Minimal State Diagrams for Controllable Behaviors over Finite Rings

In this paper we consider controllable behaviors over Zpr that admit an image representation which has a Laurent polynomial left inverse. We solve the problem of determining a minimal state diagram representation for such behaviors. Using the concept of p-basis for submodules of Zpr [z], that was introduced in (Kuijper et al., 2007), we define an expanded type of image representations for the general class of controllable behaviors over Zpr , called p-image representations. We show that every controllable behavior over Zpr admits a p-image representation H(z) whose columns form a reduced p-basis in Zpr [z] and such that the columns of H(0) form a p-basis in Zpr and use such p-image representation to determine a minimal state diagram representation for our case.