Robust stability of delta-operator discrete-time systems using frequency-domain graphical approach

There are numerous stability analysis and control design techniques for continuous-time (analog) systems. However, one needs often to transform (approximate) an analog system to its discrete-time (digital) form, such as in simulation of analog filters or controllers on digital computer, and therefore one needs to obtain stability analysis and design techniques for the transformed system. Traditionally, shift-operator (q-operator) models have been used for such an approximation. However, due to the different nature of the analog system and its q-operator digital approximation, no uniform and straightforward results have been obtained for the transformed system, i.e., the results are totally different in form and methodology. Recently, a Delta-operator approach of approximating an analog system was proposed which has several advantages over the q-operator approximation, such as superior finite-word length coefficient presentation, superior finite word length rounding error performance, superior numerical properties in calculations with digital models, and convergence of digital results and models to their continuous counterparts as the sampling rate is increased. The last property is specially important, because a uniform set of techniques for the stability and control can be obtained for both analog and discrete-time systems.