Semi-active \(\mathcal{H}_{\infty}\) damping optimization by adaptive interpolation.

*(English)*Zbl 07250715In this paper authors consider the problem of semi-active damping optimization of mechanical systems. The main problem is to determine the best damping matrix which will minimize the influence of the input to the output of the system. In order to achieve that one can use criteria based on energy that comes from the influence of input in the system (such as \(\mathcal{H}_2\) norm or \(\mathcal{H}_\infty\) norm). Since that the objective function is a non-convex function, this damping optimization problem usually requires a large number evaluations of objective function. Thus, in the paper authors investigate efficient optimization of damping viscosities for fixed damping positions using the \(\mathcal{H}_\infty\) system norm.

Their main goal was computation of a damping that is locally optimal with respect to the \(\mathcal{H}_\infty\) norm of the transfer function from the exogenous noise inputs to the performance outputs. Authors have proposed a new greedy method for computing the \(\mathcal{H}_\infty\) norm of a transfer function based on rational interpolation. The proposed approach is particularly adapted to parameter-dependent transfer functions. The interpolation leads to parametric reduced-order models that can be more efficiently optimized. At the optimizers the new interpolation points have been taken to refine the reduced-order model and to obtain updated optimizers. The numerical examples show that proposed approach normally converges fast and thus can highly accelerate the optimization procedure.

Altogether it can be concluded that with proposed new approach one has been able to perform the semi-active \(\mathcal{H}_\infty\) damping optimization for a problem with moderate internal damping with satisfactory relative accuracy, while the optimization process was considerably accelerated. On the other hand, as authors have been emphasized, the proposed method can have problems with the configurations with very small internal damping. Such problems must be treated more carefully and it is necessary to have a mindful choice of the initial sampling data as well as of the algorithm parameters. Still, the optimization problem could be solved with a satisfactory accuracy for most of the configurations of this extremely hard problem.

Their main goal was computation of a damping that is locally optimal with respect to the \(\mathcal{H}_\infty\) norm of the transfer function from the exogenous noise inputs to the performance outputs. Authors have proposed a new greedy method for computing the \(\mathcal{H}_\infty\) norm of a transfer function based on rational interpolation. The proposed approach is particularly adapted to parameter-dependent transfer functions. The interpolation leads to parametric reduced-order models that can be more efficiently optimized. At the optimizers the new interpolation points have been taken to refine the reduced-order model and to obtain updated optimizers. The numerical examples show that proposed approach normally converges fast and thus can highly accelerate the optimization procedure.

Altogether it can be concluded that with proposed new approach one has been able to perform the semi-active \(\mathcal{H}_\infty\) damping optimization for a problem with moderate internal damping with satisfactory relative accuracy, while the optimization process was considerably accelerated. On the other hand, as authors have been emphasized, the proposed method can have problems with the configurations with very small internal damping. Such problems must be treated more carefully and it is necessary to have a mindful choice of the initial sampling data as well as of the algorithm parameters. Still, the optimization problem could be solved with a satisfactory accuracy for most of the configurations of this extremely hard problem.

Reviewer: Ninoslav Truhar (Osijek)

##### MSC:

93B51 | Design techniques (robust design, computer-aided design, etc.) |

93B36 | \(H^\infty\)-control |

93B52 | Feedback control |

65D05 | Numerical interpolation |

70Q05 | Control of mechanical systems |