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access icon free Gramian-based model-order reduction of constrained structural dynamic systems

This study discusses model reduction techniques for second-order index 3 descriptor systems using the balanced truncation methods; in particular, linearised equations of motion with holonomic constraints are considered which arise in mechanics and multibody dynamics. It is shown that the index 3 system can be converted into an equivalent form of index 0 system by projecting it onto the hidden manifold. When model reduction is applied to the projected system, explicit formulation of the projected system is not required. The low-rank alternating direction implicit iteration is also discussed for solving the projected Lyapunov equations of the underlying descriptor system efficiently in an implicit way. The theoretical results are illustrated by numerical experiments.

References

    1. 1)
      • 19. Penzl, T.: ‘A cyclic low rank Smith method for large sparse Lyapunov equations’, SIAM J. Sci. Comput., 2000, 21, pp. 14011418.
    2. 2)
      • 3. Mehrmann, V., Stykel, T.: ‘Balanced truncation model reduction for large-scale systems in descriptor form’, in Benner, P., Mehrmann, V., Sorensen, D.C. (ED.): ‘Dimension reduction of large-scale systems’ (Springer-Verlag, Berlin/Heidelberg, Germany, 2005), pp. 83115.
    3. 3)
      • 36. Uddin, M.M., Saak, J., Kranz, B., et al: ‘Computation of a compact state space model for an adaptive spindle head configuration with piezo actuators using balanced truncation’, Prod. Eng., 2012, 6, pp. 577586.
    4. 4)
      • 24. Wachspress, E.L.: ‘The ADI model problem’ (Springer, New York, 2013).
    5. 5)
      • 33. Tisseur, F., Meerbergen, K.: ‘The quadratic eigenvalue problem’, SIAM Rev., 2001, 43, pp. 235286.
    6. 6)
      • 20. Glover, K.: ‘All optimal Hankel-norm approximations of linear multivariable systems and their L norms’, Int. J. Control, 1994, 39, pp. 11151193.
    7. 7)
      • 27. Wachspress, E.L.: ‘Iterative solution of the Lyapunov matrix equation’, Appl. Math. Lett., 1988, 107, pp. 8790.
    8. 8)
      • 11. Stykel, T.: ‘Gramian-based model reduction for descriptor systems’, Math. Control Signals Syst., 2004, 16, pp. 297319.
    9. 9)
      • 25. Benner, P., Kürschner, P., Saak, J.: ‘Efficient handling of complex shift parameters in the low-rank Cholesky factor ADI method’, Numer. Algorithms, 2013, 62, pp. 225251.
    10. 10)
      • 5. Riaza, R.: ‘Differential-algebraic systems. analytical aspects and circuit applications’ (World Scientific Publishing Co. Pte. Ltd., Singapore, 2008).
    11. 11)
      • 35. Uddin, M.M.: ‘Model reduction of piezo-echanical systems using balanced truncation’, Master's thesis, Stockholm University, Stockholm, Sweden, 2011. Available from: http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-78227.
    12. 12)
      • 23. Benner, P., Kürschner, P., Saak, J.: ‘An improved numerical method for balanced truncation for symmetric second order systems’, Math. Comput. Model. Dyn. Syst., 2013, 19, pp. 593615.
    13. 13)
      • 22. Benner, P., Saak, J.: ‘Efficient balancing-based MOR for large-scale second-order systems’, Math. Comput. Model. Dyn. Syst., 2011, 17, pp. 123143.
    14. 14)
      • 29. Benner, P., Mena, H., Saak, J.: ‘On the parameter selection problem in the Newton-ADI iteration for large-scale Riccati equations’, Electron. Trans. Numer. Anal., 2008, 29, pp. 136149.
    15. 15)
      • 32. Saad, Y.: ‘Numerical methods for large eigenvalue problems’ (Manchester University Press, Manchester, UK, 1992).
    16. 16)
      • 6. Craig, R.R.: ‘Structural dynamics: an introduction to computer methods’ (John Wiley & Sons Inc., New York, 1981).
    17. 17)
      • 9. Tombs, M.S., Postlethwaite, I.: ‘Truncated balanced realization of a stable non-minimal state-space system’, Int. J. Control, 1987, 46, pp. 13191330.
    18. 18)
      • 1. Wang, D., Harris, M.N.: ‘Stability analysis of the equilibrium of a constrained mechanical system’, Int. J. Control, 1994, 60, pp. 733746.
    19. 19)
      • 8. Moore, B.C.: ‘Principal component analysis in linear systems: controllability, observability, and model reduction’, IEEE Trans. Autom. Control, 1981, 26, pp. 1732.
    20. 20)
      • 12. Gugercin, S., Stykel, T., Wyatt, S.: ‘Model reduction of descriptor systems by interpolatory projection methods’, SIAM J. Sci. Comput., 2013, 35, pp. B1010B1033.
    21. 21)
      • 10. Gugercin, S., Antoulas, A., Battie, C.: ‘H2 model reduction for large-scale dynamical systems’, SIAM J. Matrix Anal. Appl., 2008, 30, pp. 609638.
    22. 22)
      • 13. Heinkenschloss, M., Sorensen, D.C., Sun, K.: ‘Balanced truncation model reduction for a class of descriptor systems with application to the Oseen equations’, SIAM J. Sci. Comput., 2008, 30, pp. 10381063.
    23. 23)
      • 16. Benner, P., Li, J.-R., Penzl, T.: ‘Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems’, Numer. Lin. Alg. Appl., 2008, 15, pp. 755777.
    24. 24)
      • 2. Eich-Soellner, E., Führer, C.: ‘Numerical methods in multibody dynamics’ (European Consortium for Mathematics in Industry, Stuttgart, 1998).
    25. 25)
      • 14. Ahmad, M.I., Benner, P.: ‘Interpolatory model reduction techniques for linear second-order descriptor systems’. Proc. in European Control Conf. (ECC), Strasbourg, France, June 2014, pp. 10751079.
    26. 26)
      • 31. Golub, G.H., Van Loan, C.F.: ‘Matrix computations’ (Johns Hopkins University Press, Baltimore, 1984, 3rd edn.1996).
    27. 27)
      • 38. Cliffe, K.A., Garratt, T.J., Spence, A.: ‘Eigenvalues of block matrices arising from problems in fluid mechanics’, SIAM J. Matrix Anal. Appl., 1994, 15, pp. 13101318.
    28. 28)
      • 18. Hammarling, S.J.: ‘Numerical solution of the stable, non-negative definite Lyapunov equation’, IMA J. Numer. Anal., 1982, 2, pp. 303323.
    29. 29)
      • 34. Bai, Z., Su, Y.: ‘SOAR: a second-order Arnoldi method for the solution of the quadratic eigenvalue problem’, SIAM J. Matrix Anal. Appl., 2005, 26, pp. 640659.
    30. 30)
      • 21. Reis, T., Stykel, T.: ‘Balanced truncation model reduction of second-order systems’, Math. Comput. Model. Dyn. Syst., 2008, 14, pp. 391406.
    31. 31)
      • 17. Stykel, T.: ‘Analysis and numerical solution of generalized lyapunov equations’, PhD thesis, Technische Universität Berlin, Berlin, 2002.
    32. 32)
      • 41. Truhar, N., Veselić, K.: ‘Bounds on the trace of a solution to the Lyapunov equation with a general stable matrix’, Syst. Cont. Lett., 2007, 56, pp. 493503.
    33. 33)
      • 40. Truhar, N., Veselić, K.: ‘An efficient method for estimating the optimal dampers’ viscosity for linear vibrating systems using Lyapunov equation', SIAM J. Matrix Anal. Appl., 2009, 31, pp. 1839.
    34. 34)
      • 37. Freitas, F., Rommes, J., Martins, N.: ‘Gramian-based reduction method applied to large sparse power system descriptor models’, IEEE Trans. Power Syst., 2008, 23, pp. 12581270.
    35. 35)
      • 28. Li, J.-R.: ‘Model reduction of large linear systems via low rank system gramians’. PhD thesis, Massachusettes Institute of Technology.
    36. 36)
      • 7. Antoulas, A.: ‘Approximation of large-scale dynamical systems’ (SIAM, Philadelphia, PA, 2005).
    37. 37)
      • 4. Brenan, K.E., Campbell, S.L., Petzold, L.R.: ‘Numerical solution of initial–value problems in differential–algebraic equations’ (Elsevier Science Publishing, North-Holland, 1989).
    38. 38)
      • 30. Saak, J.: ‘Efficient numerical solution of large scale algebraic matrix equations in PDE control and model order reduction’. PhD thesis, TU Chemnitz, July 2009.
    39. 39)
      • 15. Li, J.-R., White, J.: ‘Low rank solution of Lyapunov equations’, SIAM J. Matrix Anal. Appl., 2002, 24, pp. 260280.
    40. 40)
      • 26. Uddin, M.M.: ‘Computational methods for model reduction of large-Scale sparse structured descriptor systems’. PhD thesis, Otto-von-Guericke-Universität, Magdeburg, Germany, 2015. Available from: http://pubman.mpdl.mpg.de/pubman/item/escidoc:2164133/component/escidoc:2227420/2164133.pdf.
    41. 41)
      • 39. Benner, P., Saak, J., Uddin, M.M.: ‘Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control’, Numer. Algebra Control Optim., 2016, 6, pp. 120.
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