Iterative algorithms for reducing inversion of discrete algebraic riccati matrix equation

In practical engineering, many control problems usually can be transformed into solutions of the discrete algebraic Riccati equation (DARE), which has two matrix inverse operations formally. In this paper, first, by the relationship between properties of the matrix Schur complement and partitioned representation of inverse matrix, we change the DARE with twice inversions into an equivalent form with once inversion and propose a corresponding iterative algorithm. Next, for a special case of DARE, we deformed this DARE into a new equivalent one. For the equivalent form, we propose a new iterative algorithm in an inversion-free way. Furthermore, for these algorithms, we prove their monotone convergence and give the analysis of their errors. Last, comparing with some existing work on this topic, corresponding numerical examples are given to illustrate the superiority and effectiveness of our results.

Automated summary

In practical engineering, many control problems can be transformed into solutions of the discrete algebraic Riccati equation (DARE), which involves matrix inverse operations. This paper proposes a method to transform a DARE with multiple inversions into an equivalent form with a single inversion, and presents an iterative algorithm for solving it. The paper also introduces a new iterative algorithm for a special case of DARE, which avoids matrix inversions. The monotone convergence and error analysis of the algorithms are proven, and numerical examples validate the superiority and effectiveness of the proposed methods.