Vibration control of Timoshenko smart structures using multirate output feedback based discrete sliding mode control for SISO systems

Abstract

In this paper, the modeling and design of a multirate output feedback based discrete sliding mode control scheme application for the vibration control of a smart flexible Timoshenko cantilever beam for a single input single output (SISO) case by retaining the first two dominant vibratory modes is presented. In our work, the effect of shear and axial displacement has been considered. The algorithm uses a fast output sampling based sliding mode control strategy that would avoid the use of switching in the control input and hence avoids chattering. This method does not need the measurement of the system states for feedback as it makes use of only the output samples for designing the controller. Thus, this methodology is more practical and easy to implement. Piezoelectric patches are bonded as sensor/actuator to the master structure at different locations along the length of the beam. The beam structure is modeled in the state space form using the concept of piezoelectric theory, the Timoshenko beam theory and the FEM technique and by dividing the beam into four finite elements and placing the piezoelectric sensor/actuator at one location as a collocated pair at a time, i.e., as surface mounted sensor/actuator, say, at finite element position 1 or 2 or 3 or 4, thus giving rise to four SISO models of the same smart structure plant. Controllers are designed for the above four models of the same plant by retaining the first two dominant vibratory modes. The performance of the smart system for higher modes (say three vibratory modes) is also investigated. The piezo sensor/actuator pair is moved from the free end to the fixed end of the beam. The effect of placing the sensor/actuator at various locations along the length of the beam is observed and the conclusions are drawn for the best performance (best model) and for the smallest magnitude of the control input required to control the vibrations of the smart flexible beam.