State-space realization of multiple-frequency-scale transfer matrices

The class of linear, time-variant, singularly perturbed systems is considered in which all poles are O(1) or O(1/ε) as ε→O, with coefficient matrices that are analytic functions of ε. It has been shown previously that transfer matrices of systems in the above class are two-frequency-scale. This means that such transfer matrices behave as dynamical systems in two separate frequency ranges. It is shown that any two-frequency-scale transfer matrix can be realized in state-space form as a system in the above-mentioned class of singularly perturbed systems. The result is generalized to N frequency scales