Regressor-free adaptive vibration control of constrained smart beams with axial stretching

Abstract

This paper proposes an adaptive approximation control AAC technique (or so-called regressor-free adaptive control) for vibration regulation of constrained flexible smart beams with axial stretching. The key idea of the AAC is to control/regulate the target dynamic system while estimating the uncertainty using weighting and basis function terms with guaranteed stability based on Lyapunov theory. Accordingly, the dynamic equation of transverse vibration of the pinned–pinned smart beam is derived considering the effect of axial stretching. Due to the presence of a coupled tension-bending effect, a nonlinearly coupled cubic stiffness term appears in beam modelling making the dynamic system highly nonlinear. The resulted partial differential equation of the vibrating smart beam is discretized into definite N-mode shapes (definite degrees-of-freedom) using the Galerkin approach and a standard multi–input–multi–output ordinary differential equations system is established. Then two decoupled nonlinear control algorithms are designed based on the AAC for vibration attenuation of the nonlinear vibrating beam system. A pinned–pinned piezoelectrically-actuated/sensed flexible beam is simulated and the results show the validity of the proposed control architecture.