Singular Perturbation Margin for Nonlinear Time-Invariant systems

In this paper, Singular Perturbation Margin (SPM) is proposed as a phase margin like stability margin metric for Nonlinear (NL) systems established from the view of the singular perturbation (time-scale separation) parameter. Theorem 1 in this paper provides the SPM equivalence between Linear Time-Invariant (LTI) and Nonlinear Time-Invariant (NLTI) systems at the equilibrium point. However, unlike for linear systems, the SPM of the NL system may be reduced or even vanish when the size of Domain of Attraction (DOA) is imposed. Here, a concept, Radius of Attraction (ROA), is introduced as a conservative measure of the DOA for NL systems while taking into account the stability analysis in the neighborhood of an equilibrium point, based on which Theorem 2 offers the relationship between the SPM and the ROA for NLTI systems with the construction of Lyapunov function for the singularly perturbed model. The results developed here make it possible to develop SPM assessment methods for NLTI systems in the subsequent investigation using the corresponding LTI SPM estimating methods that have recently been developed.