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

ABSTRACT

In this paper, singular perturbation margin (SPM) and generalised gain margin (GGM) are proposed as the classical phase margin and gain margin like stability metrics for nonlinear systems established from the view of the singular perturbation and the regular perturbation, respectively. The problem of SPM and GGM assessment of a nonlinear nominal system is formulated. The SPM and GGM formulations are provided as the functions of radius of attraction (ROA), which is introduced as a conservative measure of the domain of attraction (DOA). Furthermore, the ROA constrained SPM and GGM analysis are processed through two stages: (1) the SPM and GGM assessment for nonlinear systems at the equilibrium point, based on the SPM and GGM equilibrium theorems, including time-invariant and time-varying cases (Theorem 5.3, Theorem 5.2, Theorem 5.4 and Theorem 5.5); (2) the establishment of the relationship between the SPM or GGM and the ROA for nonlinear time-invariant systems through the construction of the Lyapunov function for the singularly perturbed model (Theorem 6.1 and Section 6.2.3).

Keywords:

• singular perturbation
• regular perturbation
• stability metrics
• nonlinear time-invariant systems

Acknowledgements

The first author (Xiaojing Yang) would like to express her sincere gratitude to the Stocker Foundation of Ohio University and China Scholarship Council for the financial support during her Ph.D. study program. The authors would like to express their sincere thanks to the reviewers and the editor of this paper for their many constructive comments and suggestions of both technical and editorial nature. The material in this paper was presented in part at American Control Conference 2012 and IEEE Conference on Decision and Control 2012.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Xiaojing Yang would like to thank the Stocker Foundation of Ohio University and China Scholarship Council for the financial support during her PhD study program.