Nonlinear control of electrostatically shaped membrane with state and control constraints

A control scheme for electrostatically shaping a membrane is developed. The scheme achieves stabilization of the first mode of the membrane at a desired equilibrium, that may be open loop unstable, without violating specified state and control constraints. The ideas on the use of an "equilibrium-parametrized" family of Lyapunov functions to enforce state and control constraints in nonlinear systems go significantly beyond the specific example treated here. They can be widely applied in the context of Lyapunov-based designs, including nonlinear systems with disturbance inputs and uncertainties.