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 402

1 Introduction

The technological advance means the increase of the complexity of industrial
systems. They operate in different environments with changing conditions and
characteristics (quickly and brutally). Such as the industries aerospace, automotive,
aggro-food, process engineering, chemical process, electrical circuit, power elec-
tronic systems, Thermal The modeling of

these system types usually leads to the production of non-linear complex models of
high order. However the dynamic is in

fl fluid systems and Mechanical system. …

uenced by both discrete and continuous
event which leads to the hybrid dynamical systems which are divided into two
broad classes of hybrid systems, the

fl

first class is the multi-model system, which
assumes that it is always possible to model a complex system with simple models,
often linear models, assigning each model an operating area of the system. The
second class is the linear piecewise system or called the switched system. This class
of models is widely used for the analysis and control tools for linear systems which
are very developed and also because much of the actual process can be represented
by models from this class. Recent research on switching systems are mainly
focusing on the modeling area, design control law and stability study. However, in
the case of working for the development of the control law, the order of the
controlled system must be taken into consideration because there are several hybrid
systems of high order. These later are dif cult to manipulate and the resolution of

such models is indeed very demanding in computational resources. However,
reduction of switched systems is an important solution for these problems. In this
chapter, the iterative dual rational Krylov algorithm and the iterative SVD-dual
rational Krylov algorithm for switched linear system are presented.
The model reduction problem, focus in this work, can be stated as follows.
Given a switched linear dynamical system in state space form (Dongmei et al.
2008 ; Kouki et al. 2013a , b ; Zhendong and Shuzhi 2009 ):

E dx ð t Þ

dt ¼

ð Þþ ð Þ

A q x t B q u t

R q ¼ ð 1 Þ

y ð t Þ ¼ C q x ð t Þþ D q u ð t Þ

n n , A q 2 R n n , B q 2 R n p , C q 2 R p n , D q 2 R p p ,

In which E q 2 R

p n and q is a switching signal.
Applying the Laplace transform to the system ( 1 ), the following frequency
formulation is obtained (Kouki et al. 2014a , b ):

n p , y

u ð t Þ2 R ð t Þ2 R

sEX ð s Þ ¼ A q X ð s Þþ B q U ð s Þ

R q ¼ ð 2 Þ

Y ð s Þ ¼ C q X ð s Þþ D q U ð s Þ

where X ð s Þ , Y ð s Þ and U ð s Þ are the Laplace transform of x ð t Þ , y ð t Þ and u ð t Þ

respectively. For simplicity, supposing that the u is an impulse response, then the

transfer function of the original switched linear system is given by (Grimme 1997 ;
Kouki et al. 2013c ):

ð t Þ