On the parametrization of conditioned invariant subspaces and observer theory

The present paper is an in depth analysis of the set of conditioned invariant subspaces of a given observable pair (C,A). We do this analysis in two different ways, one based on polynomial models starting with a characterization obtained in [P.A. Fuhrmann, Linear Operators and Systems in Hilbert Space, 1981; IEEE Trans. Automat. Control AC-26 (1981) 284], the other being a state space approach. Toeplitz operators, projections in polynomial and rational models, Wiener–Hopf factorizations and factorization indices all appear and are tools in the characterizations. We single out an important subclass of conditioned invariant subspaces, namely the tight ones which already made an appearance in [P.A. Fuhrmann, U. Helmke, Systems Control Lett. 30 (1997) 217], a precursor of the present paper. Of particular importance for the study of the parametrization of the set of conditioned invariant subspaces of an observable pair (C,A) is the structural map that associates with any reachable pair, with only the input dimension constrained, a uniquely determined conditioned invariant subspace. The construction of this map uses polynomial models and the shift realization. New objects, the partial observability and reachability matrices are introduced which are needed for the state space characterizations. Kernel and image representations for conditioned invariant subspaces are derived. Uniqueness of a kernel representation of a conditioned invariant subspace is shown to be equivalent to tightness. We pass on to an analysis and derivation of the Kronecker– Hermite canonical form for full column rank, rectangular polynomial matrices. This extends the work of A.E. Eckberg (A characterization of linear systems via polynomial 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.F. Münzner and D. Prätzel-Wolters [Systems Partially supported by GIF under grant no. I-526-034. ∗ Corresponding author. Tel.: +972-7-6461616; fax: +972-7-6469343. E-mail address: [email protected] (P.A. Fuhrmann). 0024-3795/01/$ see front matter 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 3 7 9 5 ( 0 1 ) 0 0 2 4 8 8 266 P.A. Fuhrmann, U. Helmke / Linear Algebra and its Applications 332–334 (2001) 265–353 Control Lett. 1 (1981) 192]. We proceed to give a parametrization of such matrices. Based on this and utilizing insights from D. Hinrichsen et al. [Systems Control Lett. 1 (1981) 192], we parametrize the set of conditioned invariant subspaces. We relate this to image representations, making contact with the work of J. Ferrer et al. [Linear Algebra Appl. 275/276 (1998) 161; Stratification of the set of general (A,B)-invariant subspaces (1999)]. We add a new angle by being able to parametrize 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 insight into the connection to geometric control and to the stable partial realization problem. © 2001 Elsevier Science Inc. All rights reserved.