Image Processing Reference
In-Depth Information
2
4
3
5 x(k)
2
4
3
5 v(k)
100
010
001
0
:
0282 0
:
5024
0
:
2633
10 3
x(k þ
1)
¼
þ
0
:
1413 1
:
6703
0
:
9165
0
1
:
2781
1
:
2854
2
4
3
5 x(k)
100
010
001
y(k)
¼
S OLUTION
The controllability matrix is given by
¼ BABA 2 B
Q o
2
4
3
5
0
:
0282 0
:
5024
0
:
2633 0
:
0282 0
:
5024
0
:
2633 0
:
0282 0
:
5024
0
:
2633
10 3
¼
0
:
1413 1
:
6703
0
:
9165 0
:
1413 1
:
6703
0
:
9165 0
:
1413 1
:
6703
0
:
9165
01
:
2781
1
:
2854
0
1
:
2781
1
:
2854
0
1
:
2781
1
:
2854
The rank of the controllability matrix is equal to 3. That is, the developability
system with level 1 loop is fully controllable at the given nominal operating point.
In other words, there are three independent factors involved in this process (due to
rank of the Jacobian matrix), and all of them can be controlled using three
actuators in vector v(k). That means, a gain matrix K of size 3
3 can be designed
to affect all three DMAs using an arbitrary pole-placement strategy.
9.7 STEADY-STATE ERROR
An important part of the level 1 and 2 controllers is that the closed system has the
ability to give zero (or minimum) steady-state error,
theoretically,
in a single
measurement-actuation update cycle. The steady-state error is de
ned as the error
between the measured (
(filtered) values and the desired target values. Due to the use
of an integrator in the loop we show mathematically below how the steady-state error
is driven to zero with multiple iterations.
Consider the open-loop state Equation 9.21, error Equation 9.25, and the feed-
back Equation 9.26. The state-space equations are very similar in structure for both
level 1 and 2 controllers, but they have different states, Jacobian, dimensionality,
output, etc. Substituting Equation 9.26 in Equation 9.21 we get the closed-loop state
equation as follows:
x ( k þ
1
) ¼ x ( k ) þ B [ Ke ( k )]
(
9
:
35
)
Substituting the error vector, Equation 9.25, in Equation 9.35, we get
(
:
)
x ( k þ
1
) ¼ ( I BK ) x ( k ) þ BKx d
9
36
Taking the z-transform of Equation 9.36, we have
zX ( z ) zx 0 ¼ ( I BK ) X ( z ) þ BKX d ( z )
( 9 : 37 )
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