On the solvability of the constrained discrete Lyapunov and Riccati equations

In this paper we derive conditions for the existence of a solution to the discrete Lyapunov and the discrete Riccati equations subjected to linear equality constraints. These problems arise naturally in the context of output min-max robust control. It is shown that the solvability problem of the constrained discrete Riccati equation is equivalent to problem of the existence of a feedback gain that guarantees the solvability of the constrained discrete Lyapunov equation of the resulting closed loop. A simple criterion for the existence of a solution to both problems is presented. These problems are shown to be related to the discrete positive real property.