Image Processing Reference
In-Depth Information
In the case of tracking of a reference value, say x
d
, we can de
ne the error
signal as
e
(
k
) ¼
x
d
x
(
k
)
(
9
:
25
)
Then, via Equation 9.21 we have the following equations for the closed-loop system
e
(
k
þ
1
) ¼
e
(
k
)
Bv
(
k
)
v
(
k
) ¼þ
Ke
(
k
)
(
9
:
26
)
or equivalently
e
(
k
þ
1
) ¼ (
I
BK
)
e
(
k
)
(
9
:
27
)
Therefore, we end up having an autonomous linear, time-invariant state-space model
for the error and we can control the decay rate of the components of vector e(k)by
properly selecting the gain matrix K to place the eigenvalues (poles) of the matrix
(I
BK) at positions inside the unit circle in the complex z-plane to meet the required
stability speci
cations of the closed-loop system (Section 5.2.3). A robust pole-
placement algorithm for a MIMO electrostatic system is called
which can
be found in the MATLAB Control System Toolbox [12], which uses an extra
degrees of freedom to
place,
''
''
find a robust solution for gain matrix K. It minimizes the
sensitivity of closed-loop poles to uncertainties in the A and B matrices (see Example
5.5), which in our case are the uncertainties in the elements of the Jacobian matrix.
Reference [13] gives an additional procedure for assigning poles to a closed-loop
MIMO system. A simple approach described in Section 9.8 for designing level 2
controller gain matrix is also applicable to the level 1 controller gain matrix. Other
aspects of closed-loop performance can be also used in designing K (see Section
5.3), which will be discussed later in this chapter.
Example 9.2
For the open-loop electrostatic system shown in Example 9.1,
find the gain matrix K
to place the closed-loop poles within the unit circle on the real axis at 0.2 and 0.3.
S
OLUTION
0
:
9798
0
Using A ¼ I, B ¼
and P¼
[0.2 0.3] in MATLAB, command
0
:
3315 11
:
1026
K¼
place (A, B, P). The gain matrix K is computed to be
0
:
8165
0
K ¼
0
:
0244 0
:
0630
This gain matrix will give rise to stable performance.
Example 9.3
Using the gain matrix of Example 9.2 and the electrostatic model described
in Chapter 10, simulate the transient performance of
the closed-loop linear
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