Disturbance Attenuation in the Euler–Bernoulli Beam with Viscous and Kelvin–Voigt Damping via Piezoelectric Actuators

We design a state-feedback controller, applied via piezoelectric actuators, that suppresses the effect of a distributed disturbance in the Euler-Bernoulli beam with viscous and Kelvin-Voigt damping. The controller is designed to improve performance on a finite number of modes. Its effect on the remaining (infinitely many) modes is analysed by constructing an appropriate Lyapunov functional, whose properties are guaranteed by the feasibility of linear matrix inequalities (LMIs). The LMIs allow us to design suitable controller gain and estimate the induced
$L^2$
gain. A numerical example demonstrates how this modal decomposition approach leads to a controller that significantly reduces the
$L^2$
gain.