With the exception of internal descriptions, the various methods to describe linear dynamical systems are uniquely interrelated. A given step response belongs to a SISO system with a well defined transfer function if the occurance of pole-zero cancellations is ruled out. Likewise no two impulse responses correspond to the same frequency response or the same frequency response will not result from two systems with different pole- zero patterns.
This uniqueness is lost if only input-output measurements of finite accuracy are available. There is a whole class of systems of the same or of different order which exhibit the same input-output behaviour within a given error limit. Allowing e.g. only differences to the extent that all function plots lie within the bounds of indistinguishability, the class of systems exhibiting the same input-output behaviour within these bounds consists of members with considerably different pole-zero patterns. This phenomenon is demonstrated, and its influence on identification, model order reduction and controller design is discussed.
Furthermore it is shown that modal reduction techniques yield the best reduced-order model only under special circumstances.