) ¼
A
K
0
C
ð
Þ
e
(
k
)
(
9
:
104
)
The observer is in the form of a feedback system with the error dynamics satis
ed by
Equation 9.104. The matrix A
K
0
C is called the observer matrix. For the error to go
to zero asymptotically, all the eigenvalues of the observer matrix must lie inside the
unit circle in the z-domain. Determination of the gain matrix that accomplishes this is
a pole-placement task. Therefore, we chose the gain matrix K
0
to place the eigen-
values of the observer matrix within the unit circle so that the observer states
asymptotically converge to the true state. Note that, in order to place the poles of
the observer matrix, it is necessary that the observability matrix (Equation 4.126)
P
¼
C
0
A
0
C
0
:: (
A
0
)
mþ
1
1
C
0
(
9
:
105
)
be full rank. We can use the Bass
Gura formula or Ackermann formula, Equation
5.84 to design the observer gain matrix.
By the separation theorem, the observer and the state feedback designs can be
decoupled and designed independently. The dynamic TC system, Equation 9.100,
is in the linear state-space form, and we designed the state feedback control law,
u(k)
-
[K
1
K
2
K
mþ
1
] under the assumption that the state
vector, x(k), is accessible for measurement. But, because of our inability to measure
the system states, they are estimated using a linear observer (Equation 9.102) by
measuring the TC, y(k), with a TC sensor and estimating the feedforward signal,
x
mþ
2
(k
þ
¼
Kx(k) with K
¼
þ
gv(k), based on the area coverage information. After the
states are estimated, the control input is calculated using the controller gains (row
vector) and estimated states as follows:
1)
¼
x
mþ
2
(k)
u
(
k
) ¼
Kx
(
k
)
(
9
:
106
)
The system described by Equation 9.100 is a SISO system unlike our augmented
system (Equation 9.75) with the PI controller. The number of gains is equal to the
number of states that the control input u(k) affects during feedback. The gain vector
K
¼
[K
1
K
2
K
mþ
1
] can be obtained by using the pole-placement or optimal
control techniques.
Having determined the observer gain matrix and the controller gain matrix, these
processing units have to be interconnected with the TC target to derive proper control
input. The target state decomposer (shown in Figure 9.40) is designed for creating a
vectorized version of the TC target so that the output from the state feedback
controller can be subtracted correctly. It is a simple linear transformation formed
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