Modelling and control of electrostatically actuated microcantilever arrays

In this dissertation we present the design, modelling and analysis of a device consisting of an array of tightly packed electrostatically actuated microcantilevers. More precisely, each microcantilever in the array constitutes the movable plate of a capacitor and its displacement is controlled by the voltage applied across the plates. We show that, if we take the current as the measured signal, the dynamics of a single cantilever are governed by a special second order linear periodic differential equation, called the Mathieu equation. As for the array configuration, by explicitly incorporating the mechanical and electrostatic coupling into the dynamical equations, we show that the system is described by a set of coupled Mathieu equations. The structural properties of the array, with actuators and sensors distributed over a regular lattice, allow us to cast it in the class of spatially invariant distributed parameter systems, which bears important consequences in the rest of the analysis. Both models, for the single cantilever and for the array, are validated through an extensive set of experiments, which demonstrate very good agreement between theoretical predictions and experimental findings.; We propose a novel sensing scheme for the cantilevers' displacement, based on the design of an optimal state observer whose input is the current through the capacitive cantilevers. We demonstrate in simulations its excellent performance in reconstructing the cantilevers displacement within few nanometers from its actual value. We propose a procedure that uses the optimal design as an analysis tool to tune both the frequency of excitation of the cantilevers and the parameters of a reduced order observer. This way we obtain a system with the best achievable noise rejection properties, measured in terms of $H\infty$-norm of the closed loop system.; The problem of controlling tightly packed arrays of microcantilevers is presented in two case studies, in which the microcantilever arrays considered differ in the model of the coupling interactions. We consider the design a controller for the electrostatically actuated array that, by using the estimate of the cantilever displacement provided by an observer, is able to decouple the cantilevers' dynamics. Simulation results are provided to illustrate its performance. Finally, we present an optimal $H2$ control design for different array of cantilevers and discuss the synthesis of suboptimal controllers.