A proposed z-plane criterion to expedite transient-performance analyses

A simple and insightful performance criterion is described with reference to the position of a pair of complex-conjugate poles in the z-plane. Introduced as the half-cycle decay factor, this criterion is a relative stability measure that is determined directly in the z-plane without reference to a corresponding damping ratio in the s-plane. Use of the proposed criterion expedites various analysis and design procedures, and this property is illustrated. The utilization is presented with several control system applications. One example presents a time-scaling procedure as required to implement an observer-the complex-pole pairs are moved in a manner that maintains the half-cycle decay factor for each complex pair at a fixed value. Other examples are considered in which the system model is characterized by a single pair (or single dominant pair) of complex-conjugate poles. In this situation specific conditions are described for which the half-cycle decay factor can be employed in expressions to evaluate percent overshoot and settling time in response to a step input. Finally, it is shown that the concept can be applied in either the z-plane or the s-plane using either discrete-time models or continuous models, respectively