This paper reports further results on the development of a control systems theory for discrete linear repetitive processes. In particular, Roesser and Fornasini-Marchesini state space model equivalent descriptions for the dynamics of these processes are constructed and then used to develop new stability tests which involve only computations on matrices with constant entries. Also, they are used to develop a transition matrix, or fundamental matrix sequence, for these processes which is then used to define and characterise so-called local reachability and controllability properties in the form of matrix rank-based tests.