Image Processing Reference
In-Depth Information
S OLUTION
In this problemA ¼
0
:
5, B ¼
1, Q ¼
1, and R ¼
2. The algebraic Riccati equation is
0
:
25P
5P) 1 P þ
P ¼
0
:
25P
0
:
125P(1
þ
0
:
1
¼
5P þ
1
1
þ
0
:
or
P 2
þ
0
:
5P
2
¼
0
The above algebraic quadratic equation has two solutions
P ¼
1
:
1861 and
P ¼
1
:
6861
Since matrix P has to be positive de
nite, the acceptable solution is the positive
solution, therefore the constant feedback gain is
0
:
25P
0
:
25
1
:
1861
K ¼
5P ¼
1861 ¼
0
:
1861
1
þ
0
:
1
þ
0
:
5
0
:
In Chapters 7, 8, and 10, we show the bene
ts of the LQR design technique in
designing high-performance color reproduction systems.
5.4 STATE ESTIMATORS (OBSERVERS) DESIGN
5.4.1 I NTRODUCTION
In state feedback design using pole placement or optimal control using LQR, the
assumption is that all the states are available for feedback. In reality we may not have
access to all the states. A state estimator is used to estimate the states of a dynamic
system based on the output and the control signals [6,7].
5.4.2 F ULL -O RDER O BSERVER D ESIGN
In a full-order observer, every state of a dynamic system is estimated. Hence, the order
of the observer is the same as the order of the original system. Consider a MIMO
system given as
x ( k þ
1
) ¼ Ax ( k ) þ Bu ( k )
(
5
:
76
)
y ( k ) ¼ Cx ( k )
(
5
:
77
)
where
A 2 R N N
B 2 R N M
u 2 R M
y 2 R P
C 2 R P N
The block diagram of the system is shown in Figure 5.5.
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