How to decompose semi-definite discrete-time algebraic Riccati equations
Interest in the algebraic Riccati equation (ARE) was revived by the theory of \(H^\infty\) control. In this case the weight matrix in the performance index is not positive definite but usually indefinite. Even the case when the weight matrix is positive semi-definite, which is the case in standard \(H^\infty\) control, was not fully understood, and the question of how to implement good numerical solvers for these equations when they are ill posed is still a research area. The aim of the paper is to propose a way of circumventing the problems of today's numerical solvers for the semi-definite case. The idea is to decompose the potentially ill-posed equation into a trivial part and a reduced-order well-posed equation that can be solved in a numerically stable and efficient way by well-known solvers such as the Schur method or the Newton method. The proposed reduction scheme is designed for the discrete-time ARE (DARE). As a by-product a novel constructive proof for the existence of a solution to the DARE is obtained. The reduction scheme is implemented in Matlab, and it is shown in several examples that the new scheme is very promising.
scientific article; zbMATH DE number 1382825
scientific article