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 405

and

þ1

1
2

A q Þ 1 B q B q ð A q Þ 1 dw P q ¼ ð iw q I q iw q I q ð 14 Þ

P

1

Owing to the complexity dot matrix, solving of the Lyaponuv equations is very
complicated for switched linear system of high order.

2.2 The Singular Value Decomposition

n n , their singular value decomposition is de

Given a square matrix A q 2 R ned by

this relation (Antoulas 2009 ; Antoulas et al. 2001 ):

U q R q V q A q ¼ ð 15 Þ

n n , U q ¼ ð u 1 q ; u 2 q ; ...; u n q Þ2 R n n , V q ¼

where R q ¼ diag ðr 1 q ; ...; r n q Þ2 R

n n and

ð r 1 q r 1 q r n q .

Gramians matrices plays an important part in the reduction methods based on
singular value decomposition. The relationship between the singular value
decomposition and the gramians matrices is as follows (Antoulas 2009 ):

v 1 q ; v 2 q ; ...; v n q Þ2 R

q

k i q ð

r i q ¼

P q Q q Þ i q ¼

; ...; ð Þ

for 1 n 16

where, r i q presents the Hankel singular value of R q and k i q presents the eigenvalues

of the product P q Q q .

2.3 H 1 of Dynamical Switched Systems

In this work, determining the error between the original switched system and
reduced one is obtained by using the H 1 technique knowing that (Antoulas 2009 ;
Gugercin 2008 ):

ÞR q ð kR q ð jw jw Þk H 1 2 ðr ð r þ 1 Þ q þþr n q Þ ð 17 Þ