| An adaptive optics system consists mainly of a wavefront sensor to detect optical aberrations, a control system to reconstruct the wavefront and compute a correction, and a deformable mirror to apply the correction. In this dissertation, the problem of optimal control of an adaptive optics system is investigated. A direct optimal control approach is used in the controller design.; The direct optimal control methodology developed for discrete parameter systems is extended in this study to distributed parameter systems, where the Rayleigh-Ritz method is used for both spatial and temporal variables. The displacement field is written as the product of spatial functions (mode shapes for a vibrating structure, and Zernike modes for deformable mirror) and the generalized coordinates. These generalized coordinates and the control input functions (voltages) are written as simple series expansions in time in terms of selected functions and unknown coefficients. Substitution of these selected functions and their variations into Hamilton's law of varying action results in algebraic equations of motion (AEM) of the structure. These AEM are then considered as the algebraic state equations where the unknown expansion coefficients of the time series (assumed time-modes) for the generalized coordinates are recognized as the states and those of the input functions are recognized as the controls.; Using the space-time assumed mode method, the usual variational optimal control problem is transformed into an equivalent algebraic problem. Optimal solutions are then obtained in a closed form and the solution is a global optimum within the time period considered. The solution procedure does not lead to any Ricatti equation or alike. The direct method proved to be simple, computationally efficient, attractive from implementation point of view, and it is general and allows a deterministic modelling of many physical problems.; Applied to active vibration control of plates with piezoelectric transducers, the direct methodology exhibits results similar to those obtained through conventional methods. Active shape control of a deformable mirror using the direct approach results in high performance of the controller. The method allows direct control of Zernike modes, and highlights the relationship between the control inputs and Zernike modes through an algebraic controllability measurement index. Robustness of the controller is shown through simulation of smooth and severe random variations of the optical aberrations.; In the same line of thought, a space-time finite element method is developed and applied to structural optimal control problems. Finite element method is used for both spatial and temporal discretizations. The unique feature of this method is its ability to analyse the structure-control interaction in the same mathematical framework, which allows simultaneous control and structural model design iterations. However, due to its high dimensionality, the space-time finite element method is computationally less efficient than its counterpart assumed mode. |
