Dynamic stability analysis of finite element modeling of piezoelectric composite plates

The dynamic stability of negative-velocity feedback control of piezoelectric composite plates using a finite element model is investigated. Lyapunov’s energy functional based on the derived general governing equations of motion with active damping is used to carry out the stability analysis, where it is shown that the active damping matrix must be positive semi-definite to guarantee the dynamic stability. Through this formulation, it is found that imperfect collocation of piezoelectric sensor/actuator pairs is not sufficient for dynamic stability in general and that ignoring the in-plane displacements of the midplane of the composite plate with imperfectly collocated piezoelectric sensor/actuator pairs may cause significant numerical errors, leading to incorrect stability conclusions. This can be further confirmed by examining the complex eigenvalues of the transformed linear first-order state space equations of motion. To overcome the drawback of finding all the complex eigenvalues for large systems, a stable state feedback law that satisfies the second Lyapunov’s stability criteria strictly is proposed. Numerical results based on a cantilevered piezoelectric composite plate show that the feedback control system with an imperfectly collocated PZT sensor/actuator pair is unstable, but asymptotic stability can be achieved by either bonding the PZT sensor/actuator pair together or changing the ply stacking sequence of the composite substrate to be symmetric. The performance of the proposed stable controller is also demonstrated. The presented stability analysis is of practical importance for effective design of asymptotically stable control systems as well as for choosing an appropriate finite element model to accurately predict the dynamic response of smart piezoelectric composite plates.