Nonlinear Control of a Membrane Mirror Strip Actuated Axially and in Bending

The sliding mode technique is used to control the deformation of a membrane mirror strip augmented with two macrofiber composite bimorphs located near the ends of the strip. The first bimorph is actuated in bending and the second is actuated axially. The structure is modeled as an Euler–Bernoulli beam under tensile load and the macrofiber composite patches are modeled as monolithic piezoceramic wafers. To cast the system into a finitedimensional state-space form, the finite element method is used, and the model presented accounts for the dynamics of the augmented bimorphs. Themembrane strip is placed under uniform tension. Because one of the bimorphs acts axially, the resulting tension in themembrane strip is discontinuous at the location of this bimorph and, consequently, the obtained model is nonlinear. First, we validate the model experimentally by considering the system in its quasilinear state, then we consider the control problem. We formulate the regulation problem by using the sliding mode technique. Additionally, to allow coupling this system with an adaptive optics scheme, the shape-control problem is considered as well. The control law uses both actuators: the bending and axial bimorphs. However, a system singularity dictates using a switching command to avoid this singularity. Various examples are presented for the regulation and shape-control problems. The simulation results demonstrate the efficacy of the proposed control law.