An Asymmetric Instability Threshold for Fixed-Gain Sliding Mode Control Under Power-Law Contact Stiffness Mismatch

This revised preprint studies fixed-gain boundary-layer sliding-mode control under power-law contact stiffness mismatch and actuator saturation. The original closed-form boundary-layer edge threshold is retained, but its global interpretation is corrected and extended. The revised analysis distinguishes three different transitions: the local boundary-layer edge threshold, an earlier saturation-induced multistability onset, and an actuator-feasibility limit that can occur when the saturation state lies inside the boundary layer. It also corrects the negative-mismatch interpretation from “threshold-free” behavior to a finite persistence boundary followed by a stable lower switched branch. Main additions and corrections include: the saturation-induced coexistence threshold Δ kₛₐₜ; basin-dependent convergence below the conventional positive threshold; the feasibility-limited threshold Δ kfeas when x_d<x_s