Bounded real lemma for positive discrete systems

The bounded real lemma for positive discrete time-invariant systems is addressed in this study. The results about bounded real lemma for continuous time-invariant systems are extended into discrete time-invariant systems. Such systems are asymptotically stable and their H_∞ norm less than one is equivalent to that the corresponding linear matrix inequality admits a diagonal positive-definite solution. The last example in this study illustrates the result does not hold for general discrete time-invariant systems.