Singular Perturbation: When the Perturbation Parameter Becomes a State-Dependent Function

TL;DR

This paper introduces a novel framework for nonlinear singularly perturbed systems with state-dependent perturbation functions, enabling advanced stability analysis and solutions for complex control problems beyond traditional methods.

Contribution

It develops a systematic approach for systems with state-dependent perturbations, extending classical singular perturbation theory to new control applications.

Findings

Established global robust stability results.

Achieved input-to-state stability under the new framework.

Provided solutions for nonlinear integral control and formation seeking.

Abstract

This paper presents a new systematic framework for nonlinear singularly perturbed systems in which state-dependent perturbation functions are used instead of constant perturbation coefficients. Under this framework, general results are obtained for the global robust stability and input-to-state stability of nonlinear singularly perturbed systems. Interestingly, the proposed methodology provides innovative solutions beyond traditional singular perturbation theory for emerging control problems arising from nonlinear integral control, feedback optimization, and formation-based source seeking.