On the parametrization of conditioned invariant subspaces and observer theory

The present paper is an in depth analysis of the set of conditioned invaria nt subspaces of a given observable pair (l,A). We do this analysis in two d ifferent ways, one based on polynomial models starting with a characterizat ion obtained in [P.A. Fuhrmann, Linear Operators and Systems in Hilbert Spa ce, 1981; IEEE Trans. Automat. Control AC-26 (1981) 284], the other being a state space approach. Toeplitz operators, projections in polynomial and ra tional models, Wiener-Hopf factorizations and factorization indices all app ear and are tools in the characterizations. We single out an important subc lass of conditioned invariant subspaces, namely the tight ones which alread y made an appearance in [P.A. Fuhrmann, U. Helmke, Systems Control Lett. 30 (1997) 217], a precursor of the present paper. Of particular importance fo r the study of the parametrization of the set of conditioned invariant subs paces of an observable pair (C, A) is the structural map that associates wi th any reachable pair, with only the input dimension constrained, a uniquel y determined conditioned invariant subspace. The construction of this map u ses polynomial models and the shift realization. New objects, the partial o bservability and reachability matrices are introduced which are needed for the state space characterizations. Kernel and image representations for con ditioned invariant subspaces are derived. Uniqueness of a kernel representa tion of a conditioned invariant subspace is shown to be equivalent to tight ness. We pass on to an analysis and derivation of the Kronecker-Hermite can onical form for full column rank, rectangular polynomial matrices. This ext ends the work of A.E. Eckberg (A characterization of linear systems via pol ynomial matrices and module theory, Ph.D. Thesis, MIT, Cambridge, MA, 1974) , G.D. Forney [SIAM J. Control Optim. 13 (1973) 493] and D. Hinrichsen H.E Munzner and D. Pratzel-Wolters [Systems Control Left. I (1981) 192]. We pro ceed to give a parametrization of such matrices. Based on this and utilizin g insights from D. Hinrichsen et al. [Systems Control Leb. 1 (1981) 192], w e parametrize the set of conditioned invariant subspaces. We relate this to image representations, making contact with the work of J. Ferrer et al. [L inear Algebra Appl. 275/276 (1998) 161; Stratification of the set of genera l (A, B)-invariant subspaces (1999)]. We add a new angle by being able to p arametrize the set of all reachable pairs in a kernel representation, this via an embedding result for rectangular polynomial matrices in square ones. As a by product we redo observer theory in a unified way, giving a new ins ight into the connection to geometric control and to the stable partial rea lization problem. (C) 2001 Elsevier Science Inc. All rights reserved.