A polynomial characterization of (A,B)-invariant and reachability subspaces

E. Emre, M.L.J. Hautus

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Introduction The geometric approach to linear system theory has proved very succesful in solving a variety of problems (see [14] for a detailed account of this theory). The principal concepts in this theory, which are instrumental in the description of many results, are (A,B)-invariant subspaces and reachability (controllability) subspaces. An alternative approach to linear system design has been developed in [11-13]. This theory depends to a large extent on polynomial matrix techniques. It is evident that a method for translating results of one theory to another is very desirable, because such a method would yield a better understanding of the relations between the two different approaches. This would be very useful, in particular since the geometric method may be viewed as exponent of the socalled "modern control theory" and the polynomial matrix method may be considered a generalization of the classical frequency domain methods. A number of papers with the objective of translating the results of geometric control theory into polynomial matrix terms have appeared (e.g. [1-3], [8-9]). It is the purpose of this paper to show that a very useful link between the two approaches can been based on the work of P. Fuhrmann ([6-8]). Specifically, it will be shown that using the state space model associated with a system matrix, introduced by Fuhrmann, one can give characterizations of the concepts of (A,B)-invariant subspaces and reachability subspaces in terms of polynomial matrices. This will be the subject of sections 3 and 5. An application of the polynomial characterization of (A,B)-invariant subspaces will be given in section 4, where it will be shown that the disturbance decoupling problem (see [14, Ch. 4]) and the exact model matching problem (see [13], [10], [5], [2]) are equivalent problems. In section 6, the concept of row properness defined in [12-13] is used to formulate a necessary and sufficient condition for the existence of a solution of the exact model matching problem and hence of the disturbance decoupling problem in terms of degrees of polynomial matrices. Also in section 6 a constructive characterization of the supremal (A,B)-invariant subspace and reachability space contained in ker C is given. The preliminary section 2 contains a short description of Fuhrmann's state space model in addition to some auxiliary results.
Original languageEnglish
Place of PublicationEindhoven
PublisherTechnische Hogeschool Eindhoven
Number of pages23
Publication statusPublished - 1978

Publication series

NameMemorandum COSOR
ISSN (Print)0926-4493

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