Balanced realisations for infinite-dimensional discrete-time systems

For finite-dimensional systems the class of balanced realisations is defined as that whose controllability and observability gramians are both equal to the same positive diagonal matrix. The diagonal entries are in fact the singular values of the Hankel operator of the system and contain essential information about its behaviour. Balanced realisations have special structural properties and their truncations usually provide a good reduced order model of the original system. An account of balanced realisations for infinite-dimensional systems is given and its implications for the approximation of infinite-dimensional systems by finite-dimensional ones are discusse