Time-scale decoupling and order reduction for linear time-varying systems†

The class of time-varying linear systems which are two-time-scale on an interval may be decoupled by a transformation of variables into separate subsystems containing the slow and fast dynamic parts. The transformation is obtained by solving a non-symmetric Riccati differential equation forward in time and a linear matrix differential equation backward in time. Small parameters are identified which measure the strength of the time-scale separation and the stability of the fast subsystem. As these parameters go to zero, the order of the system is reduced, and a useful approximate solution to the original system is obtained. The transformation is illustrated for examples with strong and weak fast subsystem stability.

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