  This paper proposes a reduction technique for the generalised Riccati
difference equation arising in optimal control and optimal filtering. This
technique relies on a study on the generalised discrete algebraic Riccati
equation. In particular, an analysis on the eigen- structure of the
corresponding extended symplectic pencil enables to identify a subspace in
which all the solutions of the generalised discrete algebraic Riccati equation
are coin- cident. This subspace is the key to derive a decomposition technique
for the generalised Riccati difference equation that isolates its nilpotent
part, which becomes constant in a number of steps equal to the nilpotency index
of the closed-loop, from another part that can be computed by iterating a
reduced-order generalised Riccati difference equation.
