In this paper design methods are given for synchronization control of discrete-time multi-agent systems on directed communication graphs. The graph properties complicate the design of synchronization controllers due to the interplay between the eigenvalues of the graph Laplacian matrix and the required stabilizing gains. Two methods are given herein that decouple the design of the synchronizing gains from the detailed graph properties. Both are based on computation of the local control gains using Riccati design; the first is based on an H∞ type Riccati inequality and the second on an H2 type Riccati equation. Conditions are given for synchronization based on the relation of the graph eigenvalues to a bounded circular region in the complex plane that depends on the agent dynamics and the Riccati solution. The notion of ‘synchronizing region’ is used. An example shows the effectiveness of these design methods for guaranteeing synchronization in cooperative discrete-time systems.