Handling actuator magnitude and rate saturation in uncertain over-actuated systems: A modified projection algorithm approach

The paper’s Theorem 4.5 proves (i) boundedness of e and θ̃v with trajectories converging to the compact set E2 and (ii) that one can choose magnitude and rate design bounds so that u stays in Ωu whenever vs ∈ Ωv, using Y = −vseTPB, the Lyapunov function V = eTPe + tr(θ̃vTΓ−1θ̃vΛ), and a modified projection operator Projm defined in (8); the proof yields V̇ ≤ −λmin(Q)||e||2 + 2||θ̃v||2F||YMAX||F and the set E2 in (38) (and not V̇ ≤ −eTQe) . The candidate solution follows the same structure but makes key errors: it incorrectly asserts V̇ ≤ −eTQe and concludes e(t) → 0 via Barbalat (contradicting the paper’s inequality and conclusion to a compact set), misplaces Λ and Γ inside the trace (using an element-wise γij rather than the paper’s row-wise Γ) and equates Projm with a classical Proj(θ,(1−ĥ)y), which is not the piecewise definition used by the paper. Although the candidate offers a constructive way to pick θ and Y bounds, its rate-bound formula relies on γij and needs correction to the row-wise Γ used in the paper. Consequently, the paper’s result is correct, but the model’s proof contains substantive mistakes and overclaims.