Semi-active $\mathcal{H}_{\infty}$ damping optimization by adaptive interpolation

In this work we consider the problem of semi-active damping optimization of mechanical systems with fixed damper positions. Our goal is to compute a damping that is locally optimal with respect to the $\mathcal{H}_\infty$-norm of the transfer function from the exogenous inputs to the performance outputs. We make use of a new greedy method for computing the $\mathcal{H}_\infty$-norm of a transfer function based on rational interpolation. In this paper, this approach is adapted to parameter-dependent transfer functions. The interpolation leads to parametric reduced-order models that can be optimized more efficiently. At the optimizers we then take new interpolation points to refine the reduced-order model and to obtain updated optimizers. In our numerical examples we show that this approach normally converges fast and thus can highly accelerate the optimization procedure. Another contribution of this work are heuristics for choosing initial interpolation points.

[1]  P. Lancaster.  On eigenvalues of matrices dependent on a parameter , 1964 .

[2]  V. N. Bogaevski,et al.  Matrix Perturbation Theory , 1991 .

[3]  Peter Benner,et al.  A Subspace Framework for H∞-Norm Minimization , 2019, SIAM J. Matrix Anal. Appl..

[4]  M. Overton,et al.  FAST APPROXIMATION OF THE H∞ NORM VIA OPTIMIZATION OVER SPECTRAL VALUE SETS∗ , 2012 .

[5]  Peter Benner,et al.  A structured pseudospectral method for $$\mathcal {H}_\infty $$H∞-norm computation of large-scale descriptor systems , 2013, Math. Control. Signals Syst..

[6]  Ninoslav Truhar,et al.  An Efficient Approximation for Optimal Damping in Mechanical Systems , 2017 .

[7]  Peter Benner,et al.  A Greedy Subspace Method for Computing the ℒ∞‐Norm , 2017 .

[8]  Suresh M. Joshi,et al.  Advanced Structural Dynamics and Active Control of Structures , 2005, IEEE Transactions on Automatic Control.

[9]  Peter Benner,et al.  Optimal damping of selected eigenfrequencies using dimension reduction , 2013, Numer. Linear Algebra Appl..

[10]  Ninoslav Truhar,et al.  An Efficient Method for Estimating the Optimal Dampers' Viscosity for Linear Vibrating Systems Using Lyapunov Equation , 2009, SIAM J. Matrix Anal. Appl..

[11]  Serkan Gugercin,et al.  Interpolatory projection methods for structure-preserving model reduction , 2009, Syst. Control. Lett..

[12]  Peter Benner,et al.  LARGE-SCALE COMPUTATION OF L ∞ -NORMS BY A GREEDY SUBSPACE METHOD , 2017 .

[13]  Adrian S. Lewis,et al.  A Robust Gradient Sampling Algorithm for Nonsmooth, Nonconvex Optimization , 2005, SIAM J. Optim..

[14]  Michael L. Overton,et al.  Multiobjective robust control with HIFOO 2.0 , 2009, 0905.3229.

[15]  M. Overton.  NONSMOOTH OPTIMIZATION VIA BFGS , 2008 .

[16]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[17]  Adrian S. Lewis,et al.  Nonsmooth optimization via quasi-Newton methods , 2012, Mathematical Programming.

[18]  Joost Rommes,et al.  Computing Transfer Function Dominant Poles of Large-Scale Second-Order Dynamical Systems , 2008, SIAM J. Sci. Comput..

[19]  Mi-Ching Tsai,et al.  Robust and Optimal Control , 2014 .

[20]  Volker Mehrmann,et al.  Balanced Truncation Model Reduction for Large-Scale Systems in Descriptor Form , 2005 .

[21]  Peter Benner,et al.  Dimension reduction for damping optimization in linear vibrating systems , 2011 .

[22]  Franco Blanchini,et al.  Constant and switching gains in semi-active damping of vibrating structures , 2012, Int. J. Control.

[23]  S. Boyd,et al.  A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L ∞ -norm , 1990 .

[24]  Michael L. Overton,et al.  Fast Approximation of the HINFINITY Norm via Optimization over Spectral Value Sets , 2013, SIAM J. Matrix Anal. Appl..

[25]  Serkan Gugercin,et al.  Damping optimization of parameter dependent mechanical systems by rational interpolation , 2017, Adv. Comput. Math..

[26]  Peter Benner,et al.  Semi‐active damping optimization of vibrational systems using the parametric dominant pole algorithm , 2016 .

[27]  Peter Lancaster,et al.  On the Optimal Value of the Spectral Abscissa for a System of Linear Oscillators , 1999, SIAM J. Matrix Anal. Appl..

[28]  Vasile Sima,et al.  ${\cal L}_{\infty}$-Norm Computation for Continuous-Time Descriptor Systems Using Structured Matrix Pencils , 2012, IEEE Transactions on Automatic Control.

[29]  Michael L. Overton,et al.  Hybrid expansion–contraction: a robust scaleable method for approximating the H∞ norm , 2016 .

[30]  Ninoslav Truhar,et al.  Damping optimization in mechanical systems with external force , 2015, Appl. Math. Comput..

[31]  M.sc. Patrick Kürschner.  Two-sided Eigenvalue Algorithms for Modal Approximation , 2010 .

[32]  Tim Mitchell,et al.  A BFGS-SQP method for nonsmooth, nonconvex, constrained optimization and its evaluation using relative minimization profiles , 2017, Optim. Methods Softw..

[33]  M. Steinbuch,et al.  A fast algorithm to computer the H ∞ -norm of a transfer function matrix , 1990 .