Self-sustained oscillations in discrete-time nonlinear feedback systems

It is a well established fact that self-sustained oscillations in autonomous continuous time systems are periodic-the limit cycles in the state plane; for higher order continuous time systems periodic self sustained oscillations are a quite improbable phenomenon which explains the difficulty in obtaining checkable conditions for the existence of such oscillations. At this end V.A. Yakubovich has introduced a new notion of oscillation, called [-/spl alpha/, /spl beta/] output oscillation; for the existence of such solutions checkable existence conditions in the frequency characteristics language are available. The problem of self sustained oscillations for discrete-time systems is quite different from that of the continuous time case; nevertheless the [-/spl alpha/, /spl beta/] oscillation may be introduced for discrete-time systems. The present paper introduces the basic notions in [-/spl alpha/, /spl beta/] oscillations in discrete-time case and gives a frequency-domain criterion of existence for such solutions. The main idea is to obtain exponential instability in the neighborhood of the origin ("in the small") and dissipativity ("stability in the large").