An extension of the structured singular value to nonlinear systems with application to robust flutter analysis
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ORIGINAL PAPER
An extension of the structured singular value to nonlinear systems with application to robust flutter analysis Andrea Iannelli1 · Mark Lowenberg2 · Andrés Marcos2 Received: 11 January 2020 / Revised: 10 June 2020 / Accepted: 25 August 2020 © The Author(s) 2020
Abstract The paper discusses an extension of 𝜇 (or structured singular value), a well-established technique from robust control for the study of linear systems subject to structured uncertainty, to nonlinear polynomial problems. Robustness is a multifaceted concept in the nonlinear context, and in this work the point of view of bifurcation theory is assumed. The latter is concerned with the study of qualitative changes of the steady-state solutions of a nonlinear system, so-called bifurcations. The practical goal motivating the work is to assess the effect of modeling uncertainties on flutter, a dynamic instability prompted by an adverse coupling between aerodynamic, elastic, and inertial forces, when considering the system as nonlinear. Specifically, the onset of flutter in nonlinear systems is generally associated with limit cycle oscillations emanating from a Hopf bifurcation point. Leveraging 𝜇 and its complementary modeling paradigm, namely linear fractional transformation, this work proposes an approach to compute margins to the occurrence of Hopf bifurcations for uncertain nonlinear systems. An application to the typical section case study with linear unsteady aerodynamic and hardening nonlinearities in the structural parameters will be presented to demonstrate the applicability of the approach. Keywords Bifurcation · Robust control · Flutter · Modeling uncertainties
1 Introduction Flutter is a self-excited instability in which aerodynamic forces on a flexible body couple with its natural vibration modes producing an undesired and often dangerous response of the system. Therefore, flutter analysis has been widely investigated and there are several techniques representing the state-of-practice (e.g. p-k method) [32]. These often assume that the model representing the system is linear, and the classic approach is to look at the smallest speed V such that the system features a pair of purely imaginary eigenvalues. This * Andrea Iannelli [email protected] Mark Lowenberg [email protected] Andrés Marcos [email protected] 1
Department of Information Technology and Electrical Engineering, Swiss Federal Institute of Technology (ETH), 8092 Zurich, Switzerland
Department of Aerospace Engineering, University of Bristol, Bristol BS8 1TR, UK
2
speed Vf is called flutter speed and is such that for V < Vf the aeroelastic system is stable. One of the main issues related to flutter analysis using standard techniques arises from the sensitivity of this aeroelastic instability to small variations in parameter and modeling assumptions [31]. In addressing this aspect, in the last decades the so-called flutter robust analysis was proposed, which aims to quantify the gap between the prediction of the nominal stability
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