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 421

Table 3 Iterative SVDDRK-
SLS Algorithm

Iterative SVDDRK-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 q ¼ 1tor q
2/*Construction of the matrices V q by the dual rational-Krylov
based method knowing that*/
(a):V r q ¼ Span A q Þ 1 B q ; ...; ð A q Þ 1 B q ð s 1 q I q s r q I q

With V r q V r q ¼ I r q
3/*Calculate the garamian matrix of observability for each
subsystem*/
(a)g o q ¼ R 1
0

e tA q C q C q e tA dt

4/*Construction of the matrix Z q by the SVD based method
knowing that*/
(b):Z r q ¼ Q q V r q ð

V r q Þ
5/*While (the relative change in si i : 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 for i q ¼ 1:r q
(c):Construction of the matrix V q by the rational-Krylov based
method knowing that:
(d):V r q ¼ Span

A q Þ 1 B q ; ...; ð A q Þ 1 B q

ð s 1 q I q s r q I q

With Z r q V r q ¼ I r q
(e):*Construction of the matrix Z r q by the SVD based method
knowing that*/
Z r q ¼ Q q V r q ð

V r q Þ
6/*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 ¼ real ð V r q Þ , V r q ð imaginairy Þ ¼ imag 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 ,

½V r q ; rr q ¼QR½V 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
67*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 r q ; D r q ¼ D q

End Switch