Singular perturbation margin and generalised gain margin for linear time-invariant systems

Abstract

In this paper, we propose a singular perturbation margin (SPM) and a generalised gain margin (GGM) as stability metrics for single input–single output (SISO) linear time-invariant (LTI) systems from the view of singular perturbations and regular perturbations, which have bijective correspondences with the classical phase margin (PM) and the gain margin (GM), respectively. Both of the numerical and analytical time-domain SPM and GGM assessment methods are provided, and relationships between the singular perturbation parameter, PM of the perturbed system, PM and SPM of the nominal system, and the (monotonically increasing) phase of the fast system are also revealed. These results make it possible to assess the PM of the nominal system in the time domain for SISO LTI systems using the SPM with a standardised testing system called ‘PM-gauge,’ as demonstrated by examples. The concepts of SPM and GGM can be used as metrics of stability margins for linear time-varying systems and nonlinear systems.

Keywords:

• stability margin
• singular perturbation
• regular perturbation

Acknowledgements

The first author would like to express her sincere gratitude to the Stocker Foundation of Ohio University and China Scholarship Council for the financial support during her PhD study program.

Additional information

Funding

This work was supported in part by NASA [grant number NNX07AN46G] to the second and third authors.