# An accelerated technique for solving one type of discrete-time algebraic Riccati equations

## TL;DR

This paper introduces an accelerated algorithm for solving a specific class of discrete-time algebraic Riccati equations, providing faster convergence and reliable solutions in control applications.

## Contribution

It offers sufficient conditions for solution existence and a novel accelerated method with customizable convergence rate for these Riccati equations.

## Key findings

- The algorithm converges rapidly even in near-critical cases.
- It reliably computes both positive and negative definite solutions.
- Numerical experiments validate the method's high performance.

## Abstract

Algebraic Riccati equations are encountered in many applications of control and engineering problems, e.g., LQG problems and $H^\infty$ control theory. In this work, we study the properties of one type of discrete-time algebraic Riccati equations. Our contribution is twofold. First, we present sufficient conditions for the existence of a unique positive definite solution. Second, we propose an accelerated algorithm to obtain the positive definite solution with the rate of convergence of any desired order. Numerical experiments strongly support that our approach performs extremely well even in the almost critical case. As a byproduct, we provide show that this method is capable of computing the unique negative definite solution, once it exists.