Infinite-Dimensional Modelling and Control of a MEMS Deformable Mirror with Applications in Adaptive Optics

Le contrôle de déformation est un problème émergent dans les micro structures intelligentes. Une des applications type est le contrôle de la déformation de miroirs dans l'optique adaptative dans laquelle on oriente la face du miroir selon une géométrie précise en utilisant une gamme de micro-vérins afin d'éliminer la distortion lumineuse. Dans cette thèse, le problème de la conception du contrôle du suivi est considéré directement avec les modèles décrits par des équations aux dérivées partielles définies dans l'espace de dimension infinie. L'architecture du contrôleur proposée se base sur la stabilisation par retour des variables et le suivi des trajectoires utilisant la théorie des systèmes différentiellement plats. La combinaison de la commande par rétroaction et la planification des trajectoires permet de réduire la complexité de la structure du contrôleur pour que ce dernier puisse être implémentée dans les microsystèmes avec les techniques disponibles de nos jours. Pour aboutir à une architecture implémentable dans les applications en temps réel, la fonction de Green est considérée comme une fonction de test pour concevoir le contrôleur et pour représenter les trajectoires de référence dans la planification de mouvements.

Abstract

Deformation control is an emerging problem for micro-smart structures. One of its exciting applications is the control of deformable mirrors in adaptive optics systems, in which the mirror face-sheet is steered to a desired shape using an array of micro-actuators in order to remove light distortions. This technology is an enabling key for the forthcoming extremely large ground-based telescopes. Large-scale deformable mirrors typically exhibit complex dynamical behaviors mostly due to micro-actuators distributed in the domain of the system which in particular complicates control design. A model of this device may be described by a fourth-order in space/second-order in time partial differential equation for the mirror face-sheet with Dirac delta functions located in the domain of the system to represent the micro-actuators. Most of control design methods dealing with partial differential equations are performed on lumped models, which often leads to high-dimensional and complex feedback control structures. Furthermore, control designs achieved based on partial differential equation models correspond to boundary control problems. In this thesis, a tracking control scheme is designed directly based on the infinite-dimensional model of the system. The control scheme is introduced based on establishing a relationship between the original nonhomogeneous model and a target system in a standard boundary control form. Thereby, the existing boundary control methods may be applicable. For the control design, we apply the tool of differential flatness to a partial differential equation system controlled by multiple actuators, which is essentially a multiple-input multiple-output partial differential equation problem. To avoid early lumping in the motion planning, we use the properties of the Green's function of the system to represent the reference trajectories. A finite set of these functions is considered to establish a one-to-one map between the input space and output space. This allows an implementable scheme for real-time applications. Since pure feedforward control is only applicable for perfectly known, and stable systems, feedback control is required to account for instability, model uncertainties, and disturbances. Hence, a stabilizing feedback is designed to stabilize the system around the reference trajectories. The combination of differential flatness for motion planning and stabilizing feedback provides a systematic control scheme suitable for the real-time applications of large-scale deformable mirrors.