Extensions of the bounded real lemma of discrete-time systems

In this paper the bounded real lemma for discrete-time systems is extended in several directions. It is shown that an H-infinity-norm bound for a (not necessarily stable) transfer matrix T(z) combined with controllability of unimodular eigenvalues yields the existence of an unmixed solution of the a lgebraic Riccati equation (ARE) associated with T(z). Conversely, it is pro ved that the existence of a (not necessarily stabilizing) solution of the a ssociated ARE implies bounded realness of T(z). Inertia results are obtaine d and a condition for the existence of negative semidefinite solutions of t he associated ARE is given.