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# Solution of optimization problems with spatial symmetry and applications to adaptive optics

#### Abstract

The essential characteristics of large systems is their high dimensionality due to which conventional control techniques fail to give reasonable solutions with reasonable computational efforts. A number of large systems encountered in practice are composed of subsystems with similar dynamics interconnected in a symmetrical fashion. The analysis and control of a large system with these particular features must take advantage of the existing structural properties to achieve computational simplifications of the overall problem. The focus of this thesis is the feedback design and analysis of large systems possessing the property of spatial symmetry. Specifically, the problems of controller design and analysis for infinite dimensional toeplitz systems and their finite dimensional analogs, circulant systems, are studied. These spatially symmetric systems are special classes of large systems.^ The first part of this thesis is focused on the development of formal controller design methodologies which take advantage of the properties of the circulant matrices. The key to this development is the use of the FFT algorithm to diagonalize circulant matrices. The resulting controller design methodologies are computationally attractive and easily applicable to large systems with circulant symmetry. More specifically, the H$\sb2$ and H$\sb{\infty}$ controller synthesis problems are studied in detail and are shown to decompose into lower order independent problems.^ The second part of this work concentrates on proving that certain finite order toeplitz systems are asymptotically equivalent in an appropriate sense to circulant systems. This result justifies the use of circulant control design techniques for certain toeplitz systems. Moreover, the closed loop effects of controlling a toeplitz system with a controller designed for its asymptotically equivalent circulant system are analyzed.^ The application of the developed theoretical results to a realistic example is the focus of the last part of the thesis. The adaptive optics system used in this example is modeled by a transfer function matrix with toeplitz symmetry. The computational efficiency of the controller design methodologies developed in this thesis is illustrated by designing a series of controllers for this system. ^

#### Subject Area

Engineering, Aerospace|Engineering, Electronics and Electrical|Engineering, System Science

#### Recommended Citation

Nikolaos Athanasios Denis, "Solution of optimization problems with spatial symmetry and applications to adaptive optics" (January 1, 1998). Doctoral Dissertations Available from Proquest. Paper AAI9909160.
http://scholarworks.umass.edu/dissertations/AAI9909160

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