Geometric and algebraic properties of minimal bases of singular systems

Abstract

This is the accepted version of the paper.This version of the publication may differ from the final published version. Abstract For a general singular system Se[E, A, B] with an associated pencil T (S), a complete classification of the right polynomial vector pairs (x(s), u(s)), connected with the Nr{T (s)} rational vector space, is given according to the proper-nonproper property, characterising the relationship of the degrees of those two vectors. An integral part of the classification of right pairs is the development of the notions of canonical and normal minimal bases for Nr{T (s)} and Nr{R(s)} rational vector spaces, where R(s) is the state restriction pencil of Se[E, A, B]. It is shown that the notions of canonical and normal minimal bases are equivalent; the first notion characterises the pure algebraic aspect of the classification, whereas the second is intimately connected to the real geometry properties and the underlying generation mechanism of the proper and nonproper state vectors x(s). The results presented, highlight both the algebraic and geometric properties of the partitioning of the set of reachability indices; the classification of all proper and nonproper polynomial vectors x(s) induces a corresponding classification for the reachability spaces to proper-nonproper and results related to the possible dimensions feedbackspectra assignment properties of them are also given. The classification of minimal bases introduces new feedback invariants for singular systems, based on the real geometry of polynomial minimal bases, and provides an extension of the standard theory for proper systems [2] to the case of singular systems. Nicos Karcanias dedicates this paper to Alistair MacFarlane FRS who has motivated him to explore the relationships between geometric and algebraic methods in feedback control. Alistair MacFarlane was throughout his career interested in exploring the links between geometry and frequency response methods. Permanent repository link