Geometric and algebraic properties of minimal bases of singular systems

Abstract

For a general singular system with an associated pencil T(S), a complete classification of the right polynomial vector pairs , connected with the 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 and rational vector spaces, where R(s) is the state restriction pencil of . 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 . The results describe the algebraic and geometric dimensions of the invariant partitioning of the set of reachability indices of singular systems. The classification of all proper and nonproper polynomial vectors induces a corresponding classification for the reachability spaces to proper–nonproper and results related to the possible dimensions feedback-spectra 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 (Warren, M.E., & Eckenberg, A.E. (1975).

Keywords:

• singular systems
• algebraic systems theory

Acknowledgements

The paper contributes to the algebraic theory of singular systems. The author acknowledges the contribution of his research students Helen Eliopoulou and Dimitris Vafiadis; their research contributions have contributed in the shaping of ideas behind the current publication. The author dedicates this paper to Alistair MacFarlane FRS who has motivated him to explore the relationships between geometric and algebraic methods in feedback control. Throughout his career Alistair MacFarlane was interested in exploring the links between geometry and frequency response methods.