Generalized model matching and (F,G)-invariant submodules for linear systems over rings

A generalized version of the exact model matching problem (GEMMP) is considered for linear multivariable systems over an arbitrary commutative ring K with identity. Reduced forms of this problem are introduced, and a characterization of all solutions and minimal order solutions is given, both with and without the properness constraint on the solutions, in terms of linear equations over K and K-modules. An approach to the characterization of all stable solutions is presented which, under a certain Bezout condition and a freeness condition, provides a parametrization of all stable solutions. The results provide an explicit parametrization of all solutions and all stable solutions in case K is a field, without the Bezout condition. This is achieved through a very simple characterization and a generalization to an arbitrary field K of the “fixed poles” of the model matching problem in terms of invariant factors of a certain polynomial matrix. The results also show that whenever the GEMMP has a solution, there exist solutions whose poles can be chosen arbitrarily as far as they contain the “fixed poles” with the right multiplicities (in the algebraic closure of K). Implications of these results in regard to inverse systems are shown. Equivalent simpler forms (in state space form) of the problem are shown to be obtainable. A theory of finitely generated (F,G)-invariant submodules for linear systems over rings is developed, and the geometric equivalent of the model matching problem—the dynamic cover problem—is formulated, to which the results of the previous sections provide a solution in the reduced case.