Reinforcement Learning-based Disturbance Rejection Control for Uncertain Nonlinear Systems

The paper proves, under A1–A4, that the saturated ESO achieves uniform (in t ≥ T) convergence of estimation errors as ε → 0 and that the state and actor/critic weights are UUB, using a two-step argument that handles the ESO’s peaking via saturation, a scaled error system with a time-varying feedback term, Bellman-error decomposition, and a composite Lyapunov analysis with a Gramian lower bound (A4) to remove PE; see the statement of Theorem 1 and its proof, including the ESO design (9)–(10), control (23), updates (19)–(22), assumptions (24)–(25), BE decomposition (38)–(39), and Lyapunov inequality (45) with gain conditions (46) . The model’s solution mirrors this structure: it lumps uncertainties into an extended state, analyzes a high-gain saturated ESO with scaled errors, reduces the plant to the nominal loop with an O(ε) perturbation, decomposes instantaneous/extrapolated BEs, and establishes UUB of critic/actor and state via a composite Lyapunov function and A4. Differences are technical (e.g., the model streamlines the ESO analysis by not isolating the −F1ϑ2η1 feedback term, and it absorbs the Θ̃c-quadratic term in δ into bounded residuals), but do not alter the conclusion.