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. / Graduate
| Identifer | oai:union.ndltd.org:uvic.ca/oai:dspace.library.uvic.ca:1828/8251 |
| Date | 08 June 2017 |
| Creators | Abdelkader, Chellabi |
| Contributors | Stepanenko, Yury, Dost, Sadik |
| Source Sets | University of Victoria |
| Language | English, English |
| Detected Language | English |
| Type | Thesis |
| Rights | Available to the World Wide Web |
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