Autonomous behaviours decomposition and modal analysis

Abstract

In this paper, the notions of simple, very simple and indecomposable autonomous behaviour are introduced and characterized. By resorting to some recent results about the direct sum decomposition of (linear, time-invariant, differential) behaviours (Bisiacco and Valcher 2001), as well as to the well-known primary decomposition theorem for finitely generated modules (Hartley and Hawkes 1970), it is shown that every autonomous behaviour can be expressed as a direct sum of indecomposable components, which are just cyclic modules of order p v, for some irreducible polynomial p and some positive integer v . The non-uniqueness of this result is also discussed. Finally, this decomposition is interpreted in terms of modal analysis, and related to the results that can be obtained by mean of a state-space realization of the behaviour.