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

  • Peter Benner
  • Xingang CaoEmail author
  • Wil Schilders
Open Access
Part of the following topical collections:
  1. Model reduction of parametrized Systems


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.


  1. 1.
    Absil, P.A., Mahony, R., Sepulchre R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2009)zbMATHGoogle Scholar
  2. 2.
    Baars, S., Viebahn, J., Mulder, T., Kuehn, C., Wubs, F.W., Dijkstra, H.A.: Continuation of probability density functions using a generalized Lyapunov approach. J. Comput. Phys. 336, 627–643 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Benner, P., Breiten, T.: On \(\mathcal {H}_{2}\)-model reduction of linear parameter-varying systems. PAMM 11(1), 805–806 (2011)CrossRefGoogle Scholar
  4. 4.
    Benner, P., Breiten, T.: Interpolation-based \(\mathcal {H}_{2}\)-model reduction of bilinear control systems. SIAM J. Matrix Anal. Appl. 33(3), 859–885 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Benner, P., Damm, T.: Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems. SIAM J. Control. Optim. 49(2), 686–711 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483–531 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Breiten, T., Damm, T.: Krylov subspace methods for model order reduction of bilinear control systems. Syst. Control Lett. 59(8), 443–450 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bruns, A., Benner, P.: Parametric model order reduction of thermal models using the bilinear interpolatory rational Krylov algorithm. Math. Comput. Model. Dyn. Syst. 21(2), 103–129 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bruns, A.S.: Bilinear \(\mathcal {H}_{2}\)-Optimal Model Order Reduction with Applications to Thermal Parametric Systems. PhD thesis, Otto-von-Guericke Universität Magdeburg (2015)Google Scholar
  10. 10.
    Castañé Selga, R.: The Matrix Measure Framework for Projection-Based Model Order Reduction. PhD thesis, Technische Universität München (2011)Google Scholar
  11. 11.
    D’Alessandro, P., Isidori, A., Ruberti, A.: Realization and structure theory of bilinear dynamical systems. SIAM J. Control. 12(3), 517–535 (1974)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Damm, T.: Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations. Numerical Linear Algebra with Applications 15 (9), 853–871 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dorissen, H.: Canonical forms for bilinear systems. Syst. Control Lett. 13(2), 153–160 (1989)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2), 303–353 (1998)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Flagg, G., Gugercin, S.: Multipoint Volterra series interpolation and \(\mathcal {H}_{2}\) optimal model reduction of bilinear systems. SIAM J. Matrix Anal. Appl. 36(2), 549–579 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jost, J.: Riemannian Geometry and Geometric Analysis, vol. 42005. Springer, Berlin (2008)Google Scholar
  17. 17.
    Mohler, R.R.: Nonlinear Systems (Vol 2): Applications to Bilinear Control. Prentice-Hall, Inc (1991)Google Scholar
  18. 18.
    MORwiki-Community: MORwiki - Model Order Reduction Wiki. (2018)
  19. 19.
    Negri, F., Manzoni, A., Amsallem, D.: Efficient model reduction of parametrized systems by matrix discrete empirical interpolation. J. Comput. Phys. 303, 431–454 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Nijmeijer, H., Van der Schaft, A.: Nonlinear Dynamical Control Systems, vol. 175. Springer, Berlin (1990)CrossRefGoogle Scholar
  21. 21.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, USA (1999)CrossRefGoogle Scholar
  22. 22.
    Sato, H., Sato, K.: Riemannian trust-region methods for H 2 optimal model reduction. In: 2015 IEEE 54th Annual Conference on Decision and Control (CDC), pp 4648–4655, IEEE (2015)Google Scholar
  23. 23.
    Sato, H., Sato, K.: A New H 2 optimal model reduction method based on riemannian conjugate gradient method. In: 2016 IEEE 55th Annual Conference on Decision and Control (CDC), pp. 5762–5768, IEEE (2016)Google Scholar
  24. 24.
    Shank, S.D., Simoncini, V., Szyld, D.B.: Efficient low-rank solution of generalized Lyapunov equations. Numer. Math. 134(2), 327–342 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Stykel, T., Simoncini, V.: Krylov subspace methods for projected Lyapunov equations. Appl. Numer. Math. 62(1), 35–50 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Xu, Y., Zeng, T.: Fast optimal \(\mathcal {H}_{2}\) model reduction algorithms based on Grassmann manifold optimization. Int. J. Numer. Anal. Model. 10, 972–991 (2013)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Yan, W.Y., Lam, J.: An approximate approach to H 2 optimal model reduction. IEEE Trans. Autom. Control 44(7), 1341–1358 (1999)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Zhang, L., Lam, J.: On H 2 model reduction of bilinear systems. Automatica 38(2), 205–216 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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

Personalised recommendations