Flatness-based deformation control of a 1-dimensional microbeam with in-domain actuation

This paper addresses the problem of asymptotic tracking control of a micro-beam described by Euler-Bernoulli equation actuated by a one-dimensional array of micro-actuators. This system represents a one-dimensional model of deformable mirrors widely used in wavefront control and optical aberration correction in adaptive optics systems. The proposed control that allows steering the mirror along smooth reference trajectories consists in first mapping the nonhomogeneous partial differential equation model of the system into a standard boundary control form. Then, a combination of the methods of energy multiplier and flatness-based motion planning is used so that the closed-loop system is stabilized around reference trajectories. The viability and the applicability of the developed scheme is demonstrated through numerical simulations.