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

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).