Modeling and Control of a 2-D Membrane Mirror With a PZT Bimorph

The future of space satellite technology lies in ultra-large mirrors and radar apertures for significant improvements in imaging and communication bandwidths. The availability of optical-quality membranes drives a parallel effort for structural models that can capture the dominant dynamics of large, ultra-flexible satellite payloads. Unfortunately, the inherent flexibility of membrane mirrors wrecks havoc with the payload's on-orbit stability and maneuverability. One possible means of controlling these undesirable dynamics is by embedding active piezoelectric ceramics near the boundary of the membrane mirror. In doing so, active feedback control can be used to eliminate detrimental vibration, perform static shape control, and evaluate the health of the structure. In the present work, a piezoceramic wafer was attached in a bimorph configuration near the boundary of a tensioned rectangular membrane sample. A finite element model of the system was developed to capture the relevant system dynamics from 0 – 500 Hz. The finite element model was compared to experimental results with fair agreement. Using the validated finite element models, structural control using Linear Quadratic Regulator (LQR) control techniques were then used to demonstrate effective vibration control. Typical results show that less than 12 V of actuation voltage is required to eliminate detrimental vibration of the membrane samples in less than 15 ms. The functional gains of the active system are also derived and presented. These spatially descriptive control terms dictate favorable regions within the membrane domain to place sensors.