Collocative PD Control of Circular Plates with Shaped Piezoelectric Actuators/Sensors

Abstract

Abstract: In this paper, flexural vibrations of smart circular plates are considered. Distributed actuators and sensors are realized by means of spatially shaped piezoelastic layers. We use piezoelectric actuating layers shaped in order to annihilate deflections due to known external transverse forces. Such spatial shape functions correspond to the distribution of the static bending moment in the form of the so-called Marcus moment of the plate due to the external forces. When only the spatial distribution of the external forces is known, but their time evolution may be arbitrary, an automatic control system must be used in order to minimize the plate vibrations. To utilize the concept of collocated sensing, a shaped piezoelectric sensor is required that measures the so-called natural output. It is shown that the above shape function of the actuator can be used as the shape function of the sensor in order to achieve this goal. Hence, the shaped piezoelectric layer can be used as a self-sensing actuator without violating the requirements of collocated control. We develop the corresponding transfer function for the case of a clamped circular plate with a space-wise constant transverse force. This transfer function is used for the design of a self-sensing PD controller. It is proven that the energy of the closed-loop system becomes a positive definite function, its time derivative being negative semi-definite, such that the PD-controlled plate is stable. In a numerical study, output and input signals of the closed loop are discussed. This study successfully demonstrates the ability of the proposed method.