Finite‐dimensional adaptive observer design for Euler–Bernoulli beams with sensor delay

We extend the recent work of Selivanov and Fridman on control design for partial differential equations (PDEs) of Euler–Bernoulli type with viscous and Kelvin–Voigt damping. The extension is focused on adaptive observer design in the presence of sensor delay and parameter uncertainties. The considered formulation of the uncertainties makes it possible to account for the effect of structured disturbances entering the sensor and the state equations. The observer design approach combines the modal decomposition method and the decoupling transformation technique. Making use of these techniques, we design an adaptive observer that includes a (finite-dimensional) state estimator, a parameter estimator of least-squares type, and an auxiliary filter generating the regressor of the parameter estimator. The resulting estimation error system turns out to be an interconnected system driven by a signal depending on the neglected PDE modes. Exponential stability of the whole estimation error system is analyzed using suitable Lyapunov and Lyapunov–Krasovskii functionals. We show that exponential stability is guaranteed for not too large delay values, under a well-defined persistent excitation (PE) condition.