Damping optimization of parameter dependent mechanical systems by rational interpolation

Abstract

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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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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Keywords

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

Mathematics Subject Classification (2010)

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