Worst Case Uncertainty Construction via Multifrequency Gain Maximization With Application to Flutter Control

Abstract

In the analysis of uncertain systems, we often search for a worst case perturbation that drives the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_\infty $ </tex-math></inline-formula> gain of the system to the maximum over the set of allowable uncertainties. Employing the classical technique, an uncertainty sample is obtained, which, indeed, maximizes the gain but only at the single frequency where that maximum occurs. In contrast, this article considers a method to calculate a worst case perturbation that maximizes the gain of a system with mixed uncertainty at multiple frequencies simultaneously. This approach involves a nonlinear optimization that selects the worst case value of the uncertain parameters and the application of the boundary Nevanlinna–Pick interpolation to calculate the dynamic uncertainty sample. Such a perturbation can be used to augment Monte Carlo simulations of uncertain systems, especially if the system has multiple resonance frequencies. The worst case analysis of a flutter control system designed for a small flexible aircraft is provided to demonstrate the applicability of the proposed method.