Skip to main content
SpringerLink
Log in

Revisiting IRKA: Connections with Pole Placement and Backward Stability

  • Original Article
  • Published:
Vietnam Journal of Mathematics Aims and scope Submit manuscript

Abstract

The iterative rational Krylov algorithm (IRKA) is a popular approach for producing locally optimal reduced-order \({\mathscr{H}}_{2}\)-approximations to linear time-invariant (LTI) dynamical systems. Overall, IRKA has seen significant practical success in computing high fidelity (locally) optimal reduced models and has been successfully applied in a variety of large-scale settings. Moreover, IRKA has provided a foundation for recent extensions to the systematic model reduction of bilinear and nonlinear dynamical systems. Convergence of the basic IRKA iteration is generally observed to be rapid—but not always; and despite the simplicity of the iteration, its convergence behavior is remarkably complex and not well understood aside from a few special cases. The overall effectiveness and computational robustness of the basic IRKA iteration is surprising since its algorithmic goals are very similar to a pole assignment problem, which can be notoriously ill-conditioned. We investigate this connection here and discuss a variety of nice properties of the IRKA iteration that are revealed when the iteration is framed with respect to a primitive basis. We find that the connection with pole assignment suggests refinements to the basic algorithm that can improve convergence behavior, leading also to new choices for termination criteria that assure backward stability.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. For the reader’s convenience, here we actually reproduce the proof of the Sherman–Morrison determinant formula.

  2. Since the matrix is a rank one perturbation of the diagonal matrix, all eigenvalues can be computed in O(r2) operations by specially tailored algorithms.

  3. The shifts (eigenvalues) are naturally considered as equivalence classes in \(\mathbb {C}^{r}/\mathbb {S}_{r}\).

  4. We tacitly assume that throughout the iterations all shifts are simple.

  5. We should point out here that the dimensions n and r are rather small in all reported numerical experiments in [35].

  6. This is the classical backward error interpretation.

  7. Let \(\mathbf {A} \in \mathbb {C}^{n\times n}\) and \(\mathbf {b}\in \mathbb {C}^{n}\). Then, the pair (A,b) is called controllable if \(\text {rank}\left [\begin {array}{cccc}\mathbf {b} & \mathbf {A} \mathbf {b} & {\dots } & \mathbf {A}^{n-1}\mathbf {b} \end {array}\right ]=n.\)

References

  1. Ahuja, K., Benner, P., de Sturler, E., Feng, L.: Recycling BiCGSTAB with an application to parametric model order reduction. SIAM J. Sci. Comput. 37, S429–S446 (2015)

    Article  MathSciNet  Google Scholar 

  2. Ahuja, K., de Sturler, E., Gugercin, S., Chang, E.: Recycling BiCG with an application to model reduction. SIAM J. Sci. Comput. 34, A1925–A1949 (2012)

    Article  MathSciNet  Google Scholar 

  3. Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. Advances in Design and Control. SIAM, Philadelphia (2005)

  4. Antoulas, A.C., Beattie, C.A., Gugercin, S.: Interpolatory Methods for Model Reduction. Computational Science and Engineering, vol. 21. SIAM, Philadelphia (2020)

    Book  Google Scholar 

  5. Antoulas, A.C., Beattie, C.A., Gugercin, S.: Interpolatory model reduction of large-scale dynamical systems. In: Mohammadpour, J., Grigoriadis, K.M. (eds.) Efficient Modeling and Control of Large-Scale Systems, pp 2–58. Springer, Boston, MA (2010)

  6. Antoulas, A.C., Sorensen, D.C., Gugercin, S.: A survey of model reduction methods for large-scale systems. Contemp. Math. 280, 193–219 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Beattie, C., Gugercin, S.: Krylov-based minimization for optimal \({\mathscr{H}}_{2}\) model reduction. In: Proceedings of 46th IEEE Conference on Decision and Control, pp 4385–4390. IEEE, Los Alamitos (2007)

  8. Beattie, C., Gugercin, S.: A trust region method for optimal \({\mathscr{H}}_{2}\) model reduction. In: Proceedings of the 48th IEEE Conference on Decision and Control, pp 5370–5375. IEEE, Los Alamitos (2009)

  9. Beattie, C., Gugercin, S.: Realization-independent \({\mathscr{H}}_{2}\)-approximation. In: Proceedings of 51st IEEE Conference on Decision and Control, pp. 4953–4958 (2012)

  10. Beattie, C., Gugercin, S., Wyatt, S.: Inexact solves in interpolatory model reduction. Linear Algebra Appl. 436, 2916–2943 (2012)

    Article  MathSciNet  Google Scholar 

  11. Benner, P., Breiten, T.: Interpolation-based \({\mathscr{H}}_{2}\)-model reduction of bilinear control systems. SIAM J. Matrix Anal. Appl. 33, 859–885 (2012)

    Article  MathSciNet  Google Scholar 

  12. Benner, P., Goyal, P., Gugercin, S: \({\mathscr{H}}_{2}\)-quasi-optimal model order reduction for quadratic-bilinear control systems. SIAM J. Matrix Anal. Appl. 39, 983–1032 (2018)

    Article  MathSciNet  Google Scholar 

  13. Benner, P., Köhler, M., Saak, J.: Sparse-dense Sylvester equations in \({\mathscr{H}}_{2}\)-model order reduction. Tech. Rep. MPIMD/11-11, Max Planck Institute Magdeburg Preprints (2011)

  14. Benner, P., Ohlberger, M., Cohen, A., Willcox, K. (eds.): Model Reduction and Approximation. Theory and Algorithms. SIAM, Philadelphia (2017)

  15. Breiten, T., Beattie, C., Gugercin, S.: Near-optimal frequency-weighted interpolatory model reduction. Syst. Control Lett. 78, 8–18 (2015)

    Article  MathSciNet  Google Scholar 

  16. Bunse-Gerstner, A., Kubalińska, D., Vossen, G., Wilczek, D: \({\mathscr{H}}_{2}\)-norm optimal model reduction for large scale discrete dynamical MIMO systems. J. Comput. Appl. Math. 233, 1202–1216 (2010)

    Article  MathSciNet  Google Scholar 

  17. Chahlaoui, Y., Van Dooren, P.: A collection of benchmark examples for model reduction of linear time invariant dynamical systems. Tech. rep., SLICOT Working Note, 2002–2 (2002)

  18. Chahlaoui, Y., Van Dooren, P.: Benchmark examples for model reduction of linear time-invariant dynamical systems. In: Benner, P., Sorensen, D.C., Mehrmann, V (eds.) Dimension Reduction of Large-Scale Systems, pp 379–392. Springer, Berlin (2005)

  19. Draijer, W., Steinbuch, M., Bosgra, O.: Adaptive control of the radial servo system of a compact disc player. Automatica 28, 455–462 (1992)

    Article  Google Scholar 

  20. Drmač, Z., Gugercin, S., Beattie, C.: Quadrature-based vector fitting for discretized \({\mathscr{H}}_{2}\) approximation. SIAM J. Sci. Comput. 37, A625–A652 (2015)

    Article  Google Scholar 

  21. Drmač, Z., Gugercin, S., Beattie, C.: Vector fitting for matrix-valued rational approximation. SIAM J. Sci. Comput. 37, A2346–A2379 (2015)

    Article  MathSciNet  Google Scholar 

  22. Eisenstat, S., Ipsen, I.: Three absolute perturbation bounds for matrix eigenvalues imply relative bounds. SIAM J. Matrix Anal. Appl. 20, 149–158 (1998)

    Article  MathSciNet  Google Scholar 

  23. Elsner, L., Friedland, S.: Singular values, doubly stochastic matrices, and applications. Linear Algebra Appl. 220, 161–169 (1995)

    Article  MathSciNet  Google Scholar 

  24. Ferrante, A., Krajewski, W., Lepschy, A., Viaro, U.: Convergent algorithm for l2 model reduction. Automatica 35, 75–79 (1999)

    Article  Google Scholar 

  25. Flagg, G., Beattie, C., Gugercin, S.: Convergence of the iterative rational Krylov algorithm. Syst. Control Lett. 61, 688–691 (2012)

    Article  MathSciNet  Google Scholar 

  26. Flagg, G., Gugercin, S.: Multipoint Volterra series interpolation and \({\mathscr{H}}_{2}\) optimal model reduction of bilinear systems. SIAM J. Matrix Anal. Appl. 36, 549–579 (2015)

    Article  MathSciNet  Google Scholar 

  27. Goyal, P., Redmann, M.: Time-limited \({\mathscr{H}}_{2}\)-optimal model order reduction. Appl. Math. Comput. 355, 184–197 (2019)

    MathSciNet  MATH  Google Scholar 

  28. Gugercin, S.: An iterative SVD-krylov based method for model reduction of large-scale dynamical systems. Linear Algebra Appl. 428, 1964–1986 (2008)

    Article  MathSciNet  Google Scholar 

  29. Gugercin, S., Antoulas, A.C., Beattie, C: \({\mathscr{H}}_{2}\) model reduction for large-scale linear dynamical systems. SIAM J. Matrix Anal. Appl. 30, 609–638 (2008)

    Article  MathSciNet  Google Scholar 

  30. Gugercin, S., Antoulas, A.C., Bedrossian, M.: Approximation of the international space station 1R and 12A models. In: Proceedings of the 40th IEEE Conference on Decision and Control, pp 1515–1516. IEEE, Los Alamitos (2001)

  31. He, C., Laub, A., Mehrmann, V.: Placing Plenty of Poles is Pretty Preposterous. Tech. Rep., Preprint SPC 95-17 . Forschergruppe Scientific Parallel Computing, Fakultt für Mathematik, TU Chemnitz-Zwickau (1995)

  32. Hokanson, J.M., Magruder, C.C.: \({\mathscr{H}}_{2}\)-optimal model reduction using projected nonlinear least squares. arXiv:1811.11962 (2018)

  33. Hyland, D., Bernstein, D.: The optimal projection equations for model reduction and the relationships among the methods of Wilson, Skelton, and Moore. IEEE Trans. Autom. Control 30, 1201–1211 (1985)

    Article  MathSciNet  Google Scholar 

  34. Krajewski, W., Lepschy, A., Redivo-Zaglia, M., Viaro, U.: A program for solving the l2 reduced-order model problem with fixed denominator degree. Numer. Algor. 9, 355–377 (1995)

    Article  Google Scholar 

  35. Krajewski, W., Viaro, U.: Iterative-interpolation algorithms for l2 model reduction. Control Cybern. 38, 543–554 (2009)

    MATH  Google Scholar 

  36. Mehrmann, V., Xu, H.: An analysis of the pole placement problem I. The single-input case. Electron. Trans. Numer. Anal. 4, 89–105 (1996)

    MathSciNet  MATH  Google Scholar 

  37. Mehrmann, V., Xu, H.: Choosing poles so that the single-input pole placement problem is well conditioned. SIAM J. Matrix Anal. Appl. 19, 664–681 (1998)

    Article  MathSciNet  Google Scholar 

  38. Meier, L III, Luenberger, D.: Approximation of linear constant systems. IEEE Trans. Autom. Control 12, 585–588 (1967)

    Article  Google Scholar 

  39. Panzer, H.K.F., Jaensch, S., Wolf, T., Lohmann, B.: A greedy rational Krylov method for \({\mathscr{H}}_{2}\)-pseudooptimal model order reduction with preservation of stability. In: 2013 American Control Conference, pp. 5512–5517 (2013)

  40. Poussot-Vassal, C.: An iterative SVD-tangential interpolation method for medium-scale MIMO systems approximation with application on flexible aircraft. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference, pp. 7117–7122 (2011)

  41. Spanos, J.T., Milman, M.H., Mingori, D.L.: A new algorithm for l2 optimal model reduction. Automatica 28, 897–909 (1992)

    Article  Google Scholar 

  42. Van Dooren, P., Gallivan, K.A., Absil, P.A.: \({\mathscr{H}}_{2}\)-optimal model reduction of MIMO systems. Appl. Math. Lett. 21, 1267–1273 (2008)

    Article  MathSciNet  Google Scholar 

  43. Vuillemin, P., Poussot-Vassal, C., Alazard, D.: \({\mathscr{H}}_{2}\) optimal and frequency limited approximation methods for large-scale LTI dynamical systems. IFAC Proceedings Volumes 46, 719–724 (2013)

    Article  Google Scholar 

  44. žigić, D., Watson, L.T., Beattie, C.: Contragredient transformations applied to the optimal projection equations. Linear Algebra Appl. 188–189, 665–676 (1993)

    Article  MathSciNet  Google Scholar 

  45. Wilson, D.A.: Optimum solution of model-reduction problem. Proc. Inst. Electr. Eng. 117, 1161–1165 (1970)

    Article  Google Scholar 

  46. Xu, Y., Zeng, T.: Optimal \({\mathscr{H}}_{2}\) model reduction for large scale MIMO systems via tangential interpolation. Int. J. Numer. Anal. Model. 8, 174–188 (2011)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The work of Beattie was supported in parts by NSF through Grant DMS-1819110. The work of Drmač was supported in parts by the Croatian Science Foundation Grant IP-2019-04-6268 and the DARPA Contracts HR0011-16-C-0116 and HR0011-18-9-0033. The work of Gugercin was supported in parts by NSF through Grant DMS-1720257 and DMS-1819110.

Author information

Authors and Affiliations

  1. Department of Mathematics and Division of Computational Modeling and Data Analytics, Virginia Tech, Blacksburg, VA, 24061, USA

    Christopher Beattie & Serkan Gugercin

  2. Faculty of Science, Department of Mathematics, University of Zagreb, Zagreb, 10000, Croatia

    Zlatko Drmač

Authors
  1. Christopher Beattie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zlatko Drmač
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Serkan Gugercin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Serkan Gugercin.

Additional information

Dedicated to Volker Mehrmann on the occasion of his 65th birthday.

Publisher’s Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Beattie, C., Drmač, Z. & Gugercin, S. Revisiting IRKA: Connections with Pole Placement and Backward Stability. Vietnam J. Math. 48, 963–985 (2020). https://doi.org/10.1007/s10013-020-00424-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10013-020-00424-0

Keywords

Mathematics Subject Classification (2010)

Search

Navigation