The equivalence structure of descriptor representations of systems with possibly inconsistent initial conditions
The authors deal with a concrete form of transformations under external equivalence for descriptor representations when solutions are considered in the space of impulsive-smooth distributions (DR). DRs are considered in the form: \(E\dot x=Ax+Bu, y=Cx+Du\), where \(E,A,B,C,D\) are constant matrices, \(E\) may be singular or nonsquare. The study of impulse-smooth behaviour was based on the pencil representations (PR) in the form: \(G\dot z=Fz, w=Hz\), where \(z\) is a vector of external variables, by \textit{A. H. W. Geerts} and \textit{J. M. Schumacher} [Automatica 32, No. 5, 747-758 (1996; Zbl 0865.93029) and No. 6, 819-832 (1996; Zbl 0865.93030)]. A behaviour, i.e. the set of time trajectories of external variables \(y, u\), is associated to a PR in two different ways referred to as the conventional style and the unconventional style. The styles differ in the way of incorporation of initial conditions. The results are presented for both cases. Algorithms for transforming DRs to PRs, and vice versa, are given. Necessary and sufficient conditions for the minimality of DRs are derived. Minimality means that both number of rows of \(E\) and columns of \(A\) are minimal among all equivalent DRs. Equivalence relations of minimal DRs are presented.
scientific article; zbMATH DE number 1547888
scientific article