Image Processing Reference
In-Depth Information
Boundary condition:
P
(
N
) ¼
0
(
9
:
122
)
Steady-state solution:
If the control process is
finite (N
< 1
), the feedback gain K(k) is time-varying. If the
process is an in
), the feedback gain K attains a constant
value. If the steady-state form of matrix P(k)isdened as P and the steady-state form
of gain matrix K(k)isde
nite stage process (N
!1
ned as K, then P and K can be determined by solving the
following algebraic equations:
Gain matrix:
1
K
¼
R
þ
B
T
PB
B
T
PA
(
9
:
123
)
Recursive Equation:
1
P
¼
A
T
PA
A
T
P
~
BR
þ
B
T
PB
B
T
PA
þ
Q
(
9
:
124
)
Control law:
u
(
k
) ¼
Kx
(
k
)
(
:
)
9
125
From the above equations, the gains are clearly dependent on the weights. The higher
the weights, the tighter the tolerance on the control variables. Equation 9.124 is
nonlinear. Therefore, a close form solution is not feasible. Usually, instead of solving
an algebraic equation like Equation 9.124, we solve the recursive equation, in this
case, Equation 9.121, by marching backward in cycles from k
¼
N to 0. Then the
steady-state solution is P(0). Once P is found, Equation 9.123 can be solved to obtain
a steady-state gain matrix K.
Selection of weight matrices:
One way of selecting the weights in the quadratic performance equation is shown
below. Let
a
3
be the maximum percentage change deviations of the states
x
1
, x
2
, and x
3
from their nominal values. Assume matrix Q to be diagonal (which is
more realistic), then the total cost is
a
1
,
a
2
, and
x
T
Qx
¼
Q
11
x
1
þ
Q
22
x
2
þ
Q
33
x
3
(
9
:
126
)
Now, choose Q such that the overall cost is uniformly distributed between the three
states. This means that
Q
11
x
1
¼
Q
22
x
2
¼
Q
33
x
3
(
:
)
9
127
or
2
2
2
2
2
2
2
Q
11
a
1
b
¼
Q
22
a
2
b
¼
Q
33
a
3
b
¼ b
(
9
:
128
)
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