Geometric and algebraic properties of minimal bases of singular systems
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 polynomi...