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Model reduction of time-delay systems using position balancing and delay Lyapunov equations

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Abstract

Balanced truncation is a standard and very natural approach to approximate dynamical systems. We present a version of balanced truncation for model order reduction of linear time-delay systems. The procedure is based on a coordinate transformation of the position and preserves the delay structure of the system. We therefore call it (structure-preserving) position balancing. To every position, we associate quantities representing energies for the controllability and observability of the position. We show that these energies can be expressed explicitly in terms of the solutions to corresponding delay Lyapunov equations. Apart from characterizing the energies, we show that one block of the (operator) controllability and observability Gramians in the operator formulation of the time-delay system can also be characterized with the delay Lyapunov equation. The delay Lyapunov equation undergoes a contragredient transformation when we apply the position coordinate transformation and we propose to truncate it in a classical fashion, such that positions which are only weakly connected to the input and the output in the sense of the energy concepts are removed.

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References

  1. Antoulas A (2005) Approximation dynamical systems. In: Society for industrial and applied mathematics (SIAM). Philadelphia, PA

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

    Article  MathSciNet  Google Scholar 

  3. Bartels R, Stewart GW (1972) Solution of the matrix equation \(AX+XB=C\). Comm ACM 15(9):820–826

    Article  Google Scholar 

  4. Beattie C, Gugercin S (2009) Interpolatory projection methods for structure-preserving model reduction. Syst Control Lett 58(3):225–232

    Article  MathSciNet  MATH  Google Scholar 

  5. Benner P, Mehrmann V, Sorensen D (eds) (2005) Dimension reduction of large-scale systems. Springer, Berlin

    MATH  Google Scholar 

  6. Benner P, Saak J (2011) Efficient balancing based MOR for large scale second order systems. Math Comput Model Dyn Syst 17(2):123–143

    Article  MathSciNet  MATH  Google Scholar 

  7. Chahlaoui Y, Lemonnier D, Vandendorpe A, Dooren PV (2006) Second-order balanced truncation. Linear Algebra Appl 415:373–384

    Article  MathSciNet  MATH  Google Scholar 

  8. Curtain RF, Zwart H (1995) An introduction to infinite-dimensional linear systems theory. Springer, New York

    Book  MATH  Google Scholar 

  9. Hale J, Verduyn Lunel SM (1993) Introduction to functional differential equations. Springer, Berlin

    MATH  Google Scholar 

  10. Harkort C, Deutscher J (2011) Krylov subspace methods for linear infinite-dimensional systems. IEEE Trans Autom Control 56(2):441–447

    Article  MathSciNet  Google Scholar 

  11. Huesca E, Mondié S, Santos J (2009) Polynomial approximations of the Lyapunov matrix of a class of time delay systems. In: Proceedings of the 8th IFAC workshop on time-delay systems, Sinaia, Romania

  12. Jarlebring E, Vanbiervliet J, Michiels W (2011) Characterizing and computing the \({\cal H}_{2}\) norm of time-delay systems by solving the delay Lyapunov equation. IEEE Trans Autom Control 56(4):814–825

    Google Scholar 

  13. Kharitonov V (2004) Lyapunov-Krasovskii functionals for scalar time delay equations. Syst Control Lett 51(2):133–149

    Article  MathSciNet  MATH  Google Scholar 

  14. Kharitonov V (2005) Lyapunov functionals and Lyapunov matrices for neutral type time delay systems: a single delay case. Int J Control 78(11):783–800

    Article  MathSciNet  MATH  Google Scholar 

  15. Kharitonov V (2006) Lyapunov matrices for a class of time delay systems. Syst Control Lett 55(7):610–617

    Article  MathSciNet  MATH  Google Scholar 

  16. Kharitonov V, Hinrichsen D (2004) Exponential estimates for time delay systems. Syst Control Lett 53(5):395–405

    Article  MathSciNet  MATH  Google Scholar 

  17. Kharitonov V, Plischke E (2006) Lyapunov matrices for time-delay systems. Syst Control Lett 55(9):697–706

    Article  MathSciNet  MATH  Google Scholar 

  18. Kharitonov V, Zhabko AP (2003) Lyapunov-Krasovskii approach to the robust stability analysis of time-delay systems. Automatica 39(1):15–20

    Article  MathSciNet  MATH  Google Scholar 

  19. Mäkilä P, Partington J (1999) Laguerre and Kautz shift approximations of delay systems. Int J Control 72(10):932–946

    Article  MATH  Google Scholar 

  20. Mäkilä P, Partington J (1999) Shift operator induced approximations of delay systems. SIAM J Control Optim 37(6):1897–1912

    Article  MathSciNet  MATH  Google Scholar 

  21. Mastinšek M (1994) Adjoints of solution semigroups and identifiability of delay differential equations in Hilbert spaces. Acta Math Univ Comen LXIII 2:193–206

    Google Scholar 

  22. Meyer DG, Srinivasan S (1996) Balancing and model reduction for second-order form linear systems. IEEE Trans Autom Control 41(11):1632–1644

    Article  MathSciNet  MATH  Google Scholar 

  23. Michiels W (2002) Stability and stabilization of time-delay systems. Ph.D. thesis, Katholieke universiteit Leuven

  24. Michiels W, Jarlebring E, Meerbergen K (2011) Krylov-based model order reduction of time-delay systems. SIAM J Matrix Anal Appl 32(4):1399–1421

    Article  MathSciNet  MATH  Google Scholar 

  25. Niculescu SI (2001) Delay effects on stability. A robust control approach. Springer, London

    MATH  Google Scholar 

  26. Ochoa G, Kharitonov V (2005) Lyapunov matrices for neutral type time delay systems. In: Proceedings of the 2nd International Conference on Electrical and Electronics Engineering, Mexico City, Mexico

  27. Ochoa G, Velázquez-Velázquez J, Kharitonov V, Mondié S (2007) Lyapunov matrices for neutral type time delay systems. In: Proceedings of the 7th IFAC workshop on time delay systems, Nantes, France

  28. Ouellette DV (1981) Schur complements and statistics. Linear Algebra Appl 36:187–295

    Article  MathSciNet  MATH  Google Scholar 

  29. Partington J (2004) Model reduction of delay systems. In: Blondel V, Megretski A (eds) Unsolved problems in mathematical systems and control theory. Princeton university press, Princeton, pp 29–32

    Google Scholar 

  30. Plischke E (2005) Transient effects of linear dynamical systems. Ph.D. thesis, Universität Bremen

  31. Reis T, Stykel T (2008) Balanced truncation model reduction of second-order systems. Math Comput Model Dyn Syst 14(5):391–406

    Article  MathSciNet  MATH  Google Scholar 

  32. Saadvandi M, Meerbergen K, Jarlebring E (2012) On dominant poles and model reduction of second order time-delay systems. Appl Numer Math 62(1):21–34

    Article  MathSciNet  MATH  Google Scholar 

  33. Vanbiervliet J, Michiels W, Jarlebring E (2011) Using spectral discretisation for the optimal \({\cal H}_{2}\) design of time-delay systems. Int J Control 84(2):228–241

    Google Scholar 

  34. Velázquez-Velázquez J, Kharitonov V (2009) Lyapunov-Krasovskii functionals for scalar neutral type time delay equation. Syst Control Lett 58(1):17–25

    Google Scholar 

  35. Yan B, Tan S, McGaughy B (2008) Second-order balanced truncation for passive order reduction of RLCK circuits. IEEE Trans Circuits Syst II Analog Digit Signal Process 55:942–946

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Acknowledgments

This work has been supported by the Programme of Interuniversity Attraction Poles of the Belgian Federal Science Policy Office (IAP P6-DYSCO), by OPTEC, the Optimization in Engineering Center of the K.U. Leuven, and by the project STRT1-09/33 of the K.U. Leuven Research Council. The first author is supported by the Dahlquist research fellowship.

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Authors and Affiliations

  1. Department of Mathematics, KTH, 100 44 , Stockholm, Sweden

    Elias Jarlebring

  2. Technische Universität Kaiserslautern, Fachbereich Mathematik, Erwin Schrödinger Straße, 67663 , Kaiserslautern, Germany

    Tobias Damm

  3. Department of Computer Science, KU Leuven, 3001 , Heverlee, Belgium

    Wim Michiels

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  1. Elias Jarlebring
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  2. Tobias Damm
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  3. Wim Michiels
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Correspondence to Elias Jarlebring.

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Jarlebring, E., Damm, T. & Michiels, W. Model reduction of time-delay systems using position balancing and delay Lyapunov equations. Math. Control Signals Syst. 25, 147–166 (2013). https://doi.org/10.1007/s00498-012-0096-9

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  • DOI: https://doi.org/10.1007/s00498-012-0096-9

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