Versal deformation of realisable Markov parameters

摘要

Let be the set of sequences (L-1, ... , L-n), , admitting a minimal partial realisation of order d. To each , we associate two sequences of integers with r(1) >= r(2) >= ... >= r(beta) > 0 = r(beta+1) = ... = r(n) and with s(1) >= s(2) >= ... >= s(alpha) > 0 = s(alpha+1) = ... = s(n) called the partial Brunovsky column and row indices of L, respectively. Let be the subset of formed by the sequences L for which alpha + beta <= n. Let sigma(co) be the set of matrix triples with (F, G) controllable and (H, F) observable. We denote by sigma(co <=) the subset of sigma(co) formed by the triples which are minimal partial realisations of the sequences . For every xi is an element of sigma(co <=), we obtain a versal deformation of xi corresponding to the action of the group , we show a method for obtaining a minimal partial realisation xi of , and we derive a versal deformation of L from the obtained versal deformation of xi.