A reduction technique for Generalised Riccati Difference Equations

TL;DR

This paper introduces a reduction method for the generalized Riccati difference equation used in optimal control and filtering, leveraging eigen-structure analysis to simplify computations by isolating nilpotent components.

Contribution

It presents a novel decomposition technique based on eigen-structure analysis that isolates the nilpotent part of the generalized Riccati difference equation, simplifying its solution process.

Findings

Identifies a subspace where solutions of the generalized discrete algebraic Riccati equation coincide.

Decomposes the Riccati difference equation into a constant nilpotent part and a reduced-order part.

Enables more efficient computation of solutions in optimal control and filtering applications.

Abstract

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.