Nonlinear Frequency Response Analysis to Inform Aircraft Control Law Design

This paper presents the methodology and results of an initial study into generating the nonlinear frequency responses of a longitudinal airliner model – the polynomial NASA GTM – using numerical continuation with periodic forcing. The results are compared with the linear frequency responses to inform control designers of the extent to which a linearized model can adequately capture the system behavior in the frequency domain. Since the aerospace industry typically uses linearizations in controller design, both open and closed-loop behaviors are considered. When the aircraft is forced with small elevator deflections, highly nonlinear responses are observed, including period-2 and quasi-periodic motions, as well as subharmonic resonances that lead to loss of stability. Closed-loop responses of a proportional stability augmentation controller become out of phase with the linear prediction at low forcing frequencies when the aircraft operates at high angle of attack. Additionally, two-parameter continuation is used to assess the controller’s effectiveness when operated in nonlinear regions where linear controller design techniques cannot be used. Time histories are used to verify the results, which closely follow the nonlinear analysis. To illustrate the concept and validate the method for generating nonlinear frequency response, a brief numerical analysis of the Duffing equation is also presented.