Sliding Mode Vibration Suppression Control Design for a Smart Beam

Abstract

Active vibration control is the main problem in different structure. Smart material like piezoelectric make a structure smart, adaptive and self-controlling so, they are effective in active vibration control. In this paper piezoelectric elements are used as sensors and actuators in flexible structures for sensing and actuating purposes, and to control the vibration of a cantilever beam by using sliding mode control. The sliding mode controller (SMC) is designed to attenuate the vibration induced by initial tip displacement which is equal to 15 mm.  It is designed based on the balance realization reduction method where three states are selected for the reduced model from the 24^th states that describe the cantilever beam according to the FEM. These states are most controllable and observable. The stability and control performance for the proposed SMC are proved using candidate Lyapunov function and the equivalent control concept. The control spillover, which is the sources of instability, is completely avoided as ensured within the control performance proof.

            Numerical simulations are preformed to test the vibration attenuation ability of the proposed SMC. For 15 mm initial tip displacement, the piezoelectric actuator was found able to reduce the tip displacement to about (0.2) mm within (2.5 s), while it is equal to (3.5) mm with the open loop case. Moreover, the induced chattering in system response, due to the discontinuous control action, is removed by approximating the signum function by a continuous arctan function. As a result  a smoother response are obtained with the same control performance as can be shown in the sliding variable, the control input voltage and the tip displacement plots.