Dry-friction oscillators are mechanical systems with dry friction and stick-slip vibrations. In the context of control theory, the stability analysis of this type of dynamical systems is important since they exhibit nonsmooth bifurcations, or most famously a sliding–grazing bifurcation inducing abrupt chaos. This paper develops a Lyapunov-based framework to study the so-called structural stability of the system, predicting the onset of such unique bifurcations. To achieve this, the nonlinear system is first represented as a nonsmooth Takagi–Sugeno (TS) fuzzy model, and the structural stability is then formulated as linear matrix inequalities (LMI) feasibility problems with less conservative formulation. Solving the resulting LMI problem, the onset of sliding–grazing bifurcation can be accurately predicted.