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

• Published in: Systems & control letters Vol. 201; p. 106096
• Main Authors: Wu, Ruixin, Xiao, Yu, Xu, Xiaodong
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.07.2025
• Subjects: Adaptive observer; Euler–Bernoulli beam; Nonlinear dynamics; Parameter uncertainty; Nonlinear dynamics; Euler–Bernoulli beam; Adaptive observer; Parameter uncertainty
• ISSN: 0167-6911
• DOI: 10.1016/j.sysconle.2025.106096

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. •A novel adaptive boundary observer is developed for a class of Euler–Bernoulli beam equations with unknown parameters and interior nonlinearity.•The adaptive boundary observer for the Euler–Bernoulli beam system is designed based on the finite-dimensional backstepping-like transformation.•The proposed observer consists of three main parts: a state observer; a parameter adaptive update law; and an auxiliary filter. The exponential convergence of the estimation errors is obtained.