# Algebraic and Geometric Methods and Problems for Implicit Linear Systems

Vafiadis, Dimitris (1995). Algebraic and Geometric Methods and Problems for Implicit Linear Systems. (Unpublished Doctoral thesis, City, University of London)

## Abstract

This thesis investigates a number of problems of the implicit linear systems framework.

First, the problem of realisations of nonproper transfer functions is considered. The main result obtained here is the generalisation of the realisation method from MFDs to the case of the nonproper transfer functions. The obtained realisations are singular systems. The method treats both finite and infinite frequency behaviour in a unified way and generalises the results related to the minimality of the realisation and coprimeness and column reducedness of the MFD. Furthermore, it displays transparently the relation between the extended MacMillan degree of the transfer function and the minimal realisation.

The next problem considered is the problem of canonical forms of minimal singular systems under restricted system equivalence transformations. For systems with outputs a canonical form is obtained and it is shown that it is directly related to the echelon form of the composite matrix of an MFD of the transfer function of the system. This result is a direct generalisation of the results of Popov and Forney for strictly proper systems. The canonical form obtained is of Popov type and may be considered as a direct generalisation of the well known form for strictly proper systems. The second canonical form is for systems without outputs. A Popov type canonical form for a class of these systems is obtained. This class is that of systems with equal reachability indices. For both canonical forms, the sequence of the transformations yielding the canonical description is described in detail. In the general case of systems without outputs a semi canonical Popov type form is obtained.

Another problem considered in the thesis is the problem of first order realisations of autoregressive equations within the external equivalence framework. An alternative to the existing methods is provided; in fact, the proposed method is simpler than the existing ones and allows the derivation of the realisation by inspection of the autoregressive equations. A generalisation of the observability indices is proposed for nonsquare descriptor systems and their connection to the autoregressive equations is established.

The problem of model matching for implicit systems is considered next. This is a generalisation of the model matching problem for systems described by transfer functions. Here a controller is interconnected to the given plant such that the overall system has a desired external behaviour. The problem is studied within the framework of external and A-external (input-output) equivalence. Necessary as well as sufficient conditions for the solvability of the problem are derived and the equations of the controllers are found in a constructive way.

The last problem considered here is the generalised dynamic cover problem of geometric theory i.e. the problem of finding the family of (A, B)-invariant subspaces covering a given subspace. This problem is formulated here by using the matrix pencil approach of the geometric theory. This approach allows the unification of the problem for state-space and nonsquare descriptor systems. An extension of the problem to the case of infinite spectrum spaces is also obtained. The solution of the above problems is reduced to the solution of appropriately defined systems of linear equations. Finally, an alternative method for the solution involving systems of multilinear equations is proposed using the mathematical tool of Groebner bases.

Publication Type: Thesis (Doctoral) Q Science > QA Mathematics School of Science & Technology > Engineering > Electrical & Electronic EngineeringDoctoral ThesesSchool of Science & Technology > School of Science & Technology Doctoral Theses  Preview
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