There are two different ways of describing uynamicu systems: (i) bymeans of state variables and (ii) by input/output relations. The first method may be regarded as an axiornatization of Newton’s laws of mechanics and is taken to be the basic definition of a system.It is then shown (in the linear case) that the input/output relations determine only one part of a system, that which is completely observable and completely controllable. Using the theory of controllability and observability, methods are given for calculating irreducible realization of a given impulse-response matrix. In particular, an explicit procedure is given to determine the minimal number of state variables necessary to realize a given transfer-function matrix. Difficulties arising from the use of reducible realizations are discussed briefly.