Transformations between discrete-time and continuous-time algebraic Riccati equations

We introduce a transformation between the discrete-time and continuous-time algebraic Riccati equations. We show that under mild conditions the two algebraic Riccati equations can be transformed from one to another, and both algebraic Riccati equations share common Hermitian solutions. The transformation also sets up the relations about the properties, commonly in system and control setting, that are imposed in parallel to the coefficient matrices and Hermitian solutions of two algebraic Riccati equations. The transformation is simple and all the relations can be easily derived. We also introduce a generalized transformation that requires weaker conditions. The proposed transformations may provide a unified tool to develop the theories and numerical methods for the algebraic Riccati equations and the associated system and control problems.