Observer-Based Adaptive Scheme for Fixed-Time Frequency Estimation of Biased Sinusoidal Signals

The candidate solution follows the paper’s exact strategy: derive |y⃛| = w²|ẏ| to obtain ζγ1 = γ2, use the finite‑time exact observer to make γ̂i = γi after T1 + r, define eγ = ζ̂γ̂1 − γ̂2, pick the same adaptive law, obtain the scalar error dynamics ėγ = −α1⌊eγ⌉^{1+q/p} − β1⌊eγ⌉^{1−q/p}, and conclude fixed‑time convergence and exact identification. The paper proves fixed‑time via a Lyapunov lemma and states existence of a uniform Tmax, while the model adds a standard explicit bound T ≤ (p/q)(1/α1 + 1/β1). Minor discrepancies are notational (the paper’s exponents allow 1 − q/p ≤ 0 despite defining ⌊·⌉^α only for α > 0), but the core argument and result coincide. See the paper’s definitions, estimator, and proof steps in (1), (3)–(7), (12)–(15), and Theorem 1, as well as Propositions 1–2 for the observer and PE claims.