Gain/phase relationships for discrete-time systems

Abstract

The classical gain/phase relations obtained by Bode for minimum-phase continuous-time systems can be used to establish bounds on the performance of feedback systems. It is shown that completely analogous relations can be obtained for discrete-time systems, and can be used for the same purpose. In particular, the limitations imposed by plant zeros and poles which lie outside the unit circle are investigated, for both genuine discrete-time systems (namely those defined by difference equations) and for sampled-data systems. One of the results shows that sampled-data systems may practically be unstabilizable if their unstable poles lie outside a particular circular region. The intended application of these results is to allow preliminary screening of proposed specifications for feasibility (in a CAD system), before embarking on a detailed design.