Almost Self-Bounded Controlled-Invariant Subspaces and Almost Disturbance Decoupling

The objective of this contribution is to characterize the so-called finite fixed poles of the Almost Disturbance Decoupling Problem by state feedback (ADDP) ′ . The most important step towards this result relies on the extension to almost invariant subspaces of the key notion of self-boundedness, as initially introduced by Basile and Marro for perfect controlled-invariants, namely, we introduce the Almost Self-Bounded Controlled-Invariant subspaces. We recall the pole placement flexibilities and constraints that both exist when using a particular almost invariant subspace as a support for the construction of specific (including high gain) feedbacks, and we show, when (ADDP) ′ is solvable, what is the “best” almost invariant subspace to choose, in order to achieve (ADDP) ′ and simultaneously place the “largest possible” set of finite poles for the closed loop solution. We finally characterize the set of fixed finite poles for (ADDP) ′ , in terms of some finite zero structures.