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 404

Z 1

e A q t C q C q e A q t dt Q q ¼ ð 7 Þ

0

Z 1

e A q t B q B q e A q t dt P q ¼ ð 8 Þ

0

The solution of the stable switched linear system in the sense of Lyapunov is
obtained by solving the following two equations for each subsystem (Andres et al.
2013 ; Antoulas 2009 ; Kouki et al. 2013b ; Mehrmann et al. 2012 ):

P q A q þ B q B q ¼ A q P q þ 0 ð 9 Þ

A q Q q þ C q C q ¼ Q q A q þ 0 ð 10 Þ

The solutions of these two equations are P and Q (Antoulas 2009 ; Gugercin
2008 ). P q 2 R

n n and Q q 2 R n n are called the reachability and the observability

gramians matrices, respectively.
The Eq. ( 9 ) is proved by the following formulation (Bao et al. 2006 ):

Z 1

½A q e A q t B q B q e A q t þ e A q t B q B q e A q t A T

A q P q þ P q A q ¼

dt

0

ð 11 Þ

Z 1

e A q t B q B q e A q t

B q B q

¼ d ð Þ

0

The Eq. ( 10 ) is proved by the following formulation:

Z 1

A q e A q t C q C q e A q t e A q t C q C q e A q t A A q Q q þ

Q q A q ¼ ½ þ dt

0

ð 12 Þ

Z 1

e A q t C q C q e A q t C q C q

¼ d ð Þ

0

nite gramians in the frequency time of the system ( 1 ) are obtained by
these relationships:

The in

þ1

1
2

A q Þ 1 C q C q ð A q Þ 1 dw Q q ¼ ð iw q I q iw q I q ð 13 Þ

P

1