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Guaranteed Accuracy Computations in Systems and Control

Kanno M.

December 2003

Abstract

The purpose of this dissertation is to introduce and explore the idea of `guaranteed accuracy' or `validated numerical methods' in systems and control and to demonstrate that it is possible to make progress towards it. Computer algebra systems and interval methods are major tools. Algorithms with guaranteed accuracy are developed for some analysis indices and some controller synthesis problems. Firstly, the necessity and importance of guaranteed accuracy computation are discussed. Also conceptual and algorithmic framework for the guaranteed accuracy solution is developed. A formal definition of the guaranteed accuracy computation is presented. Further some techniques, e.g., square-free factorisation, polynomial root localisation, interval methods, which have potential for guaranteed accuracy are reviewed. An algorithm for the H_2 norm computation with guaranteed accuracy is realised by rewriting a Lyapunov equation into a set of linear equations and then solving it. It is shown that the L_\infty norm can be computed with guaranteed accuracy by way of square-free factorisation and guaranteed accuracy real root localisation. The \nu-gap computation relies on the Routh-Hurwitz test and the L_\infty norm computation which can be carried out with guaranteed accuracy and thus a guaranteed accuracy algorithm can be implemented. This dissertation further suggests algorithms for the H_2-optimal controller synthesis problem and the gap-optimal and suboptimal controller synthesis problems. A feature of the H_2-optimal controller useful to develop an algorithm with guaranteed accuracy is firstly shown and the H_2-optimal controller is shown to be computable with guaranteed accuracy by using the guaranteed accuracy polynomial spectral factorisation and the Bezout identity. In the the gap-optimal controller synthesis problem, it is shown that the underlying Nehari problem can be solved as an eigenvalue/eigenvector problem and additionally that the achievable generalised stability margin is computable with guaranteed accuracy. An algorithm is then suggested which calculates the gap-optimal controller with guaranteed accuracy. A method of dealing with discontinuities is also discussed. Finally, a gap-suboptimal controller computation algorithm with guaranteed accuracy is developed which partially exploits state space formulae. An alternative way of calculating the achievable generalised stability margin with guaranteed accuracy is also developed. Further research directions are also discussed.

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BibTex Entry

@PhdThesis{,: author = {Kanno M.},; school = {University of Cambridge},; title = {Guaranteed Accuracy Computations in Systems and Control},; year = {2003},; month = {December}
}



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