Parametrization of (C,A)-invariant subspaces

On observability subspaces

“…Proposition 4 is the key to the parametrization of all conditioned invariant subspaces of a given observable pair (C, A), that can be taken, without loss of generality, to be in dual Brunovsky form. Again, the basic results are those of Hinrichsen et al (1981) with extensions given in Fuhrmann and Helmke (2001). As a result of the above, all information, up to similarity, on the conditioned invariant subspace is, in principle, derivable from the polynomial matrices D(z) and H(z).…”

“…In the matrix case, the degree conditions are replaced by conditions on the Wiener-Hopf factorization indices. In order to overcome the nonuniqueness issue, we look for a submodule of F½z p that is uniquely determined by V. This can be done and in this we follow Hinrichsen et al (1981), see also the discussion in Fuhrmann and Helmke (2001) from which the following is quoted. …”

“…Corollary 1 then yields In order to gain some intuition, we consider a relatively simple example. We use the parametrization of the set of all conditioned invariant subspaces of X D , given in Hinrichsen et al (1981) or Fuhrmann and Helmke (2001). In this approach the set of conditioned invariant is decomposed into cells depending on ordered, reduced observability indices.…”

“…Furthermore, there is no uniqueness in such a representation. Using techniques originating in Hinrichsen et al (1981), and further developed in Fuhrmann and Helmke (2001), we can actually parametrize all tight subspaces for which the transversal intersection representation (43) holds. In Fuhrmann and Helmke (2001), tight conditions invariant subspaces were introduced and, with respect to the (C, A) defined via the shift realization associated with the non-singular polynomial matrix D 2 F½z pÂp , several alternative characterizations of tightness were given.…”

Conditioned Invariant Subspaces and the Geometry of Nilpotent Matrices

“…spectral factorization, linear quadratic control, H ∞ and game theory, as well as observer theory, filtering and estimation. However, it is only until recently, that first attempts have been made towards a better understanding of the geometry of the set of conditioned invariant subspaces Inv k (C, A); see [9], [10], [19], [6], [17]. The recent Ph.D. thesis [29] contains a comprehensive summary.…”