Discrete-time timescale analysis via a new separation ratio and unified approach

In this paper, we first propose a modified definition for timescale separation ratio for the discrete-time system. Then, based on this new timescale separation ratio, a timescale analysis is presented to decompose the system into fast and slow subsystems. Next, using the difference (delta) operator, a timescale analysis is presented using a unified formulation (applicable to both continuous-time and discrete-time systems). Furthermore, it is shown that by using simple transformations, one can obtain the discrete-time results we developed earlier, and by similarly setting the sampling interval to zero, we obtain the continuous-time result developed previously by others. Thus, with this unified approach to a timescale system, a single technique is sufficient to develop the techniques for the continuous-time and discrete-time systems. Having a pair of routines for each task emphasizes the differences between discrete-time and continuous-time theory rather than the similarities. An F-8 fighter aircraft is given as an example to illustrate the methodology.