Boundary input-to-state stabilization of a damped Euler-Bernoulli beam in the presence of a state-delay

TL;DR

This paper develops a boundary feedback control strategy for stabilizing a damped Euler-Bernoulli beam with time-varying delays, using spectral reduction and LMIs to ensure stability and robustness against disturbances.

Contribution

It introduces an LMI-based method for boundary stabilization of delayed Euler-Bernoulli beams, including robustness analysis with disturbances.

Findings

Stability guaranteed for maximum delay amplitude using LMIs

Exponential stability of the infinite-dimensional system established

Robustness against distributed and boundary perturbations demonstrated

Abstract

This paper is concerned with the point torque boundary feedback stabilization of a damped Euler-Bernoulli beam model in the presence of a time-varying state-delay. First, a finite-dimensional truncated model is derived by spectral reduction. Then, for a given stabilizing state-feedback of the delay-free truncated model, an LMI-based sufficient condition on the maximum amplitude of the state-delay is employed to guarantee the stability of the closed-loop state-delayed truncated model. Second, we assess the exponential stability of the resulting closed-loop infinite-dimensional system under the assumption that the number of modes of the original infinite-dimensional system captured by the truncated model has been selected large enough. Finally, we consider in our control design the possible presence of a distributed perturbation, as well as additive boundary perturbations in the control inputs. In this case, we derive for the closed-loop system an exponential input-to-state estimate with fading memory of the distributed and boundary disturbances.