Cross-Gramian-based dominant subspaces

  • Peter Benner
  • Christian HimpeEmail author
Open Access
Part of the following topical collections:
  1. Model reduction of parametrized Systems


A standard approach for model reduction of linear input-output systems is balanced truncation, which is based on the controllability and observability properties of the underlying system. The related dominant subspaces projection model reduction method similarly utilizes these system properties, yet instead of balancing, the associated subspaces are directly conjoined. In this work, we extend the dominant subspace approach by computation via the cross Gramian for linear systems, and describe an a-priori error indicator for this method. Furthermore, efficient computation is discussed alongside numerical examples illustrating these findings.


Controllability Observability Cross Gramian Model reduction Dominant subspaces HAPOD DSPMR 

Mathematics Subject Classification (2010)

93A15 93B11 93B20 



This work is dedicated to the late Thilo Penzl, who wrote the preprint version of [32] 20 years (at this time of writing) ago, in 1999, and, moreover, 2019 marks the year of his 20th death anniversary. Thilo Penzl died December 17, 1999, but his work and ideas inspire researchers in model reduction and matrix equations to date.

The authors thank the two anonymous reviewers for their helpful feedback and comments.

Funding information

Open access funding provided by Max Planck Society. This study is supported by the German Federal Ministry for Economic Affairs and Energy (BMWi), in the joint project: “MathEnergy – Mathematical Key Technologies for Evolving Energy Grids,” sub-project: Model Order Reduction (Grant No. 0324019B).

Code availability section

The source code of the presented numerical examples can be obtained from: and is authored by: Christian Himpe.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Antoulas, A. C.: Approximation of Large-Scale Dynamical Systems, Adv. Des. Control, vol. 6. SIAM Publications, Philadelphia (2005). CrossRefGoogle Scholar
  2. 2.
    Baur, U., Benner, P., Feng, L.: Model order reduction for linear and nonlinear systems: a system-theoretic perspective. Arch. Comput. Methods Eng. 21 (4), 331–358 (2014). MathSciNetCrossRefGoogle Scholar
  3. 3.
    Benner, P.: Solving large-scale control problems. IEEE Control Syst Mag. 14 (1), 44–59 (2004). Google Scholar
  4. 4.
    Benner, P., Himpe, C., Mitchell, T.: On reduced input-output dynamic mode decomposition. Adv. Comput. Math. 44(6), 1821–1844 (2018). MathSciNetCrossRefGoogle Scholar
  5. 5.
    Benner, P., Kürschner, P.: Computing real low-rank solutions of Sylvester equations by the factored ADI method. Comput. Math. Appl. 67(9), 1656–1672 (2014). MathSciNetCrossRefGoogle Scholar
  6. 6.
    Benner, P., Kürschner, P., Saak, J.: Self-generating and efficient shift parameters in ADI methods for large Lyapunov and Sylvester equations. Electron. Trans. Numer. Anal. 43, 142–162 (2014). MathSciNetzbMATHGoogle Scholar
  7. 7.
    Benner, P., Li, R. C., Truhar, N.: On the ADI method for Sylvester equations. J. Comput. Appl. Math. 233(4), 1035–1045 (2009). MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bond, B. N., Daniel, L.: Guaranteed stable projection-based model reduction for indefinite and unstable linear systems. In: 2008 IEEE/ACM International Conference on Computer-Aided Design (2008).
  9. 9.
    Bru, R., Coll, C., Thome, N.: Symmetric singular linear control systems. Appl. Math. Lett. 15(6), 671–675 (2002). MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chahlaoui, Y., Van Dooren, P.: A collection of benchmark examples for model reduction of linear time invariant dynamical systems. Tech. Rep. 2002–2, SLICOT Working Note. Available from (2002)
  11. 11.
    Davidson, A.: Balanced systems and model reduction. Electron. Lett. 22(10), 531–532 (1986). CrossRefGoogle Scholar
  12. 12.
    Fernando, K. V., Nicholson, H.: Minimality of SISO linear systems. Proc. IEEE 70(10), 1241–1242 (1982). CrossRefGoogle Scholar
  13. 13.
    Fernando, K. V., Nicholson, H.: On the structure of balanced and other principal representations of SISO systems. IEEE Trans. Autom. Control 28(2), 228–231 (1983). MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gardiner, J. D., Laub, A. J., Amato, J. J., Moler, C. B.: Solution of the Sylvester matrix equation AXB + CXD = E. ACM Trans. Math. Softw. 18 (2), 223–231 (1992). MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gugercin, S., Antoulas, A. C., Beattie, C.: \({\mathscr{H}}_{2}\) model reduction for large-scale linear dynamical systems. SIAM J. Matrix Anal. Appl. 30(2), 609–638 (2008). MathSciNetCrossRefGoogle Scholar
  16. 16.
    Himpe, C.: emgr – the Empirical Gramian Framework. Algorithms 11(7), 91 (2018). MathSciNetCrossRefGoogle Scholar
  17. 17.
    Himpe, C.: emgr – EMpirical GRamian framework (version 5.7) (2019).
  18. 18.
    Himpe, C., Leibner, T., Rave, S.: Hierarchical approximate proper orthogonal decomposition. SIAM J. Sci. Comput. 40(5), A3267–A3292 (2018). MathSciNetCrossRefGoogle Scholar
  19. 19.
    Himpe, C., Leibner, T., Rave, S., Saak, J.: Fast low-rank empirical cross Gramians. Proc. Appl. Math. Mech. 17(1), 841–842 (2017). CrossRefGoogle Scholar
  20. 20.
    Himpe, C., Ohlberger, M.: A note on the cross Gramian for non-symmetric systems. Syst. Sci. Control Eng. 4(1), 199–208 (2016). CrossRefGoogle Scholar
  21. 21.
    Himpe, C., Rave, S.: HAPOD – hierarchical approximate proper orthogonal decomposition (version 2.0). (2019)
  22. 22.
    Jiang, Y. L., Qi, Z. Z., Yang, P.: Model order reduction of linear systems via the cross Gramian and SVD. IEEE Trans. Circ. Syst. II: Express Briefs 66(3), 422–426 (2019). Google Scholar
  23. 23.
    Li, J. R., White, J.: Efficient model reduction of interconnect via approximate system Gramians. In: 1999 IEEE/ACM International Conference on Computer-Aided Design. Digest of Technical Papers, pp. 380–383 (1999).
  24. 24.
    Li, J.R., White, J.: Reduction of large circuit models via low rank approximate Gramians. Int. J. Appl. Math. Comput. Sci. 11(5), 1151–1171 (2001). MathSciNetzbMATHGoogle Scholar
  25. 25.
    The MathWorks, Inc., MATLAB
  26. 26.
    Mirsky, L.: A trace inequality of John von Neumann. Monat. Math. 79(4), 303–306 (1975). MathSciNetCrossRefGoogle Scholar
  27. 27.
    Moore, B. C.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control AC–26(1), 17–32 (1981). MathSciNetCrossRefGoogle Scholar
  28. 28.
    Moosmann, C., Greiner, A.: Convective thermal flow problems. In: Dimension Reduction of Large-Scale Systems, vol. 45, pp. 341–343. Springer (2005). Google Scholar
  29. 29.
    Opmeer, M. R., Reis, T.: A lower bound for the balanced truncation error for MIMO systems. IEEE Trans. Autom. Control 60(8), 2207–2212 (2015). MathSciNetCrossRefGoogle Scholar
  30. 30.
    Or, A. C., Speyer, J. L., Kim, J.: Reduced balancing transformations for large nonnormal state-space systems. J. Guid. Control Dyn. 35(1), 129–137 (2012). CrossRefGoogle Scholar
  31. 31.
    Peng, L., Mohseni, K.: Symplectic model reduction of Hamiltonian systems. SIAM J. Sci. Comput. 38(1), A1–A27 (2016). MathSciNetCrossRefGoogle Scholar
  32. 32.
    Penzl, T.: Algorithms for model reduction of large dynamical systems. Linear Algebra Appl. 415(2–3), 322–343 (2006). (Reprint of Technical Report SFB393/99-40, TU Chemnitz, 1999.)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Perev, K.: The unifying feature of projection in model order reduction. Inf. Technol. Control 12(3–4), 17–27 (2016). Google Scholar
  34. 34.
    Rahrovani, S., Vakilzadeh, M.K., Abrahamsson, T.: On Gramian-based techniques for minimal realization of large-scale mechanical systems. In: Topics in Modal Analysis, vol. 7, pp. 797–805 (2014). Google Scholar
  35. 35.
    Redmann, M., Kürschner, P.: An output error bound for time-limited balanced truncation. Syst. Control Lett. 121, 1–6 (2018). MathSciNetCrossRefGoogle Scholar
  36. 36.
    Rowley, C. W.: Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. Bifurcat. Chaos 15(3), 997–1013 (2005). MathSciNetCrossRefGoogle Scholar
  37. 37.
    Saak, J.: Efficient numerical solution of large scale algebraic matrix equations in PDE control and model order reduction. Dissertation, Technische Universität Chemnitz, Chemnitz. (2009)
  38. 38.
    Shaker, H. R.: Generalized cross-Gramian for linear systems. In: Proceedings of IEEE Conf. Ind. Electron. Appl., pp. 749–751 (2012).
  39. 39.
    Shi, G., Shi, C. R. J.: Model-order reduction by dominant subspace projection: error bound, subspace computation, and circuit applications. IEEE Trans. Circ. Syst. I: Reg. Papers 52(5), 975–993 (2005). MathSciNetCrossRefGoogle Scholar
  40. 40.
    Sorensen, D. C., Antoulas, A. C.: The Sylvester equation and approximate balanced reduction. Numer. Lin. Alg. Appl. 351–352, 671–700 (2002). MathSciNetCrossRefGoogle Scholar
  41. 41.
    Stykel, T.: Gramian-based model reduction for descriptor systems. Math. Control Signal. Syst. 16(4), 297–319 (2004). MathSciNetCrossRefGoogle Scholar
  42. 42.
    The MORwiki Community: MORwiki - Model Order Reduction Wiki.
  43. 43.
    Toscano, R.: Structured controllers for uncertain systems. Advances in industrial control. Springer, London (2013). CrossRefGoogle Scholar
  44. 44.
    Wang, X., Yu, M.: The error bound of timing domain in model order reduction by Krylov subspace methods. J. Circ. Syst. Comput. 27(6), 1850093 (2018). MathSciNetCrossRefGoogle Scholar
  45. 45.
    Willcox, K., Peraire, J.: Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40(11), 2323–2330 (2002). CrossRefGoogle Scholar
  46. 46.
    Wolf, T., Panzer, H., Lohmann, B.: Gramian-based error bound in model reduction by Krylov subspace methods. IFAC Proc. Vol. (Proc. 18th IFAC World Congress) 44(1), 3587–3592 (2011). Google Scholar
  47. 47.
    Wong, N.: Efficient positive-real balanced truncation of symmetric systems via cross-Riccati equations. IEEE Trans. Comput.-Aided Des. Integr. Circ. Syst. 27(3), 470–480 (2008). CrossRefGoogle Scholar

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© 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.Faculty of MathematicsOtto von Guericke University MagdeburgMagdeburgGermany

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