A structured pseudospectral method for \(\mathcal {H}_\infty \)-norm computation of large-scale descriptor systems

Abstract

In this paper, we discuss the problem of computing the \({\mathcal {H}}_\infty \)-norm of transfer functions associated to large-scale descriptor systems. We exploit the relationship between the \({\mathcal {H}}_\infty \)-norm and the structured complex stability radius of a corresponding matrix pencil. To compute the structured stability radius we consider so-called structured pseudospectra. Namely, we have to find the pseudospectrum touching the imaginary axis. Therefore, we set up an iteration over the real part of the rightmost pseudoeigenvalue. For that, we use a new fast iterative scheme which is based on certain rank-1 perturbations of a matrix pencil. Finally, we analyze the performance of our algorithm by using real-world examples. In particular we compare our method with different other algorithms including a recently and independently derived method from Guglielmi, Gürbüzbalaban and Overton.

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Notes

  1. 1.

    http://www.mpi-magdeburg.mpg.de/mpcsc/software/infnorm/.

  2. 2.

    http://sites.google.com/site/rommes/software.

  3. 3.

    http://www.slicot.org/index.php?site=benchmodred.

  4. 4.

    http://www.slicot.org/.

  5. 5.

    http://cims.nyu.edu/~mert/software/hinfinity.html.

References

  1. 1.

    Benner P, Byers R, Mehrmann V, Xu H (2002) Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils. SIAM J Matrix Anal Appl 24:165–190

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Benner P, Saak J, Schieweck F, Skrzypacz P, Weichelt HK (2012) A non-conforming composite quadrilateral finite element pair for feedback stabilization of the Stokes equations. J Numer Math. Accepted, also available as Max Planck Institute Magdeburg, Preprint MPIMD/12-19

  3. 3.

    Benner P, Sima V, Voigt M (2012) \({\cal L}_\infty \)-norm computation for continuous-time descriptor systems using structured matrix pencils. IEEE Trans Automat Control 57:233–238

    Google Scholar 

  4. 4.

    Benner P, Sima V, Voigt M (2012) Robust and efficient algorithms for \({\cal L}_\infty \)-norm computation for descriptor systems. In: Proceedings of the 7th IFAC Symposium on Robust Control Design, Aalborg, Denmark, IFAC, pp 195–200

  5. 5.

    Benner P, Voigt M (2012) \({\cal {H}}_\infty \)-norm computation for large and sparse descriptor systems. Proc Appl Math Mech 12:797–800

    Article  Google Scholar 

  6. 6.

    Benner P, Voigt M (2012) Numerical computation of structured complex stability radii od large-scale matrices and pencils. In: Proceedings of the 51st IEEE Conference on Decision and Control. Maui, Hawaii, pp 6560–6565

  7. 7.

    Boyd S, Balakrishnan V (1990) A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its \(L_{\infty }\)-norm. Syst Control Lett 15:1–7

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Boyd S, Balakrishnan V, Kabamba P (1989) A bisection method for computing the \(H_\infty \) norm of a transfer matrix and related problems. Math Control Signals Syst 2:207–219

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Bruinsma NA, Steinbuch M (1990) A fast algorithm to compute the \(H_{\infty }\)-norm of a transfer function matrix. Syst Control Lett 14:287–293

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Burke JV, Lewis AS, Overton ML (2003) Robust stability and a criss-cross algorithm for pseudospectra. IMA J Numer Anal 23:359–375

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Byers R (1988) A bisection method for measuring the distance of a stable matrix to the unstable matrices. SIAM J Sci Stat Comput 9:875–881

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Byers R, Nichols NK (1993) On the stability radius of generalized state-space systems. Linear Algebra Appl 188–189:113–134

    Google Scholar 

  13. 13.

    Chahlaoui Y, Gallivan K, Van Dooren P (2004) \({\cal H}_\infty \)-norm calculations of large sparse systems. In: Proceedings of the International Symposium of Mathematical Theory of Networks and Systems. Leuven, Belgium

  14. 14.

    Chahlaoui Y, Gallivan K, Van Dooren P (Oct. 2007) Calculating the \({\cal H}_\infty \) norm of a large sparse system via Chandrasekhar iterations and extrapolation. In: ESAIM Proceedings, vol 20. Rabat, Algeria

  15. 15.

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

  16. 16.

    Dai L (1989) Singular control systems, vol. 118 of Lecture Notes in Control and Inform. Sci. Springer, Heidelberg

  17. 17.

    Du NH (2008) Stability radii of differential algebraic equations with structured perturbations. Syst Control Lett 57:546–553

    Article  MATH  Google Scholar 

  18. 18.

    Du NH, Linh VH, Mehrmann V (2013) Robust stability of differential-algebraic equations. In Surveys in Differential-Algebraic Equations I. In: Ilchmann A, Reis T (eds) Differential-Algebraic Equations Forum. Springer, Berlin, Heidelberg, ch. 2, pp 63–95

  19. 19.

    Du NH, Thuan DD, Liem NC (2011) Stability radius of implicit dynamic equations with constant coefficients on time scales. Syst Control Lett 60:596–603

    Article  MATH  MathSciNet  Google Scholar 

  20. 20.

    Eich-Soellner E, Führer C (1998) Numerical methods in multibody dynamics. B. G. Teubner, Stuttgart

    Google Scholar 

  21. 21.

    Freitas F, Rommes J, Martins N (2008) Gramian-based reduction method applied to large sparse power system descriptor models. IEEE Trans Power Syst 23:1258–1270

    Article  Google Scholar 

  22. 22.

    Guglielmi N, Gürbüzbalaban M, Overton ML (2013) Fast approximation of the \(H_\infty \) norm via optimization of spectral value sets. SIAM J Matrix Anal Appl 34:709–737

    Article  MATH  MathSciNet  Google Scholar 

  23. 23.

    Guglielmi N, Kressner D, Lubich C (2013) Low-rank differential equations for Hamiltonian matrix nearness problems, Oberwolfach Preprint OWP 2013–01, Mathematisches Forschungsinstitut Oberwolfach, Jan. 2013. Available from http://www.mfo.de/scientific-programme/publications/owp/2013/OWP2013_01.pdf

  24. 24.

    Guglielmi N, Lubich C (2013) Low-rank dynamics for computing extremal points of real pseudospectra. SIAM J Matrix Anal Appl (in press)

  25. 25.

    Guglielmi N, Overton ML Local convergence analysis of [22]- private communication with M. L. Overton

  26. 26.

    Guglielmi N, Overton ML (2011) Fast algorithms for the approximation of the pseudospectral abscissa and pseudospectral radius of a matrix. SIAM J Matrix Anal Appl 32:1166–1192

    Article  MATH  MathSciNet  Google Scholar 

  27. 27.

    Heinkenschloss M, Sorensen DC, Sun K (2008) Balanced truncation model reduction for a class of descriptor systems with application to the Oseen equations. SIAM J Sci Comput 30:1038–1063

    Article  MATH  MathSciNet  Google Scholar 

  28. 28.

    Hinrichsen D, Pritchard AJ (1986) Stability radii of linear systems. Syst Control Lett 7:1–10

    Google Scholar 

  29. 29.

    Hinrichsen D, Pritchard AJ (1986) Stability radius for structured perturbations and the algebraic Riccati equation. Syst Control Lett 8:105–113

    Article  MATH  MathSciNet  Google Scholar 

  30. 30.

    Hinrichsen D, Pritchard AJ (1990) Real and complex stability radii: a survey. In: Progress in systems and Control Theory, vol 6. Birkhäuser, Boston, pp 119–162

  31. 31.

    Kunkel P, Mehrmann V (2006) Differential-Algebraic Equations—Analysis and Numerical Solution. In: Textbooks in Mathematics. European Mathematical Society, Zürich

  32. 32.

    Leibfritz F (2004) Compl\(_{e}\)ib: Constraint matrix-optimization problem library—a collection of test examples for nonlinear semidefinite programs, control system design and related problems. Technical report, 2004. Available from http://www.friedemann-leibfritz.de/COMPlib_Data/COMPlib_Main_Paper.pdf

  33. 33.

    Leibfritz F, Lipinski W (2004) Compl\(_{e}\)ib 1.0 - user manual and quick reference. Technical report, 2004. Available from http://www.friedemann-leibfritz.de/COMPlib_Data/COMPlib_User_Guide.pdf

  34. 34.

    Martins N, Pellanda PC, Rommes J (2007) Computation of transfer function dominant zeros with applications to oscillation damping control of large power systems. IEEE Trans Power Syst 22:1657–1664

    Article  Google Scholar 

  35. 35.

    Mehrmann V, Schröder C, Simoncini V (2012) An implicitly-restarted Krylov subspace method for real symmetric/skew-symmetric eigenproblems. Linear Algebra Appl 436:4070–4087

    Article  MATH  MathSciNet  Google Scholar 

  36. 36.

    Mehrmann V, Stykel T (2005) Balanced truncation model reduction for large-scale systems in descriptor form. In: Benner P, Mehrmann V, Sorensen D (eds) Dimension Reduction of Large-Scale Systems, vol 45 of Lecture Notes Comput. Sci. Eng. Springer, Berlin, Heidelberg, New York, ch. 3, pp 89–116

  37. 37.

    Overton ML, Van Dooren P (2005) On computing the complex passivity radius. In: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, pp 7960–7964

  38. 38.

    Reis T (2010) Circuit synthesis of passive descriptor systems—a modified nodal approach. Int J Circ Theor Appl 38:44–68

    Article  MATH  Google Scholar 

  39. 39.

    Reis T, Stykel T (2010) PABTEC: Passivity-preserving balanced truncation for electrical circuits. IEEE Trans Computer-Aided Design Integr Circuits Syst 29:1354–1367

    Google Scholar 

  40. 40.

    Reis T, Stykel T (2011) Lyapunov balancing for passivity-preserving model reduction of RC circuits. SIAM J Appl Dyn Syst 10:1–34

    Article  MATH  MathSciNet  Google Scholar 

  41. 41.

    Riedel KS (1994) Generalized epsilon-pseudospectra. SIAM J Numer Anal 31:1219–1225

    Article  MATH  MathSciNet  Google Scholar 

  42. 42.

    Rommes J (2008) Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems \(Ax=\lambda Bx\) with singular \(B\). Math Comp 77:995–1015

    Article  MATH  MathSciNet  Google Scholar 

  43. 43.

    Rommes J, Martins N (2006) Efficient computation of multivariate transfer function dominant poles using subspace acceleration. IEEE Trans Power Syst 21:1471–1483

    Article  Google Scholar 

  44. 44.

    Rommes J, Martins N (2006) Efficient computation of transfer function dominant poles using subspace acceleration. IEEE Trans Power Syst 21:1218–1226

    Article  Google Scholar 

  45. 45.

    Rommes J, Sleijpen GLG (2008) Convergence of the dominant pole algorithm and Rayleigh quotient iteration. SIAM J Matrix Anal Appl 30:346–363

    Article  MATH  MathSciNet  Google Scholar 

  46. 46.

    Stewart GW, Sun J-G (1990) Matrix Perturbation Theory. In: Computer Science and Scientific Computing. Academic Press, London

  47. 47.

    Weickert J (1997) Applications of the Theory of Differential-Algebraic Equations to Partial Differential Equations of Fluid Dynamics. PhD thesis, Chemnitz University of Technology, Department of Mathematics, Germany

  48. 48.

    Wright TG (2002) Eigtool, 2002. Available from http://www.comlab.ox.ac.uk/pseudospectra/eigtool/

  49. 49.

    Zhou K, Doyle JD (1998) Essentials of Robust Control, 1st edn. Prentice Hall, Upper Saddle River

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Acknowledgments

We thank Volker Mehrmann from TU Berlin for pointing at the difficulties arising if the perturbations make the transfer functions improper or not well-defined. We gratefully acknowledge the work done by our former intern Maximilian Bremer from the University of Texas at Austin by implementing a MATLAB code for the visualization of structured pseudospectra. Furthermore, we thank the two anonymous reviewers for their valuable and constructive comments that helped to significantly improve the quality of the paper.

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Correspondence to Matthias Voigt.

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Benner, P., Voigt, M. A structured pseudospectral method for \(\mathcal {H}_\infty \)-norm computation of large-scale descriptor systems. Math. Control Signals Syst. 26, 303–338 (2014). https://doi.org/10.1007/s00498-013-0121-7

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Keywords

  • Descriptor systems
  • \({\mathcal {H}}_\infty \) control
  • Iterative methods
  • Pseudospectra
  • Sparse matrices
  • Stability of linear systems