Image Processing Reference
In-Depth Information
The state feedback control law is given by
u
(
k
) ¼
K
(
k
)
x
(
k
)
(
5
:
47
)
where K
(
k
)
is the controller feedback gain at time k. The closed-loop system
dynamic is governed by the state equation given as
x
(
k
þ
1
) ¼ (
A
BK
(
k
))
x
(
k
)
(
5
:
48
)
Let us form a quadratic performance function as follows
2
P
N
1
k
¼
0
x
T
(
k
)
Qx
(
k
) þ
u
T
(
k
)
Ru
(
k
)
1
2
x
T
(
N
)
Sx
(
N
) þ
1
J
¼
(
5
:
49
)
where
S and Q are positive or positive semide
nite matrices
R is a positive de
nite matrix
The
first term in the performance index penalizes the deviation of the
final state from
the origin. The term x
T
(
k
)
Qx
(
k
)
penalizes deviation of the states from the origin during
the control process and the term u
T
(
k
)
Qu
(
k
)
is a measure of energy used by the control
signal. The goal is to minimize performance index J with respect to the control signal
u
(
k
)
subject to the constraints given by the state equation of the plant (Equation 5.46).
5.3.2 S
OLUTION OF THE
LQR P
ROBLEM
The optimal control problem is to minimize the objective function
2
P
N
1
1
2
x
T
(
N
)
Sx
(
N
) þ
1
0
x
T
(
k
)
Qx
(
k
) þ
u
T
(
k
)
Ru
(
k
)
J
¼
(
5
:
50
)
k
¼
subject to the constraint given by the state equations
x
(
k
þ
1
) ¼
Ax
(
k
) þ
Bu
(
k
)
(
5
:
51
)
Using Lagrange multipliers, we form a new performance index H as follows
2
P
N
1
k
¼
0
x
T
(
k
)
Qx
(
k
) þ
u
T
(
k
)
Ru
(
k
)
1
2
x
T
(
N
)
Sx
(
N
) þ
1
H
¼
þ l
T
(
k
þ
1
)
x
(
k
þ
½
1
)
Ax
(
k
)
Bu
(
k
)
(
5
:
52
)
T
.
where
l(
k
) ¼ l
1
(
k
) l
2
(
k
)
½
l
N
(
k
)
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