(A, B)-invariant subspaces and polynomial matrix algebra— towards a more integrated approach. Part I: Square systems

Abstract

The algebra of (A, B)-invariant subspaces 𝒱 having null intersection with range B = ℬ is examined from a viewpoint of the algebra of controllability subspaces in their polynomial vector parametrization and decoupling theory. Thus a characterization of such subspaces is arrived at in terms of an equivalence class of numerator matrices associated with the Wolovich-Falb canonical matrix fraction decomposition of a linear reachable time-invariant multivariable system. A simple and useful formula is obtained for reading off the maximal (A, B)-invariant subspace 𝒱 max ⊆ ker C in terms of any related numerator matrix. It is also shown how to obtain the important class of feedback F(𝒱) of interest to typical geometric theory calculations by an inspection method akin to that used in the solution of the restricted decoupling problem. The set of all (A, B)-invariant subspaces in some fixed subspace, ker C say, is classified in terms of suitably defined divisor classes of the equivalence class of numerator matrices b...