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Z
r
q
V
r
q
¼
I
r
ð
23
Þ
Theorem 2 Take a stable switched linear system
R
q
with the transfer functions
f
q
ð
s
Þ
as in (
1
) and fix the the interpolation points s
i
q
, Let
R
r
q
be an rth reduced
subsystems with transfer
fixed stable reduced poles
k
1
q
; ...; k
r
q
. Then the error between each original subsystem and reduced one is
minimized if and only if (Grimme
1997
):
f
q
ð
functions f
r
q
ð
s
Þ
having
s
Þ
¼
f
r
q
ð
s
Þ
for
s
¼
k
1
q
; ...; k
r
q
ð
24
Þ
The reduced order model is de
ned by these relationships:
Z
r
q
A
q
V
r
q
;
Z
r
q
B
q
;
A
r
q
:
¼
B
r
q
:
¼
C
r
q
:
¼
C
q
V
r
q
;
D
r
q
:
¼
D
q
:
ð
25
Þ
The different steps of the Iterative SVD-Dual Rational Krylov Algorithm for
switched linear system can be found in Table
3
(Gugercin
2008
; Lee et al.
2006
;
Quarteroni et al.
2007
):
The main steps of Iterative SVD-Dual Rational Krylov algorithm for switched
linear system are:
Step 1: Choose the interpolation points for each subsystem by the use of
criterion poles, the number of interpolation points must be equal to the order
of reduced subsystem.
Step 2: Use the based method of dual rational Krylov for constructing the V
r
q
basis knowing that the orthogonality condition is satis
ð
V
r
q
V
r
q
¼ I
r
q
Þ
.
Step 3: Calculate the gramian matrix of observability g
o
q
for each subsystem.
Step 4: Construct the matrix Z
r
q
using the observability matrix gramian and the
matrix V
r
q
.
Step 5: Calculate the reduced matrices states A
r
q
and the corresponding
eigenvalues. Determine the eigenvalues of these matrices to re-initialize the
interpolation points of each subsystem. Then, recalculate the orthonormal basis
V
r
q
and the matrix Z
r
q
. Repeat these instructions until the satisfaction of a
stoping criterion in the interpolation points.
The stoping criterion is a tolerance which was set at the beginning, that denote the
relative change between two successive interpolation points.
Step 6: Generate the real orthogonal matrices V
r
q
using the reduced QR fac-
torization if there exists any complex interpolation points.
Step 7: The reduced order model is de
ed
ned as:
Z
r
q
A
q
V
r
q
;
Z
r
q
B
q
;
A
r
q
¼
B
r
q
¼
C
r
q
¼
C
q
V
r
q
;
D
r
q
¼
D
q
:
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