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In this paper we discuss how to decompose the constrained generalizeddiscrete-time algebraic Riccati equation arising in optimal control and optimalfiltering problems into two parts corresponding to an additive decompositionX=X0+D of each solution X: The first part is an explicit expression of theaddend X0 which is common to all solutions, and does not depend on theparticular X. The second part can be either a reduced-order discrete-timeregular algebraic Riccati equation whose associated closed-loop matrix isnon-singular, or a symmetric Stein equation.

The Discrete-Time Generalized Algebraic Riccati Equation: Order Reduction and Solutions' Structure