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Regularity of the Kronecker stratification of realisable Markov parameters. (English) Zbl 1461.93090

Summary: Let \(\mathcal{L}_n(d)^\leq\) be the set of sequences of real matrices \(L=(L_1,\dots,L_n)\), \(L_i\in\mathbb{R}^{p\times m}\), \(i=1,\dots,n\), 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 \(\Sigma^{co}\) 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 \(\mathcal{L}_n(d)^\leq\) will be proved. The study of generic families of elements in \(\mathcal{L}_n(d)^\leq\) or in \(\Sigma^{co}\) is also included.

MSC:

93B25 Algebraic methods
93B05 Controllability
93B07 Observability
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