Transfer functions for piezoelectric control of a flexible beam

Abstract

Designers of lightweight flexible space structures are faced with the significant challenge of achieving position and shape control while avoiding structural vibration. A potentially attractive approach to the vibration and shape control problems lies in the use of piezoelectric films as actuators and sensors. Piezoelectric materials deform when exposed to an electric field, or conversely, when deformed they produce an electric field. This, property considered along with the fact that they are lightweight, inexpensive and exhibit a very wide dynamic response bandwidth, make piezoelectric films attractive as actuators and sensors for certain applications. In this paper, some fundamental relationships for beams incorporating piezoelectric film actuators and sensors are examined. The differential equation of motion for a beam with piezoelectric film bonded to both sides is used to develop Laplace domain transfer function models of the system. These transfer functions are exact Laplace domain representations of the system equations of motion. The transfer functions are cast into a closed rational form using Maclaunn series expansions representing a specific number of modes. In this form, the transfer functions lend themselves to classical control analysis. It is shown that the transfer function relating a voltage applied to a full coverage actuating layer, to the voltage induced in a full coverage sensing layer on the opposite beam face, behaves like a classic colocated system with alternating poles and zeros and accordingly the system is easy to stabilize with low order compensation. In contrast to this result, it is shown that in spite of the effective colocation of actuator and sensor in the case of the transfer function from actuating voltage to tip position, the desirable alternating pole/zero pattern is not exhibited due to incompatibility of actuating and sensing signals. This result is verified experimentally.