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 419

100

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Fig. 16 Poles distribution of beam reduced switched system (order 24) with DRK-SLS method

plane of each subsystem is depicts in Fig. 16 , noting the existence of positive real
part poles, then the subsystems are unstable.
It is obvious the results obtained by this method is better than that obtained by
the previous method, but this method does not guarantee the stability of reduced
system. To solve this problem, a new method that minimizes the H 1 error between
the original switched linear system and the reduced one and guarantee the stability
of reduced switched system is proposed in the next section.

5 Iterative SVD-Dual Rational Krylov for Switched Linear
Systems

While IDRK-SLS algorithm do not always guarantee stability of the each reduced
subsystem, Iterative SVD-Dual Rational Krylov algorithm for linear switched
system gives a reduced model with guaranteed stability and minimize the error
between the original system and reduced one for each sub-system. Hence, Iterative
SVD-Dual Rational Krylov algorithm for linear switched system combines the
advantages of the dual rational Krylov based method and the singular value
decomposition based method, the use of SVD provide stability for reduced system.
This method can generate two matrices, one matrix generated by the Dual Rational
Krylov method

depends on the observability gramian and the other generated
by the singular value decomposition

ð V r q Þ

es the
following orthogonality relation (Gugercin 2008 ; Gugercin and Antoulas 2006 ;
Gugercin et al. 2003 ; Quarteroni et al. 2007 ):

ð Z r q Þ . The two matrices Z r q and V r q satis