A reduction technique for discrete generalized algebraic and difference Riccati equations

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 eigenstructure of the corresponding extended symplectic pencil enables to identify a subspace in which all the solutions of the generalised discrete algebraic Riccati equation are coincident. This subspace is the key to derive a decomposition technique for the generalised Riccati difference equation. This decomposition isolates a ``nilpotent" part, which converges to a steady-state solution in a finite number of steps, from another part that can be computed by iterating a reduced-order generalised Riccati difference equation.