Damping optimization of parameter dependent mechanical systems by rational interpolation


We consider an optimization problem related to semi-active damping of vibrating systems. The main problem is to determine the best damping matrix able to minimize influence of the input on the output of the system. We use a minimization criteria based on the \(\mathcal {H}_{2}\) system norm. The objective function is non-convex and the associated optimization problem typically requires a large number of objective function evaluations. We propose an optimization approach that calculates ‘interpolatory’ reduced order models, allowing for significant acceleration of the optimization process. In our approach, we use parametric model reduction (PMOR) based on the Iterative Rational Krylov Algorithm, which ensures good approximations relative to the \(\mathcal {H}_{2}\) system norm, aligning well with the underlying damping design objectives. For the parameter sampling that occurs within each PMOR cycle, we consider approaches with predetermined sampling and approaches using adaptive sampling, and each of these approaches may be combined with three possible strategies for internal reduction. In order to preserve important system properties, we maintain second-order structure, which through the use of modal coordinates, allows for very efficient implementation. The methodology proposed here provides a significant acceleration of the optimization process; the gain in efficiency is illustrated in numerical experiments.

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  1. 1.

    Afanasiev, K., Hinze, M.: Adaptive control of a wake flow using proper orthogonal decomposition. Shape Optimization and Optimal Design, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 216, 317–332 (2001)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Alexandrov, N., Dennis, J., Lewis, R., Torczon, V.: A trust-region framework for managing the use of approximation models in optimization. Struct. Multidiscip. Optim. 15(1), 16–23 (1998)

    Article  Google Scholar 

  3. 3.

    Antoulas, A.: Approximation of Large-Scale dynamical systems. SIAM publications, Philadelphia (2005)

    Google Scholar 

  4. 4.

    Antoulas, A., Beattie, C., Gugercin, S.: Interpolatory model reduction of large-scale dynamical systems. In: Efficient modeling and control of large-scale systems, pp. 3–58. Springer (2010)

  5. 5.

    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  Article  Google Scholar 

  6. 6.

    Arian, E., Fahl, M., Sachs, E.: Trust-region proper orthogonal decomposition for flow control. ICASE Technical Report (2000)

  7. 7.

    Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H.: Templates for the solution of algebraic eigenvalue problems: a practical guide. SIAM, Philadelphia (2000)

    Google Scholar 

  8. 8.

    Bai, Z., Su, Y.: Dimension reduction of second order dynamical systems via a second-order Arnoldi method. SIAM J. Sci. Comput. 5, 1692–1709 (2005)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Bashir, O., Willcox, K., Ghattas, O., van Bloemen Waanders, B., Hill, H.: Hessian-based model reduction for large-scale systems with initial condition inputs. Internat. J. Numer. Methods Engrg. 73, 844–868 (2008)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Baur, U., Beattie, C.A., Benner, P., Gugercin, S.: Interpolatory projection methods for parameterized model reduction. SIAM J. Sci. Comput. 33(5), 2489–2518 (2011)

    MathSciNet  Article  Google Scholar 

  11. 11.

    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 

  12. 12.

    Beattie, C., Gugercin, S.: Krylov-based model reduction of second-order systems with proportional damping. In: Proceedings of the 44Th IEEE conference on decision and control, pp. 2278–2283 (2005)

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 14.

    Beattie, C., Gugercin, S.: Model reduction by rational interpolation. In: Benner, P., Cohen, A., Ohlberger, M., Willcox, K. (eds.) To appear in model reduction and approximation: theory and algorithms. Available as http://arxiv.org/abs/1409.2140. SIAM, Philadelphia (2017)

  15. 15.

    Benner, P., Breiten, T.: Interpolation-based \(\mathcal {H}_{2}\)-model reduction of bilinear control systems. SIAM J. Matrix Anal. Appl. 33(3), 859–885 (2012)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Benner, P., Breiten, T., Damm, T.: Generalized tangential interpolation for model reduction of discrete-time mimo bilinear systems. Int. J. Control. 84(8), 1398–1407 (2011)

    Article  Google Scholar 

  17. 17.

    Benner, P., Cohen, A., Ohlberger, M., Willcox, K.: Model reduction and approximation: theory and algorithms. SIAM (2005)

  18. 18.

    Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483–531 (2015)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Benner, P., Kürschner, P., Tomljanović, Z., Truhar, N.: Semi-active damping optimization of vibrational systems using the parametric dominant pole algorithm. J. Appl. Math. Mech., 1–16. https://doi.org/10.1002/zamm201400158 (2015)

  20. 20.

    Benner, P., Mehrmann, V., Sorensen, D.: Dimension reduction of Large-Scale systems. Lecture notes in computational science and engineering. Springer, Berlin (2005)

    Google Scholar 

  21. 21.

    Benner, P., Sachs, E., Volkwein, S.: Model order reduction for PDE constrained optimization. Trends in PDE Constrained Optimization, Springer, 303–326 (2014)

  22. 22.

    Benner, P., Tomljanović, Z., Truhar, N.: Dimension reduction for damping optimization in linear vibrating systems. Z. Angew. Math. Mech. 91(3), 179–191 (2011). https://doi.org/10.1002/zamm.201000077

    MathSciNet  Article  Google Scholar 

  23. 23.

    Benner, P., Tomljanović, Z., Truhar, N.: Optimal damping of selected eigenfrequencies using dimension reduction. Numer. Linear Algebr. 20(1), 1–17 (2013). https://doi.org/10.1002/nla.833

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Blanchini, F., Casagrande, D., Gardonio, P., Miani, S.: Constant and switching gains in semi-active damping of vibrating structures. Int. J. Control 85(12), 1886–1897 (2012)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Bonin, T., Faßbender, H., Soppa, A., Zaeh, M.: A fully adaptive rational global arnoldi method for the model-order reduction of second-order mimo systems with proportional damping. Math. Comput. Simul. 122, 1–19 (2016)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Breiten, T.: Structure-preserving model reduction for integro-differential equations. SIAM J. Control. Optim. 54(6), 2992–3015 (2016)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Bunse-Gerstner, A., Kubalinska, D., Vossen, G., Wilczek, D.: \(\mathcal {H}_{2}\)-norm optimal model reduction for large scale discrete dynamical {MIMO} systems. J. Comput. Appl. Math. 233(5), 1202–1216 (2010). https://doi.org/10.1016/j.cam.2008.12.029. http://www.sciencedirect.com/science/article/pii/S0377042709001824. Special Issue Dedicated to William B. Gragg on the Occasion of His 70th Birthday

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Burl, J.B.: Linear optimal control: \(\mathcal {H}_{2}\) and \(\mathcal {H}_{\infty }\) Methods, 1st edn. Addison-wesley Longman Publishing Co., Inc., Boston (1998)

    Google Scholar 

  29. 29.

    Fehr, J., Fischer, M., Haasdonk, B., Eberhard, P.: Greedy-based approximation of frequency-weighted gramian matrices for model reduction in multibody dynamics. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift fź,r Angewandte Mathematik und Mechanik 93(8), 501–519 (2013). https://doi.org/10.1002/zamm.201200014

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

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

    MathSciNet  Article  Google Scholar 

  31. 31.

    Freitas, P., Lancaster, P.: On the optimal value of the spectral abscissa for a system of linear oscillators. SIAM J. Matrix Anal. Appl. 21(1), 195–208 (1999)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Gallivan, K., Vandendorpe, A., Van Dooren, P.: Model reduction of MIMO systems via tangential interpolation. SIAM J. Matrix Anal. Appl. 26(2), 328–349 (2005)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Gawronski, W.: Advanced structural dynamics and active control of structures. Springer, New York (2004)

    Google Scholar 

  34. 34.

    Golub, G., Loan, C.V.: Matrix computations, 3rd edn. J. Hopkins University Press, Baltimore (1996)

    Google Scholar 

  35. 35.

    Gratton, S., Vicente, L.N.: A surrogate management framework using rigorous trust-region steps. Optimization Methods Software 29(1), 10–23 (2014)

    MathSciNet  Article  Google Scholar 

  36. 36.

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

    MathSciNet  Article  Google Scholar 

  37. 37.

    Gumussoy, S., Henrion, D., Millstone, M., Overton, M.L.: Multiobjective robust control with HIFOO 2.0. In: Proceedings of 6th IFAC symposium on robust control design Haifa (2009)

    Article  Google Scholar 

  38. 38.

    Hesthaven, J.S., Rozza, G., Stamm, B.: Certified reduced basis methods for parametrized partial differential equations. Springer (2016)

  39. 39.

    Inman, D.J., Andry, A.N.J.: Some results on the nature of eigenvalues of discrete damped linear systems. ASME J. Appl. Mech. 47, 927–930 (1980)

    Article  Google Scholar 

  40. 40.

    Kanno, Y.: Damper placement optimization in a shear building model with discrete design variables: a mixed-integer second-order cone programming approach. Earthq. Eng. Struct. Dyn. 42, 1657–1676 (2013)

    Article  Google Scholar 

  41. 41.

    Kuzmanović, I., Tomljanović, Z., Truhar, N.: Optimization of material with modal damping. Appl. Math. Comput. 218, 7326–7338 (2012)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Meier III, L., Luenberger, D.: Approximation of linear constant systems. IEEE Trans. Autom. Control 12(5), 585–588 (1967)

    Article  Google Scholar 

  43. 43.

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

    MathSciNet  Article  Google Scholar 

  44. 44.

    Mitchell, T., Overton, M.L.: Hybrid expansion-contraction: a robust scaleable method for approximating the \(\mathcal {H}_{\infty }\) Norm. IMA J. Numer. Anal. 36, 985–1014 (2016)

    MathSciNet  Article  Google Scholar 

  45. 45.

    Müller, P., Schiehlen, W.: Linear vibrations. martinus nijhoff publishers (1985)

  46. 46.

    Nour-Omid, B., Regelbrugge, M.E.: Lanczos method for dynamic analysis of damped structural systems. Earthq. Eng. Struct. Dyn. 18, 1091–1104 (1989)

    Article  Google Scholar 

  47. 47.

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

    MathSciNet  Article  Google Scholar 

  48. 48.

    Saadvandi, M., Meerbergen, K., Desmet, W.: Parametric Dominant Pole Algorithm for Parametric Model Order Reduction. Tech. rep., KU Leuven, Department of Computer Science (2013)

  49. 49.

    Su, T.J., Craig, Jr, R.: Model reduction and control of flexible structures using krylov vectors. J. Guid. Control. Dyn. 14(2), 260–267 (1991)

    Article  Google Scholar 

  50. 50.

    Takewaki, I.: Optimal damper placement for minimum transfer functions. Earthq. Eng. Struct. Dyn. 26, 1113–1124 (1997)

    Article  Google Scholar 

  51. 51.

    Truhar, N., Tomljanović, Z., Puvača, M.: An efficient approximation for optimal damping in mechanical systems. Int. J. Numer. Anal. Model. 14(2), 201–217 (2017)

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Truhar, N., Tomljanović, Z., Veselić, K.: Damping optimization in mechanical systems with external force. Appl. Math. Comput. 250, 270–279 (2015)

    MathSciNet  MATH  Google Scholar 

  53. 53.

    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. 31(1), 18–39 (2009)

    MathSciNet  Article  Google Scholar 

  54. 54.

    Van Dooren, P., Gallivan, K.: Absil, P.: \(\mathcal {H}_{2}\)-optimal model reduction of MIMO systems. Appl. Math. Lett. 21(12), 1267–1273 (2008)

    MathSciNet  Article  Google Scholar 

  55. 55.

    Veselić, K.: Damped oscillations of linear systems. Springer Lecture Notes in Mathematics. Springer, Berlin (2011)

    Google Scholar 

  56. 56.

    Wyatt, S.: Issues in interpolatory model reduction: inexact solves, second-order systems and daes. Dissertation, Virginia Polytechnic Institute and State University, Blacksburg (2012)

    Google Scholar 

  57. 57.

    Zhou, K., Doyle, J., Glower, K.: Robust and optimal control. Prentice Hall, Upper Saddle River (1996)

    Google Scholar 

  58. 58.

    Yue, Y., Meerbergen, K.: Parametric model order reduction of damped mechanical systems via the block Arnoldi process. Appl. Math. Lett. 26, 643–648 (2013)

    MathSciNet  Article  Google Scholar 

  59. 59.

    Yue, Y., Meerbergen, K.: Accelerating optimization of parametric linear systems by model order reduction . SIAM J. Optim. 23(2), 1344–1370 (2013)

    MathSciNet  Article  Google Scholar 

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Corresponding author

Correspondence to Zoran Tomljanović.

Additional information

The work of Zoran Tomljanović was supported in part by the Croatian Science Foundation under the project Optimization of parameter dependent mechanical systems, Grant No. IP-2014-09-9540 and project Control of Dynamical Systems, Grant No. IP-2016-06-2468. The work of Christopher Beattie was supported in part by the Einstein Foundation Berlin. The work of Serkan Gugercin was supported in part by the Alexander von Humboldt Foundation and by the NSF through Grant DMS-1522616.

Communicated by: Peter Benner

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Tomljanović, Z., Beattie, C. & Gugercin, S. Damping optimization of parameter dependent mechanical systems by rational interpolation. Adv Comput Math 44, 1797–1820 (2018). https://doi.org/10.1007/s10444-018-9605-9

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  • Model reduction
  • Interpolation
  • Second-order systems
  • Semi-active damping

Mathematics Subject Classification (2010)

  • 49J15
  • 74P10
  • 70Q05
  • 41A05