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This paper studies the disturbance decoupling problem by dynamic outputfeedback with required closed-loop stability, in the general case ofnonstrictly-proper systems. We will show that the extension of the geometricsolution based on the ideas of self boundedness and self hiddenness, which isthe one shown to maximize the number of assignable eigenvalues of theclosed-loop, presents structural differ- ences with respect to the strictlyproper case. The most crucial aspect that emerges in the general case is theissue of the well-posedness of the feedback interconnection, which obviouslyhas no counterpart in the strictly proper case. A fundamental property of thefeedback interconnection that has so far remained unnoticed in the literatureis investigated in this paper: the well-posedness condition is decoupled fromthe remaining solvability conditions. An important consequence of this fact isthat the well-posedness condition written with respect to the supremal outputnulling and infimal input containing subspaces does not need to be modifiedwhen we consider the solvability conditions of the problem with internalstability (where one would expect the well-posedness condition to be expressedin terms of supremal stabilizability and infimal detectability subspaces), andalso when we consider the solution which uses the dual lattice structures ofBasile and Marro.

On the well-posedness in the solution of the disturbance decoupling by dynamic output feedback with self bounded and self hidden subspaces