# zbMATH — the first resource for mathematics

Damping optimization of parameter dependent mechanical systems by rational interpolation. (English) Zbl 1416.90038
Summary: 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.

##### MSC:
 90C26 Nonconvex programming, global optimization 74P10 Optimization of other properties in solid mechanics 70Q05 Control of mechanical systems 41A05 Interpolation in approximation theory
##### Software:
HIFOO; JDQR; JDQZ
Full Text:
##### References:
 [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) · Zbl 1013.76028 [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, 16-23, (1998) [3] Antoulas, A.: Approximation of Large-Scale dynamical systems. SIAM publications, Philadelphia (2005) · Zbl 1112.93002 [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) · Zbl 1229.65103 [5] Antoulas, AC; Sorensen, DC; Gugercin, S., A survey of model reduction methods for large-scale systems, Contemp. Math., 280, 193-219, (2001) · Zbl 1048.93014 [6] Arian, E., Fahl, M., Sachs, E.: Trust-region proper orthogonal decomposition for flow control. ICASE Technical Report (2000) [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) · Zbl 0965.65058 [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) · Zbl 1078.65058 [9] Bashir, O.; Willcox, K.; Ghattas, O.; 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) · Zbl 1195.76311 [10] Baur, U.; Beattie, CA; Benner, P.; Gugercin, S., Interpolatory projection methods for parameterized model reduction, SIAM J. Sci. Comput., 33, 2489-2518, (2011) · Zbl 1254.93032 [11] Baur, U.; Benner, P.; Feng, L., Model order reduction for linear and nonlinear systems: a system-theoretic perspective, Arch. Comput. Methods Eng., 21, 331-358, (2014) · Zbl 1348.93075 [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] Beattie, C.; Gugercin, S., Interpolatory projection methods for structure-preserving model reduction, Syst. Control Lett., 58, 225-232, (2009) · Zbl 1159.93317 [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] Benner, P.; Breiten, T., Interpolation-based $$\mathcal {H}_{2}$$ ℋ2-model reduction of bilinear control systems, SIAM J. Matrix Anal. Appl., 33, 859-885, (2012) · Zbl 1256.93027 [16] Benner, P.; Breiten, T.; Damm, T., Generalized tangential interpolation for model reduction of discrete-time mimo bilinear systems, Int. J. Control., 84, 1398-1407, (2011) · Zbl 1230.93010 [17] Benner, P., Cohen, A., Ohlberger, M., Willcox, K.: Model reduction and approximation: theory and algorithms. SIAM (2005) · Zbl 1378.65010 [18] Benner, P.; Gugercin, S.; Willcox, K., A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev., 57, 483-531, (2015) · Zbl 1339.37089 [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] Benner, P., Mehrmann, V., Sorensen, D.: Dimension reduction of Large-Scale systems. Lecture notes in computational science and engineering. Springer, Berlin (2005) · Zbl 1066.65004 [21] Benner, P., Sachs, E., Volkwein, S.: Model order reduction for PDE constrained optimization. Trends in PDE Constrained Optimization, Springer, 303-326 (2014) · Zbl 1327.49043 [22] Benner, P.; Tomljanović, Z.; Truhar, N., Dimension reduction for damping optimization in linear vibrating systems, Z. Angew. Math. Mech., 91, 179-191, (2011) · Zbl 1370.70041 [23] Benner, P.; Tomljanović, Z.; Truhar, N., Optimal damping of selected eigenfrequencies using dimension reduction, Numer. Linear Algebr., 20, 1-17, (2013) · Zbl 1289.70028 [24] Blanchini, F.; Casagrande, D.; Gardonio, P.; Miani, S., Constant and switching gains in semi-active damping of vibrating structures, Int. J. Control, 85, 1886-1897, (2012) · Zbl 1253.93137 [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) [26] Breiten, T., Structure-preserving model reduction for integro-differential equations, SIAM J. Control. Optim., 54, 2992-3015, (2016) · Zbl 1350.93025 [27] Bunse-Gerstner, A.; Kubalinska, D.; Vossen, G.; Wilczek, D., $$\mathcal {H}_{2}$$ℋ2-norm optimal model reduction for large scale discrete dynamical MIMO systems, J. Comput. Appl. Math., 233, 1202-1216, (2010) · Zbl 1178.93032 [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) [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, 501-519, (2013) · Zbl 1331.70003 [30] Flagg, G.; Gugercin, S., Multipoint Volterra series interpolation and $$\mathcal {H}_{2}$$ ℋ2 optimal model reduction of bilinear systems, SIAM J. Matrix Anal. Appl., 36, 549-579, (2015) · Zbl 1315.93036 [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, 195-208, (1999) · Zbl 0942.34069 [32] Gallivan, K.; Vandendorpe, A.; Dooren, P., Model reduction of MIMO systems via tangential interpolation, SIAM J. Matrix Anal. Appl., 26, 328-349, (2005) · Zbl 1078.41016 [33] Gawronski, W.: Advanced structural dynamics and active control of structures. Springer, New York (2004) · Zbl 1064.93001 [34] Golub, G., Loan, C.V.: Matrix computations, 3rd edn. J. Hopkins University Press, Baltimore (1996) · Zbl 0865.65009 [35] Gratton, S.; Vicente, LN, A surrogate management framework using rigorous trust-region steps, Optimization Methods Software, 29, 10-23, (2014) · Zbl 1282.65067 [36] Gugercin, S.; Antoulas, AC; Beattie, C., $$\mathcal {H}_{2}$$ℋ2 model reduction for large-scale linear dynamical systems, SIAM J. Matrix Anal. Appl., 30, 609-638, (2008) · Zbl 1159.93318 [37] Gumussoy, Suat; Henrion, Didier; Millstone, Marc; Overton, Michael L., Multiobjective Robust Control with HIFOO 2.0, IFAC Proceedings Volumes, 42, 144-149, (2009) [38] Hesthaven, J.S., Rozza, G., Stamm, B.: Certified reduced basis methods for parametrized partial differential equations. Springer (2016) · Zbl 1329.65203 [39] Inman, DJ; Andry, ANJ, Some results on the nature of eigenvalues of discrete damped linear systems, ASME J. Appl. Mech., 47, 927-930, (1980) · Zbl 0452.70022 [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) [41] Kuzmanović, I.; Tomljanović, Z.; Truhar, N., Optimization of material with modal damping, Appl. Math. Comput., 218, 7326-7338, (2012) · Zbl 1427.70045 [42] Meier, L.; Luenberger, D., Approximation of linear constant systems, IEEE Trans. Autom. Control, 12, 585-588, (1967) [43] Meyer, D.; Srinivasan, S., Balancing and model reduction for second-order form linear systems, IEEE Trans. Autom. Control, 41, 1632-1644, (1996) · Zbl 0859.93015 [44] Mitchell, T.; Overton, ML, Hybrid expansion-contraction: a robust scaleable method for approximating the $$\mathcal {H}_{\infty }$$ ℋ8 Norm, IMA J. Numer. Anal., 36, 985-1014, (2016) · Zbl 1433.93100 [45] Müller, P., Schiehlen, W.: Linear vibrations. martinus nijhoff publishers (1985) [46] Nour-Omid, B.; Regelbrugge, ME, Lanczos method for dynamic analysis of damped structural systems, Earthq. Eng. Struct. Dyn., 18, 1091-1104, (1989) [47] Reis, T.; Stykel, T., Balanced truncation model reduction of second-order systems, Math. Comput. Model. Dyn. Syst., 14, 391-406, (2008) · Zbl 1151.93010 [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) · Zbl 1291.93063 [49] Su, TJ; Craig, R., Model reduction and control of flexible structures using krylov vectors, J. Guid. Control. Dyn., 14, 260-267, (1991) [50] Takewaki, I., Optimal damper placement for minimum transfer functions, Earthq. Eng. Struct. Dyn., 26, 1113-1124, (1997) [51] Truhar, N.; Tomljanović, Z.; Puvača, M., An efficient approximation for optimal damping in mechanical systems, Int. J. Numer. Anal. Model., 14, 201-217, (2017) · Zbl 1391.70057 [52] Truhar, N.; Tomljanović, Z.; Veselić, K., Damping optimization in mechanical systems with external force, Appl. Math. Comput., 250, 270-279, (2015) · Zbl 1328.70016 [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, 18-39, (2009) · Zbl 1303.70021 [54] Dooren, P.; Gallivan, K., Absil, P.: $$\mathcal {H}_{2}$$ℋ2-optimal model reduction of MIMO systems, Appl. Math. Lett., 21, 1267-1273, (2008) · Zbl 1182.93034 [55] Veselić, K.: Damped oscillations of linear systems. Springer Lecture Notes in Mathematics. Springer, Berlin (2011) · Zbl 1232.37004 [56] Wyatt, S.: Issues in interpolatory model reduction: inexact solves, second-order systems and daes. Dissertation, Virginia Polytechnic Institute and State University, Blacksburg (2012) [57] Zhou, K., Doyle, J., Glower, K.: Robust and optimal control. Prentice Hall, Upper Saddle River (1996) [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) · Zbl 1261.93025 [59] Yue, Y.; Meerbergen, K., Accelerating optimization of parametric linear systems by model order reduction, SIAM J. Optim., 23, 1344-1370, (2013) · Zbl 1273.35279
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.