Regularity of the Kronecker stratification of realisable Markov parameters

Let L n ( d ) <= be the set of sequences of real matrices L = ( L 1 , horizontal ellipsis , L n ) , L i is an element of R p x m , i = 1 , horizontal ellipsis , n , admitting a minimal partial realisation of order d and such that the sum of their largest partial row and column Kronecker indices is smaller than or equal to n. This set is a differentiable manifold which can be stratified according to the partial row and column Kronecker indices of the sequences L. Let sigma c o be the set of triples ( F , G , H ) such that ( F , G ) is controllable and ( H , F ) is observable. This set can also be stratified according to the Brunovsky indices of controllability and observability of ( F , G ) and ( H , F ) , respectively. In this paper we show that this stratification is Whitney regular and from it the regularity of the considered stratification of L n ( d ) <= will be proved. The study of generic families of elements in L n ( d ) <= or in sigma c o is also included.