The discrete-time generalized algebraic Riccati equation: Order reduction and solutions’ structure

In this paper we discuss how to decompose the constrained generalized discrete-time algebraic Riccati equation arising in optimal control and optimal filtering problems into two parts corresponding to an additive decomposition X = X_0 + D of each solution X: The first part is trivial, in the sense that it is an explicit expression of the addend X_0 which is common to all solutions, so that it does not depend on the particular X. The second part can be - depending on the structure of the considered generalized Riccati equation - either a reduced-order discrete-time regular algebraic Riccati equation whose associated closed-loop matrix is non-singular, or a symmetric Stein equation. The proposed reduction is explicit, so that it can be easily implemented in a software package that uses only standard linear algebra procedures.