Image Processing Reference
In-Depth Information
Since the Jacobian matrix is a function of the actuator for the level 2 system, the
gain matrix can be computed for different regions of the actuator operating space.
That is, the gain matrix will be actuator dependent, which can be written as
K j U j ¼ B 1 j U j s
(
9
:
47
)
where suf
x j represents the discrete nature of the actuator vector. It also means that
the operating actuator space is quantized to j ¼
1, 2, 3, . . . , J regions.
This type of pole assignment will work when the Jacobian matrix is invertible. If
faster convergence is required, then it is normal practice to assign the poles (p 1 , p 2 , p 3 )
close to 0. If the poles are assigned equal to zero, then
I, and the performance will
be theoretically
That means, due to the multivariable nature of the
system, the theoretical minimum for convergence to zero steady-state error will be
''
''
dead beat.
''
(i.e., one measurement-actuation update). If the poles are assigned
closer to unity (but not exactly unity, because unit pole values will lead to marginal
stability, meaning no convergence), then number of iterations will be greater than
three (about 7 to 8). As a result, the performance of the closed-loop system in the
presence of system or sensor noise can be improved.
one iteration
''
Example 9.6
Let the desired closed-loop poles for level 2 controller be located at p 1 , p 2 , p 3 .
(i) Using Equation 9.45 for the Jacobian matrix of Example 9.5
nd the gain
matrix that will provide equal poles p 1 ¼p 2 ¼p 3 ¼
0.3. Use the MATLAB
pole-placement algorithm from Ref. [12]. Find the gain matrix using both
methods for poles at p 1 ¼
0.7. What do you see?
(ii) Calculate the step responses for case (i). Compare the results.
0.3; p 2 ¼
0.3; p 3 ¼
S OLUTION
Case (i): Gain matrix from Equation 9.45 for pole values p 1 ¼p 2 ¼p 3 ¼
0.3 is
given by
2
4
3
5
4
:
223
1
:
3384
0
:
0891
10 4
K ¼
0
:
7863
0
:
1570
0
:
0492
0
:
7818
0
:
1561
0
:
1033
The MATLAB pole-placement algorithm gives the same gain values for equal poles.
Equation 9.45 and the MATLAB pole-placement algorithm give the same gain
matrix for pole values p 1 ¼
0.3; p 2 ¼
0.3; p 3 ¼
0.7.
2
4
3
5
10 4
4
:
223
1
:
3384
0
:
0382
K ¼
0
:
7863
0
:
1570
0
:
0211
0
:
7818
0
:
1561
0
:
0433
Since the system matrix, A,is3
3 and is equal to identity, Equation 9.45 gives a
robust gain matrix with all the bene
ts of the pole-placement algorithm, place().
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