Iterative Dual Rational Krylov and Iterative SVD-Dual Rational Krylov Model Reduction for Switched Linear Systems - Complex System Modelling and Control Through Intelligent Soft Computations - page 414

40

20

0

−20

−40

−160 −140 −120 −100 −80 −60 −40 −20 0 20

Real part

20

10

0

−10

−20

−40 −35 −30 −25 −20 −15 −10 −5 0

Real part

Fig. 10 Poles distribution of beam reduced switched system (order 24) with DRK-SLS method

4 Iterative Dual Rational Krylov for Linear Switched
Systems

In this section the proposed method is given, the iterative dual rational Krylov
model reduction for switched linear system, is an extended version of the dual
rational Krylov method for switched linear system. Iterative dual rational Krylov is
a connection between the Krylov-based reduction method and the interpolation of
the expansion points. Given a stable switched linear system as the form ( 1 ) and
using the eigenvalues criterion in the choice of the interpolation points (Flagg et al.
2012 ; Kouki et al. 2013a , 2014a , b ). However, this method generated two Krylov
subspaces V r q and Z r q for each subsystem, the generation of the two Krylov sub-
spaces is performed iteratively until the satisfaction of the stopping criterion

ðð and guarantees the biorthogonalithy condition of the two

Krylov subspaces for each subsystem (i.e. Z r q V r q ¼

s ð i þ 1 Þ q s i q Þ= s ð i þ 1 Þ q Þ

I r q where r q is the order of
reduced system) (Druskin and Simoncini 2011 ; Gallivan et al. 1996 ; Quarteroni
et al. 2007 ). Theorem 1 summarizes this result:

Theorem 1 Take a switched linear system as a form ( 1 ) and the interpolation point
f

n r n r

s i q g for i q ¼ 1 ; ...; r q . Let V r q 2 R and Z r q 2 R be obtained as follows

(Grimme 1997 ):

(

A q Þ 1 B q
Z r q ¼ Span ð s 1 q I A q Þ T C q ; ...; ð s r q I A q Þ T C q

A q Þ 1 B q ; ...; ð V r q ¼ Span ð s 1 q I s r q I

ð 21 Þ

with Z r q V r q ¼ I r q .