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

2021/07/25 by Yu, Bo, Dong, Ning
#15A24 #65F30 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2107.11708

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.

Related

• Decoupled Structure-Preserving Doubling Algorithm with Truncation for Large-Scale Algebraic Riccati Equations
• A modified large-scale structure-preserving doubling algorithm for a large-scale Riccati equation from transport theory
• Low-rank alternating direction doubling algorithm for solving large-scale continuous time algebraic Riccati equations
• Highly accurate decoupled doubling algorithm for large-scale M-matrix algebraic Riccati equations
• A decoupled form of the structure-preserving doubling algorithm with low-rank structures