Semi‐active ℋ∞ damping optimization by adaptive interpolation

Tomljanović, Zoran; Voigt, Matthias

doi:10.1002/nla.2300

Cited by 7 publications

(4 citation statements)

References 28 publications

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“…We consider external disturbance forces that attack at the sequential masses from m 471 to m 480 . Hence, in the input matrix B the values at positions 471 to 480 are set to be B(471 : 480, 1 : 10) = diag (10,20,30,40,50,50,40,30,20,10).…”

Section: Examplementioning

confidence: 99%

“…In [2], a sampling-free approach is presented that reduces the system (1) for all admissible parameters. Alternatively, in [40] the H ∞ -norm of the transfer function G is minimized. In [6,41,43], the authors present different reduction techniques to optimize the related problem of minimizing the total average energy for the system (1) with no input.…”

Section: Introductionmentioning

confidence: 99%

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Semi-active damping optimization of vibrational systems using the reduced basis method

Przybilla,

Pontes Duff,

Benner

2024

Adv Comput Math

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In this article, we consider vibrational systems with semi-active damping that are described by a second-order model. In order to minimize the influence of external inputs to the system response, we are optimizing some damping values. As minimization criterion, we evaluate the energy response, that is the $$\mathcal {H}_2$$ H 2 -norm of the corresponding transfer function of the system. Computing the energy response includes solving Lyapunov equations for different damping parameters. Hence, the minimization process leads to high computational costs if the system is of large dimension. We present two techniques that reduce the optimization problem by applying the reduced basis method to the corresponding parametric Lyapunov equations. In the first method, we determine a reduced solution space on which the Lyapunov equations and hence the resulting energy response values are computed approximately in a reasonable time. The second method includes the reduced basis method in the minimization process. To evaluate the quality of the approximations, we introduce error estimators that evaluate the error in the controllability Gramians and the energy response. Finally, we illustrate the advantages of our methods by applying them to two different examples.

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Section: Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Semi-active damping optimization of vibrational systems using the reduced basis method

Przybilla,

Pontes Duff,

Benner

2024

Adv Comput Math

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“…For multiple input, multiple output systems, one can also optimize damping in second‐order systems by minimizing standard systems norms, such as the

\mathcal {H}_2

\mathcal {H}_\infty

norms; see Refs. [5–9].…”

Section: Introductionmentioning

confidence: 99%

Fast optimization of viscosities for frequency‐weighted damping of second‐order systems

et al. 2022

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We consider frequency-weighted damping optimization for vibrating systems described by a second-order differential equation. The goal is to determine viscosity values such that eigenvalues are kept away from certain undesirable areas on the imaginary axis. To this end, we present two complementary techniques.First, we propose new frameworks using nonsmooth constrained optimization problems, whose solutions both damp undesirable frequency bands and maintain the stability of the system. These frameworks also allow us to weight, which frequency bands are the most important to damp. Second, we also propose a fast new eigensolver for the structured quadratic eigenvalue problems (QEPs) that appear in such vibrating systems. In order to be efficient, our new eigensolver exploits special properties of diagonal-plus-rank-one (DPR1) complex symmetric matrices, which we leverage by showing how each QEP can be transformed into a short sequence of such linear eigenvalue problems. The result is an eigensolver that is substantially faster than standard techniques. By combining this new solver with our new optimization frameworks, we obtain our overall algorithm for fast computation of optimal viscosities. The efficiency and performance of our new approach are verified and illustrated on several numerical examples.

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“…In [24], the authors consider energy over arbitrary time, while [35] considers the case where the excitation function is periodic. For multiple input, multiple output systems, one can also optimize damping in second-order systems by minimizing standard systems norms, such as the H 2 or H ∞ norms; see [3,5,10,31,32].…”

Section: Introductionmentioning

confidence: 99%

Fast optimization of viscosities for frequency-weighted damping of second-order systems

Stor¹,

Mitchell²,

Tomljanović³

et al. 2021

Preprint

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We consider frequency-weighted damping optimization for vibrating systems described by a second-order differential equation. The goal is to determine viscosity values such that eigenvalues are kept away from certain undesirable areas on the imaginary axis. To this end, we present two complementary techniques. First, we propose new frameworks using nonsmooth constrained optimization problems, whose solutions both damp undesirable frequency bands and maintain stability of the system. These frameworks also allow us to weight which frequency bands are the most important to damp. Second, we also propose a fast new eigensolver for the structured quadratic eigenvalue problems that appear in such vibrating systems. In order to be efficient, our new eigensolver exploits special properties of diagonal-plus-rank-one complex symmetric matrices, which we leverage by showing how each quadratic eigenvalue problem can be transformed into a short sequence of such linear eigenvalue problems. The result is an eigensolver that is substantially faster than standard techniques. By combining this new solver with our new optimization frameworks, we obtain our overall algorithm for fast computation of optimal viscosities. The efficiency and performance of our new methods are verified and illustrated on several numerical examples.

show abstract

Semi‐active ℋ∞ damping optimization by adaptive interpolation

Cited by 7 publications

References 28 publications

Semi-active damping optimization of vibrational systems using the reduced basis method

Semi-active damping optimization of vibrational systems using the reduced basis method

Fast optimization of viscosities for frequency‐weighted damping of second‐order systems

Fast optimization of viscosities for frequency-weighted damping of second-order systems

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