Algebraic characterizations of almost invariance

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Abstract
This paper stresses the algebraic (as opposed to analytic) nature of the concept of almost invarianoe, which has been introduced by Willems (1980) as a means of studying asymptotic aspects of linear systems. Based on an interpretation' in terms of difference equations, it is shown that the basic properties and a number of characterizations of tho four fundamental classes of subspaces can be derived in a relatively simple and purely algebraic fashion. Among other things, this paper also extends the results of Hautus on the frequency-domain interpretation of ordinary ( A, B)-invariant subspaces, re-derives the pencil characterization of Jaffe and Karcanias (1981), and obtains a new ‘ hybrid ’ type of characterization. The last result also loads to a rank test for almost invariance.

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