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 406

2.4 Krylov Subspace

Given a square matrix A q and a vector b q , the spanned by the vectors

A m 1
q

f b q ; A q b q ; ...; b q g is called a standard Krylov subspace of dimension m denoted

K m q f A q ; b q g for each sub-matrix (Awais et al. 2007 ; Heyouni and Jbilou 2006 ):

K m q f A q ; b q g ¼span f b q ; A q b q ; ...; A m 1

b q g ð 18 Þ

q

An effective reduced model in the form ( 2 ) by the projection onto the Krylov
subspace of the states matrices of system ( 1 ) is obtained. But there is another
method to generate the Krylov subspace, which is more ef cient that called rational

Krylov subspace de ned as :

Y

m q
j q ð

s 1 q I q Þ 1 b q ; ...; s j q I q Þ 1 b q g

K m q f A q ; b q ; s q g ¼ span fð A q A q ð 19 Þ

where, s q ¼ ð s 1 q ; s 2 q ; ...; s m q Þ

3 Dual Rational Krylov for Switched Linear System

In this section, the details of the Dual Rational Krylov algorithm for computing of
two projection matrices V r q and Z r q for each subsystem according to switching
signal q are brie y recalled. Dual Rational Krylov is among the best approaches to

reduce the large-scale linear switched systems. It is easy to implemented, numer-
ically stable and to avoid the dif

fl

culties in the constructing of the two projections
matrices. V r q and Z r q are constructed column by column during the iteration process
using a Gram Schmidt techniques in orthogonalization procedure, such as the
condition of biorthogonalithy is satis ed Z r q V r q ¼ I r q . Take a switched linear sys-

tem as a form ( 1 ) and assume that a sequence of expansion points

s r q g
is given, with r is the order of reduced subsystem. These expansion points are
interspersed. For each expansion point of each subsystem a two column vectors are
generated, i.e in the first iteration uses s 1 q , the second iteration uses s 2 q until
rth iteration.
The details of the Dual Rational Krylov algorithm for switched linear system can
be found in Table 1 (Antoulas 2009 ; Druskin and Simoncini 2011 ; Flagg et al.
2012 ; Zhanga et al. 2008 ).
The main steps of this method are:

f s 1 q ; s 2 q ; ...;

Step 1: Choose the interpolation points for each subsystem by the use of the
eigenvalues criterion (Gugercin 2008 ).