Fixed poles of the disturbance decoupling problem by dynamic output feedback for biproper systems

TL;DR

This paper extends geometric conditions for disturbance decoupling via dynamic output feedback to biproper systems with feedthrough matrices, ensuring well-posedness and maximizing eigenvalue assignability without eigenspace computations.

Contribution

It generalizes the geometric approach to systems with non-zero feedthrough matrices, addressing well-posedness issues and preserving eigenvalue assignment capabilities.

Findings

The geometric condition based on self-boundedness and self-hiddenness applies to biproper systems.

The solution maximizes the number of assignable eigenvalues.

Well-posedness conditions are well-behaved when expressed via self-bounded and self-hidden subspaces.

Abstract

This paper investigates the disturbance decoupling problem by dynamic output feedback in the general case of systems with possible input-output feedthrough matrices. In particular, we aim to extend the geometric condition based on self-boundedness and self-hiddenness, which as is well-known enables to solve the decoupling problem without requiring eigenspace computations. We show that, exactly as in the case of zero feedthrough matrices, this solution maximizes the number of assignable eigenvalues of the closed-loop. Since in this framework we are allowing every feedthrough matrix to be non-zero, an issue of well-posedness of the feedback interconnection arises, which affects the way the solvability conditions are structured. We show, however, that the further solvability condition which originates from the problem of well-posedness is well-behaved in the case where we express such condition in terms of self bounded and self hidden subspaces.