A bilinear \(\mathcal {H}_{2}\) model order reduction approach to linear parameter-varying systems

  • Peter Benner
  • Xingang CaoEmail author
  • Wil Schilders
Open Access


This paper focuses on the model reduction problem for a special class of linear parameter-varying systems. This kind of systems can be reformulated as bilinear dynamical systems. Based on the bilinear system theory, we give a definition of the \(\mathcal {H}_{2}\) norm in the generalized frequency domain. Then, a model reduction method is proposed based on the gradient descent on the Grassmann manifold. The merit of the method is that by utilizing the gradient flow analysis, the algorithm is guaranteed to converge, and further speedup of the convergence rate can be achieved as well. Two numerical examples are tested to demonstrate the proposed method.


Model order reduction Linear parameter-varying systems Bilinear dynamical systems Gradient descent Grassmann manifold 

Mathematics Subject Classification (2010)

14M15 34C20 65K10 93C10 



The authors would like to thank the European Cost Action: TD1307-European Model Reduction Network (EU-MORNET) for the funding of a short-term scientific mission, which leads to this work.


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© The Author(s) 2019

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Authors and Affiliations

  1. 1.Computational Methods in Systems and Control TheoryMax Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany
  2. 2.Centre for Analysis, Scientific Computing and ApplicationsEindhoven University of TechnologyEindhovenThe Netherlands

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