A unified setting for decoupling with preview and fixed-lag smoothing in the geometric context

Exact decoupling with preview, perfect tracking of previewed references, unknown-input state observation with fixed lag, and left inversion with fixed lag are considered from a unifying perspective, where exact decoupling with preview is the basic problem. Necessary and sufficient constructive conditions for decoupling with relative-degree preview are proved in the geometric framework. Structural and stabilizability conditions are considered separately and the use of self-bounded controlled invariant subspaces enables the minimal order dynamic solution to be straightforwardly derived. A steering along zeros technique is devised to solve decoupling in the presence of unstable unassignable dynamics of the minimal self-bounded controlled invariant satisfying the structural constraint.