Convergence Theorem of the Iterative Algorithm for Identifying Two‐Time‐Scale Systems Using Adaptive PreFilters

This article presents a rigorous proof of convergence for an iterative algorithm that identifies Two‐Time‐Scale Systems (TTSS) using adaptive prefilters. The proof is grounded in the Banach Fixed‐Point Theorem (Contraction Mapping Principle) and establishes that the iterative mapping of the algorithm is a contraction on the compact, convex space of cutoff‐frequency pairs . Three complementary results constitute the main contribution: (i) a quantitative bound on the off‐diagonal terms of the Jacobian in terms of the spectral‐separation ratio , showing they are and therefore negligible; (ii) an explicit product‐of‐sensitivities estimate for the diagonal terms, proving and uniformly over ; and (iii) an extension from a local to a global contraction via the convexity of and a Mean Value Theorem argument. The framework is validated on the longitudinal dynamics model of the Lockheed F‐104G aircraft, a classical TTSS, where the contraction factors are computed explicitly as and , yielding an overall contraction constant . These results provide the essential theoretical foundation for the practical deployment of the algorithm in complex TTSS identification problems.