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 407

Table 1 DRK-SLS
Algorithm

DRK-SLS Algorithm:(input:I q ; A q ; B q ; C q ; D q ; S q ;
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 and Z q by the dual rational-
Krylov based method*/
for k q =1tor q
if k q :=1
v0 q ¼ ðð A q s q I q Þ 1 B q

v0 q ¼ v0 q = norm ð v0 q ; 0 fro 0 Þ
V q ð:; Þ ¼ v0 q

z0 q ¼ ðð

1

I q Þ T

A q s q C q

z0 q ; 0 fro 0 Þ z0 q ¼ z0 q = norm ð

Z q ð:; 1 Þ ¼ z0 q
else
v q ð:; k Þ ¼ ðð A q s q I q Þ 1

B q
v q ð:; k Þ ¼ v q ð:; k Þ V q ð:; k 1 Þ V q ð:; k 1 Þ

T

v q ð:; k Þ

V q ð:; k Þ ¼v q ð:; k Þ= norm ð v q ð:; k Þ; 0 fro 0 Þ
z q ð:; k Þ ¼ ðð A q s q I q Þ T

C q

T

z q ð:; k Þ ¼ z q ð:; k Þ Z q ð:; k 1 Þ Z q ð:; k 1 Þ z q ð:; k Þ

Z q ð:; k Þ ¼ z q ð:; k Þ= norm ð z q ð:; k Þ; 0 fro 0 Þ
End if
End for
End Switch

Step 2: Compute the V r q and Z r q bases with Rational Krylov subspaces, such as
the condition of biorthogonalithy is satis ed :

V r q Þ 1 Z r q Þ

Z r q

ðð V r q ¼ I r q : ð 20 Þ

The parameters of the reduced system can be obtained by the congruence
transformation:

A r q ¼ Z1A q V q ; B r q ¼ Z1B q ; C r q ¼ C q V q ; D r q ¼ D q

Z q V q Þ ð 1 Þ Z q Þ: where ; Z1 ¼ ðð