A New Least Squares Parameter Estimator for Nonlinear Regression Equations with Relaxed Excitation Conditions and Forgetting Factor

Both the paper and the candidate solution derive the key interlaced identity Y(t) = Δ(t) G(θ) from the LS dynamics and use strong monotonicity (A1) plus interval excitation (A2) to obtain exponential decay of the parameter error. The candidate gives a slightly sharper, more explicit argument: an invertible closed form for F^{-1}, a clean formula H(t)=αA(t)[f0I+αA(t)]^{-1}, monotonicity of Δ(t), and an explicit post-transient lower bound Δ(t)≥δ>0 that yields an explicit rate. The paper’s proof sketches the same mechanism—LS identity (10) ⇒ extended NLPRE (11) ⇒ scalar regressor (12) ⇒ Lyapunov decay—and argues Δ(t) is PE under IE, which is sufficient for exponential convergence; it is correct though less explicit about δ and monotonicity of Δ. Hence both are correct and essentially the same proof at core, with the model providing additional details and constants (Proposition 1 and its proof, including equations (8)–(12), and the PE discussion around (9)–(11) in the paper confirm this structure).