Identifying limits of linear control design validity in nonlinear systems: a continuation-based approach

Duc H Nguyen*, Mark H Lowenberg , Simon A Neild

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

2 Citations (Scopus)
42 Downloads (Pure)

Abstract

It is well-known that a linear-based controller is only valid near the point from which the linearised system is obtained. The question remains as to how far one can move away from that point before the linear and nonlinear responses differ significantly, resulting in the controller failing to achieve the desired performance. In this paper, we propose a method to quantify these differences. By appending a harmonic oscillator to the equations of motion, the frequency responses at different operating points of a nonlinear system can be generated using numerical continuation. In the presence of strong nonlinearities, subtle differences exist between the linear and nonlinear frequency responses, and these variations are also reflected in the step responses. A systematic way of comparing the discrepancies between the linear and the nonlinear frequency responses is presented, which can determine whether the controller performs as predicted by linear-based design. We demonstrate the method on a simple fixed-gain Duffing system and a gain-scheduled reduced-order aircraft model with a manoeuvre-demand controller; the latter presents a case where strong nonlinearities exist in the form of multiple attractors. The analysis is then expanded to include actuator rate saturation, which creates a limit-cycle isola, coexisting multiple solutions (corresponding to the so-called ‘flying qualities cliff’), and chaotic motions. The proposed method can infer the influence of these additional attractors even when there is no systematic way to detect them. Finally, when severe rate saturation is present, reducing the controller gains can mitigate – but not eliminate – the risk of limit-cycle oscillation.
Original languageEnglish
Pages (from-to)901-921
Number of pages21
JournalNonlinear Dynamics
Volume104
Issue number2
Early online date18 Mar 2021
DOIs
Publication statusPublished - Apr 2021

Bibliographical note

Funding Information:
The first author is partially funded by the University of Bristol?s Alumni Grant. We would like to thank Professor Mario di Bernardo for the technical discussion. His ideas and recommendations laid the foundation for the work presented in this paper. We also gratefully acknowledge QinetiQ for the HHIRM model, Professor Thomas S. Richardson for advices on using the HHIRM controller, as well as Mr Thomas R. O. Wainwright and Mr Duc M. Nguyen (University of Warwick) for provision of additional computing resources for this research.

Funding Information:
The first author is partially funded by the University of Bristol’s Alumni Grant. We would like to thank Professor Mario di Bernardo for the technical discussion. His ideas and recommendations laid the foundation for the work presented in this paper. We also gratefully acknowledge QinetiQ for the HHIRM model, Professor Thomas S. Richardson for advices on using the HHIRM controller, as well as Mr Thomas R. O. Wainwright and Mr Duc M. Nguyen (University of Warwick) for provision of additional computing resources for this research.

Publisher Copyright:
© 2021, The Author(s).

Keywords

  • Frequency response
  • Bifurcation analysis
  • Dynamical systems
  • Flight dynamics
  • Rate saturation
  • Gain scheduling

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