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Step 1: Choose the interpolation points for each sub-system by the use of the
eigenvalues criterion (Gugercin
2008
).
Step 2: Compute the V
r
q
and Z
r
q
bases with Dual Rational Krylov knowing that
the orthogonality condition is satis
V
r
q
Þ
1
Z
r
q
Þ
Z
r
q
.
Step 3: Calculate the reduced states matrices A
r
q
and the corresponding
eigenvalues. Using the mirror of these eigenvalues as interpolation points and
recalculate the new bases V
r
q
and Z
r
q
applying again the Dual Rational Krylov
method. Repeat these instructions until the satisfaction of a stopping criterion in
the expansion frequency.
ed
ðð
V
r
q
¼
I
r
q
Þ
4.1 Numerical Example
To evaluate this approach, the same models used previously are taken with the same
switching signal. fixing the largest singular value of the frequency response of each
original subsystem and the reduced one, the variation of absolute error between each
original subsystem and the reduced one is presented and the poles distribution of the
reduces subsystems are given (Chahlaoui and Dooren
2005
; Diepold and Eid
2011
).
The Fig.
11
presents the largest singular value of the frequency response of the
original switched linear system (order 1006) and reduced one (order 10) to a
frequency range by IDRK-SLS method. Note that when a correlation over the entire
frequency range shape with a low error rate. The Fig.
12
shows the variation of the
singular value of the absolute error between the original switched linear system and
Fom Model Reduced System (sub−system1)
50
Exact
IRK−SLS
0
−50
10
−1
10
0
10
1
10
2
10
3
10
4
Frequency (Hz)
Fom Model Reduced System (sub−syste
m2)
50
Exact1
IRK−SLS
0
−50
10
−1
10
0
10
1
10
2
10
3
10
4
Frequency (Hz)
Fig. 11 Largest singular value of the frequency response of the original switched system of order
(1006) and reduced one of order (10) to a frequency range with IDRK-SLS method
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