On discrete algebraic Riccati equations: A rank characterization of solutions

• Published in: Linear algebra and its applications 2017-08, Vol.527, p.184-215
• Main Authors: Dilip, A. Sanand Amita, Pillai, Harish K., Jungers, Raphaël M.
• Format: Article
• Language: English
• Subjects: Algebra; Algebraic group theory; Difference Riccati equation; Discrete algebraic Riccati equation; Discrete Lyapunov equation; Integral equations; Integrals; Invariants; Linear algebra; Mathematical analysis; Matrix; Non-symmetric algebraic Riccati equation; Riccati equation; Schur complement; Subspaces
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2017.04.001

We give a rank characterization of the solution set of discrete algebraic Riccati equations (DAREs) involving real matrices. Our characterization involves the use of invariant subspaces of the system matrix. We observe that a DARE can be considered as a specific type of a non-symmetric continuous-time algebraic Riccati equation. We show how to extract the low rank solutions from the full rank solution of a homogeneous DARE. We also study the eigen-structure of various feedback/closed loop matrices associated with solutions of a DARE. We show as a consequence of our observations that a solution X of a DARE for controllable systems and certain specific types of uncontrollable systems satisfies the inequality Xmin≤X≤Xmax. We characterize conditions under which the solution set of DARE is bounded/unbounded. We use the method of invariant subspaces to identify some invariant sets of a difference Riccati equation.