Robust vibration control of a functionally graded cracked beam

In this paper، vibration control of a cracked، functoinally graded، uncertain beam allocated in a thermal environment has been investigated. For this purpose، piezoelectric patches are used as sensors to measure the displacement of the beam and also as actuators to apply control forces. In this way، firstly، partial differential equation governing the dynamics of the system is derived by considering the Euler-Bernoulli assumption using Lagrange method. Approximate solution of eigenvalue equation is achieved using Rayleigh–Ritz method. After that، time dependent ordinary differential equations is obtained using Galerkin projection scheme and then represented in the state-space form. Based on this model، a robust observer-based output feedback controller is designed for this continuous-time model. In this regard، controller and observer gains are designed by a Lyapunov-based method. This procedure is done by solving a set on linear matrix inequalities. Simulation studies show the effectiveness of the proposed method.