Control of constrained nonlinear systems: a case study

Presents a Lyapunov function approach to the control of nonlinear systems that are subject to pointwise-in-time constraints on state and control. This approach is applied to an electromechanical system that serves as a prototype for the first mode of an electrostatically shaped membrane. Electrostatically shaped membranes have been proposed as mirrors and antennas since the early 1960s because they can be used as lightweight reflectors for radar, radio, and optics applications. Lightweight reflectors are in demand, for example, in spacecraft applications where launch weight is a significant constraint. A thin, electrically conducting membrane is formed into a desired shape by electrostatic forces that are controlled by varying the electrical potential between the membrane and an electrode mounted below it. Because the membrane is under lateral in-plane tension and a uniform normal stress due to the electrostatic potential, it assumes a paraboloidal shape for optics applications. Since the focal length can be varied by changing the gap distance, electrostatically controlled membranes are particularly suitable for adaptive optics applications. Small focal lengths needed for many applications can be achieved if the gap distance between the membrane and the fixed plate is made sufficiently small.