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Multipoint interpolation of Volterra series and \(\mathcal{H}_2\)-model reduction for a family of bilinear descriptor systems. (English) Zbl 1350.93024

Summary: In this paper, we investigate interpolatory model order reduction for large-scale bilinear descriptor systems. Recently, it was shown in S. Gugercin et al. [SIAM J. Sci. Comput. 35, No. 5, B1010-B1033 (2013; Zbl 1290.41001)]Wyatt et al. (2013) for linear descriptor systems that directly extending the standard rational interpolation conditions used in \(\mathcal{H}_2\) optimal model reduction to descriptor systems, in general, yields an unbounded error in the \(\mathcal{H}_2\)-norm. This is due to the possible mismatch of the polynomial part of the original and reduced-order systems. This conclusion holds for nonlinear systems as well. In this paper, we deal with bilinear descriptor systems and aim to pay attention to the polynomial part of the bilinear descriptor system along with interpolation. To this end, we have shown in P. Goyal, M.I. Ahmad, P. Benner [“Model reduction of quadratic-bilinear descriptor systems via Carleman bilinearization”, in: Proc. of European Control Conference, pp. 1171-1176 (2015)] how to determine the polynomial part of each subsystem of the bilinear descriptor system explicitly, by assuming special structures of the system matrices. Considering the same structured bilinear descriptor systems, in this paper we first show how to achieve multipoint interpolation of the underlying Volterra series of bilinear descriptor systems while retaining the polynomial part of each subsystem of the bilinear system. Then, we extend the interpolation based first-order necessary conditions for \(\mathcal{H}_2\) optimality to bilinear descriptor systems and propose an iterative scheme to obtain an \(\mathcal{H}_2\) optimal reduced-order system. By means of various numerical examples, we demonstrate the efficiency of the proposed model order reduction technique and compare it with reduced bilinear systems obtained by using linear IRKA (Iterative Rational Krylov Algorithm), the Loewner method for bilinear systems and POD-based approximations.

MSC:

93B11 System structure simplification
93A15 Large-scale systems
93C10 Nonlinear systems in control theory

Citations:

Zbl 1290.41001

Software:

Loewner
PDFBibTeX XMLCite
Full Text: DOI

References:

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