A note on the driving variable realization of behaviors

The problem of realizing a discrete-time behavior by means of a driving variable (DV) state-space representation is here afforded. An algebraic relation between kernel descriptions and driving variable representations is first derived and later used for obtaining a minimal DV realization of a given behavior. Internal properties of the behavior, in particular controllability, are related to the properties of its DV realizations. Finally, a different technique for obtaining a minimal DV realization in case of a controllable behavior is described.