The Discrete-Time Generalized Algebraic Riccati Equation: Order Reduction and Solutions' Structure

TL;DR

This paper presents a method to decompose solutions of the generalized discrete-time algebraic Riccati equation into a common part and a reduced-order part, simplifying analysis and computation in control and filtering.

Contribution

It introduces a decomposition approach for the generalized Riccati equation, separating solutions into a universal component and a reduced-order component for easier analysis.

Findings

Explicit expression for the common part of solutions.

Reduction to a regular algebraic Riccati equation or a symmetric Stein equation.

Simplified structure for solutions of the generalized Riccati equation.

Abstract

In this paper we discuss how to decompose the constrained generalized discrete-time algebraic Riccati equation arising in optimal control and optimal filtering problems into two parts corresponding to an additive decomposition X=X0+D of each solution X: The first part is an explicit expression of the addend X0 which is common to all solutions, and does not depend on the particular X. The second part can be either a reduced-order discrete-time regular algebraic Riccati equation whose associated closed-loop matrix is non-singular, or a symmetric Stein equation.