Vibration control of a piezoelectric cantilever smart beam by ℒ1 adaptive control system

In this paper, the modelling and design of an $ \mathcal {L}_1 $ state feedback control system applied for vibration control of a smart piezoelectric flexible Euler–Bernoulli cantilever beam is presented. For a Single-Input Single-Output (SISO) case by retaining the first two dominant vibratory modes, the dynamics of the system is presented. Classical $ \mathcal {L}_1 $ adaptive control law, with time-varying parameters and in presence of disturbance, is employed to suppress the vibration of the beam. Three Piezoelectric patches, two as actuators and the other as a sensor, are bonded to the structure at the support of the beam and along the length of the beam. The beam structure is modelled in the state space form using the concept of piezoelectric theory, the Euler–Bernoulli beam theory and the Finite Element technique. Also, for comparing the performance of the proposed controller, PID and LQR control systems are applied. Simulation results, first for constant parameters and then for time-varying parameters and disturbance, represent the better performance of the proposed control system in comparison to the other mentioned controllers