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 403

A q Þ 1 B q þ f q ð s Þ ¼ C q ð sI q D q ð 3 Þ

r q r q , A r q 2 R r q r q , B r q 2 R r q p ,

The problem consists in approximating: E r q 2 R

p r q , D r q 2 R p p p r q ,

C r q 2 R and y r ð

Þ2 R

t the matrices of the each reduced

subsystem of order r q , where r q n.

The state space representation of reduction switched dynamical linear systems is
as follows (Benner et al. 2003 ; Gaoa et al. 2006 ; Grimme 1997 ; Kouki et al. 2013a ,
b ; Tulpule et al. 2011 ):

(

E r q dx r q ð t Þ

¼

A r q x q ð

Þþ ð Þ

t B r q u t

R q ¼

ð 4 Þ

dt

y r q ð

Þ ¼

C r q x q ð

Þþ ð Þ

t t D r q u t

The Laplace transform is applied to the system ( 4 ), this relation is obtained:

sE r q X r q ð s Þ ¼A r q X r q ð s Þþ B r q U ð s Þ
Y r q ð

R q ¼

ð 5 Þ

t Þ ¼ C r q X r q ð s Þþ D r q U ð s Þ

where X r q ð s Þ and Y r q ð s Þ represents the Laplace transform of the reduces x r q ð t Þ and

y r q ð t Þ . The transfer function of the reduced linear switched system is as follows:

A r q Þ 1 B r q þ

f r q ð s Þ ¼ C r q ð sI r q D r q ð 6 Þ

y presents an overview of
the Lyapunov equations and the H 1 error. In Sect. 3 , the Dual Rational Krylov is
presented. Section 4 , the Iterative Dual Rational Krylov method for switched linear
systems, will be presented with application on the numerical examples. In Sect. 5 ,
the Iterative SVD-Dual Rational Krylov method for switched linear systems is
detailed and evaluated by the use of the numerical examples. In Sect. 6 , a com-
parative study between the Iterative Dual Rational Krylov method and the Iterative
SVD-Dual Rational Krylov method is given. The last section is dedicated to con-
clude this paper.

This chapter is organized as follows. Section 2 , brie fl

2 Preliminaries

2.1 Lyapunov Equations

Let a switched linear stable system as in ( 1 ). The in nite observability and

reachability gramians in the continuous time of this system are obtained by this two
relations (Antoulas 2009 ; Diepold and Eid 2011 ):