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 415

The transfer function f r q ð s Þ of the reduced switched system ( 2 ) is matched with

the transfer function f q ð s Þ of original switched linear system in ( 1 ):

f q ð s i q Þ ¼f r q ð s i q Þ for i q ¼ 1 ; ...; r q and s i q ¼ k i q ð A r q Þ ð 22 Þ

where, k i q is the eigenvalues of A r q :

The details of the Iterative Dual Rational Krylov algorithm for switched linear
system (IDRK-SLS) can be found in Table 2 (Druskin and Simoncini 2011 ;
Gallivan et al. 1996 ; Quarteroni et al. 2007 ):
The main steps of this method are:

Table 2 IDRK-SLS
Algorithm

IDRK-SLS Algorithm:(input:I q ; A q ; B q ; C q ; D q ; S q ; tol;
output:V r q ; Z r q Þ

Switch q
1/*Choose the Initial Interpolation points*/
s i q for i=1 to r
2/*Construction of the matrices V q and Z q by the dual rational-
Krylov based method knowing that*/
(a):V r q ¼ Span

ð s 1 q I q A q Þ 1 B q ; ...; ð s r q I q A q Þ 1 B q

A q Þ T C q
With Z r q V r q ¼ I r q , where Z r q ¼ ð Z r q V r q Þ 1 Z r q
3/*While (the relative change in si i

A q Þ T C q ; ...; ð (b):Z r q ¼ Span ð s 1 q I q s r q I q

: ðð s i þ 1 s i Þ= s i Þ tol

(a):A r q ¼ Z r q A q V r q
(b):s i q ¼ k i q ð A r q Þ : r q

(c):Construction of the matrices V r q and Z r q by the rational-
Krylov based method knowing that:
(d):V r q ¼ Span ð s 1 q I q A q Þ 1 B q ; ...; ð s r q I q A q Þ 1 B q
(e):Z r q ¼ Span

for i ¼ 1

A q Þ T C q
With Z r q V r q ¼ I r q , where Z r q ¼ ð Z r q V r q Þ 1 Z r q
( 4 )/*Generate real V r q and Z r q for complex interpolation point*/
if there exist any is i q is not a real number
then V r q ð real Þ ¼real ð V r q Þ , V r q ð imaginairy Þ ¼imag ð V r q Þ ,
Z r q ð real Þ ¼ real ð Z r q Þ , Z r q ð imaginairy Þ ¼ imag ð Z r q Þ ,
½ V r q ;

A q Þ T C q ; ...; ð ð s 1 q I q s r q I q

rr q ¼ QR ½ V r q ð real Þ; V r q ð imaginairy Þ ,

½Z r q ; rr q ¼QR½Z r q ð real Þ; Z r q ð imaginairy Þ ,

end if
( 5 )/*parameters of reduced model*/
A r q ¼ Z r q A q V r q ; B r q ¼ Z r q B q ; C r q ¼ C q V q ; D r q ¼ D q

End Switch