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

This paper introduces a new class of singularly perturbed systems in which the small, but constant, perturbation coefficient in standard singular perturbation theory is replaced by a state-dependent function. This generalization is aimed at broadening the applicability of singular perturbation theory in practice. For this class of singularly perturbed systems, it is assumed that the boundary-layer subsystem is globally asymptotically stable (GAS) at the origin and the reduced subsystem is input-to-state stable (ISS) with respect to the state of the boundary-layer subsystem. Under a mild monotonicity condition, sufficient conditions on the perturbation functions are given under which the singularly perturbed system is GAS at the origin. ISS and nonlinear small-gain techniques are exploited in the stability analysis. The efficacy of the proposed theoretical result is validated via its applications to tackling integral control and feedback optimization problems.