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Algebrogeometric and topological methods in control theory

Leventides, J. (1993). Algebrogeometric and topological methods in control theory. (Unpublished Doctoral thesis, City, University of London)

Abstract

The aim of this thesis is to provide a unifying framework and tools for the study of a number of Control Theory problems of the determinantal type. These problems are known as Frequency Assignment Problems (FRT) and they include the constant, dynamic, pole, zero assignment by centralised as well as decentralised output feedback and the zero assignment problems via squaring down. It has been shown [Kar.1],[Gia.2] that all such problems may be formulated under the unifying framework of the Determinantal Assignment Problem (DAP), and it can be studied using tools from exterior algebra and algebraic geometry. The main objective of this thesis is to develop further the DAP framework, unify it with other algebrogeometric approaches and develop issues related to computation and parametrisation of solutions when such solutions exist.

The natural setup for the study of solutions of the DAP framework has been the intersection theory of projective varieties. This has been extended by developing the topological properties of the pole, zero placement maps and introducing an equivalent formulation for real intersection based on cohomology theory. The properties of this map, with respect to standard system invariants are also established. This approach allows the derivation of new conditions for constant pole, zero assignment with centralised and decentralised controllers, using conditions based on the height of an appropriate cohomology class. Affine algebraic geometry methods are also used for the derivation of partial results for the dynamic case corresponding to PI and OBD controllers.

An entirely new approach for the study of solvability of DAP, as well as computation of solutions is introduced in terms of the notion of global linearisation of the corresponding pole, zero assignment map around a degenerate point. This is based on the special “blow up” property of the pole placement map at degenerate feedbacks and permits the reduction of the overall DAP to a globally linear problem, the solvability of which is defined by the properties of a new local invariant, the “blow up” matrix.

Publication Type: Thesis (Doctoral)
Subjects: Q Science > QA Mathematics
Departments: School of Science & Technology > Mathematics
School of Science & Technology > School of Science & Technology Doctoral Theses
Doctoral Theses
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