Asymptotic stability of infinite dimensional discrete-time balanced realizations

The question of the power and asymptotic stability of infinite-dimensional discrete-time state-space systems is investigated. By relating balanced realizations to restricted shift realizations, it is shown that every balanced realization is asymptotically stable. In general, input normal and output normal realizations do not have the same stability properties as balanced realizations, but necessary and sufficient conditions can also be given for them to be asymptotically and/or power stable. It turns out that an input normal or output normal realization is power stable if and only if its transfer function is rational, whereas the power stability of a par-balanced realization is more complicated to characterize in terms of the properties of the transfer function