Large-Scale Algebraic Riccati Equations with High-Rank Nonlinear Terms and Constant Terms

TL;DR

This paper introduces a specialized doubling algorithm for solving large-scale discrete-time algebraic Riccati equations with high-rank nonlinear and constant terms, leveraging the banded-plus-low-rank structure to improve computational efficiency.

Contribution

It develops a factorized structure-preserving doubling algorithm (FSDA) tailored for large-scale Riccati equations with high-rank terms, incorporating partial truncation and a deflation process.

Findings

Efficient computation of stabilizing solutions for large-scale Riccati equations.

Reduction in computational complexity through structure-preserving techniques.

Enhanced algorithm stability and convergence with the proposed methods.

Abstract

For large-scale discrete-time algebraic Riccati equations (DAREs) with high-rank nonlinear and constant terms, the stabilizing solutions are no longer numerically low-rank, resulting in the obstacle in the computation and storage. However, in some proper control problems such as power systems, the potential structure of the state matrix -- banded-plus-low-rank, might make the large-scale computation essentially workable. In this paper, a factorized structure-preserving doubling algorithm (FSDA) is developed under the frame of the banded inverse of nonlinear and constant terms. The detailed iterations format, as well as a deflation process of FSDA, are analyzed in detail. A partial truncation and compression technique is introduced to shrink the dimension of columns of low-rank factors as much as possible. The computation of residual, together with the termination condition of the structured version, is also redesigned.