Adaptive Parameters Estimation Using Modulating Functions and the KKL Observer

Abstract

This work addresses the problem of estimating unmeasured states and identifying parameters of a nonlinear system using only one measurable signal. To achieve this, we employ a KKL observer (Kazantzis–Kravaris–Luenberger), which transforms the original system x˙ = f(x, u) into a latent linear system, where the states can be reconstructed via an invertible transformation learned by a neural network. In parallel, the Modulating Functions (MF) method enables parameter estimation from differential models without requiring numerical differentiation. By multiplying the system’s equation with a test function and integrating over a time interval, the problem becomes a robust algebraic system, well-suited for noisy measurements. The proposed approach is validated on a Hill-type isometric muscle model driven by real EMG signals, including both healthy and pathological cases.