Adaptive boundary observer for Euler–Bernoulli beam equations with nonlinear dynamics and parameter uncertainties

Abstract

This paper focuses on designing boundary adaptive observers for systems that can be modeled as Euler–Bernoulli beams and are represented by fourth-order partial differential equations. Specifically, the objective is to estimate the entire state of the beam solely based on measurements taken at its boundaries. The difficulty of this study lies not only in the uncertain parameters contained in the boundary and domain of the beam but also in taking into account the interior nonlinearity in-domain. Unlike many observers in adaptive control frameworks that do not require accurate estimation of unknown parameters, our approach is dedicated to accurately estimating both the system state and the unknown parameters. The crucial element in the design process of the adaptive observer is the introduction of a kind of finite-dimensional backstepping-like transformation, based on which we can transform the observer error system into the desired system. Then, we can use common parameter estimation methods, allowing the design of the parameter adaptive law to be decoupled from the choice of the state estimator. Using Lyapunov stability analysis, we show that the observer converges exponentially under persistent excitation conditions. Numerical simulations also demonstrate the effectiveness of the observer.