Versal deformation of realisable Markov parameters

Abstract

ABSTRACTLet be the set of sequences (L1,… , Ln), , admitting a minimal partial realisation of order d. To each , we associate two sequences of integers with r1 ≥ r2 ≥ … ≥ rβ > 0 = rβ+1 = … = rn and with s1 ≥ s2 ≥ … ≥ sα > 0 = sα+1 = … = sn called the partial Brunovsky column and row indices of L, respectively. Let be the subset of formed by the sequences L for which α + β ≤ n. Let Σco be the set of matrix triples with (F, G) controllable and (H, F) observable. We denote by Σco ≤ the subset of Σco formed by the triples which are minimal partial realisations of the sequences . For every ξ ∈ Σco ≤, we obtain a versal deformation of ξ corresponding to the action of the group , we show a method for obtaining a minimal partial realisation ξ of , and we derive a versal deformation of L from the obtained versal deformation of ξ.