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=X0+ΔX=X0+Δ of each solution XX: The first part is trivial, in the sense that it is an explicit expression of the addend X0X0 which is common to all solutions, so that it does not depend on the particular XX. 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.