Cross-Gramian-based dominant subspaces


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.


  1. 1.

    Antoulas, A. C.: Approximation of Large-Scale Dynamical Systems, Adv. Des. Control, vol. 6. SIAM Publications, Philadelphia (2005).

    Book  Google 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).

    MathSciNet  Article  Google 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).

    MathSciNet  Article  Google 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).

    MathSciNet  Article  Google 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).

    MathSciNet  MATH  Google 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).

    MathSciNet  Article  Google 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).

    MathSciNet  Article  Google 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).

    Article  Google Scholar 

  12. 12.

    Fernando, K. V., Nicholson, H.: Minimality of SISO linear systems. Proc. IEEE 70(10), 1241–1242 (1982).

    Article  Google 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).

    MathSciNet  Article  Google 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).

    MathSciNet  Article  Google 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).

    MathSciNet  Article  Google Scholar 

  16. 16.

    Himpe, C.: emgr – the Empirical Gramian Framework. Algorithms 11(7), 91 (2018).

    MathSciNet  Article  Google 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).

    MathSciNet  Article  Google 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).

    Article  Google 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).

    Article  Google 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).

    MathSciNet  MATH  Google 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).

    MathSciNet  Article  Google 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).

    MathSciNet  Article  Google 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).

    MathSciNet  Article  Google 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).

    Article  Google Scholar 

  31. 31.

    Peng, L., Mohseni, K.: Symplectic model reduction of Hamiltonian systems. SIAM J. Sci. Comput. 38(1), A1–A27 (2016).

    MathSciNet  Article  Google 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.)

    MathSciNet  Article  Google 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).

    MathSciNet  Article  Google Scholar 

  36. 36.

    Rowley, C. W.: Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. Bifurcat. Chaos 15(3), 997–1013 (2005).

    MathSciNet  Article  Google 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).

    MathSciNet  Article  Google 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).

    MathSciNet  Article  Google Scholar 

  41. 41.

    Stykel, T.: Gramian-based model reduction for descriptor systems. Math. Control Signal. Syst. 16(4), 297–319 (2004).

    MathSciNet  Article  Google 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).

    Book  Google 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).

    MathSciNet  Article  Google Scholar 

  45. 45.

    Willcox, K., Peraire, J.: Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40(11), 2323–2330 (2002).

    Article  Google 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).

    Article  Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to Christian Himpe.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by: Anthony Nouy

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Benner, P., Himpe, C. Cross-Gramian-based dominant subspaces. Adv Comput Math 45, 2533–2553 (2019).

Download citation


  • Controllability
  • Observability
  • Cross Gramian
  • Model reduction
  • Dominant subspaces

Mathematics Subject Classification (2010)

  • 93A15
  • 93B11
  • 93B20