Input-output description of linear systems with multiple time-scales†

We study the multiple time-scale structure of linear systems of the form x = A o(∊)x + B o(∊)u, y = C o(∊)x with a view to obtaining ’ approximate ’ lower-order transfer functions valid at different time-scales. Our development includes the classical two time-scale case as well. We use our results to study the positive realness of linear systems with multiple time-scales in terms of the positive realness of the reduced-order transfer functions.