A Characterization of Solutions of the ARE and ARI

Abstract

This article is about a characterization of the solution set of algebraic Riccati equation (ARE) and the algebraic Riccati inequality (ARI) over the reals, for both controllable and uncontrollable systems. We characterize these solutions using simple linear algebraic arguments. It turns out that solutions of ARE of maximal rank have lower rank solutions encoded within it. We demonstrate how these lower rank solutions are encoded within the full rank solution and how one can retrieve the lower rank solutions from the maximal rank solution. We also obtain a parametrization for solutions of certain specific ARIs. We generalize Willems’ result \(K_{min}\le K\le K_{max}\) for ARI arising out of controllable systems to some specific kind of uncontrollable systems.