Image Processing Reference
In-Depth Information
The scalar parameter
designer to
constrain all the eigenvalues of the overall system to lie inside a circle of radius
a
is a positive weight that is chosen by the customer
=
b
1
in the z-domain, where
b¼
1
=a
. The index N indicates the length of a
finite process.
This will guarantee that the largest eigenvalue will be less than
b
in magnitude or the
k
. N can be chosen to be equal to number of
cycles over which the optimization function is to be minimized. The importance of
states and actuators is adjusted by changing the matrices, Q and R, respectively. When
we increase the weights in Q corresponding to the state x
3
—
transient response will decay faster than
b
the high density state will
be emphasized more and affected more (Problem 9.14).
Let
k
x(k) and
k
u(k). Then the new performance index becomes
x(k)
¼a
u(k)
¼a
2
P
N
1
k
¼
0
x
T
(
k
)
Qx
(
k
) þ
u
T
(
k
)
Ru
(
k
)
J
¼
1
½
(
9
:
115
)
k
þ
1
x
(
k
þ
) ¼ a
x
(
k
þ
)
(
:
)
1
1
9
116
nition of x(k),
Substituting Equation 9.112 in Equation 9.116 and using the de
we get
x
(
k
þ
1
) ¼ a
Ax
(
k
) þ a
Bu
(
k
)
¼
Ax
(
k
) þ
Bu
(
k
)
(
9
:
117
)
where Ã
¼a
A and B
¼a
B. Optimal gain can be obtained by minimizing J subject to
the constraint shown by system model (Equation 9.117). This type of minimization
problem will yield the feedback law
u
(
k
) ¼
K
(
k
)
x
(
k
)
(
9
:
118
)
This can be solved using the standard optimization method involving a Lagrange
multiplier. We do not show a proof of this method, since the basic techniques are
already described in Section 5.3.2. Instead, the
final set of recursive algorithms are
shown below.
Control law:
u
(
k
) ¼
K
(
k
)
x
(
k
)
(
9
:
119
)
Gain matrix:
1
)
A
K
(
k
) ¼
R
þ
B
T
P
(
k
þ
)
B
B
T
P
(
k
þ
1
1
(
9
:
120
)
Recursive equation:
1
P
(
k
) ¼
A
T
P
(
k
þ
)
A
A
T
P
(
k
þ
)
A
þ
Q
(
)
~
BR
þ
B
T
P
(
k
þ
)
B
B
T
P
(
k
þ
1
1
1
1
9
:
121
)
Search WWH ::
Custom Search