Solvability conditions and design for synchronization of discrete-time multiagent systems

This paper provides solvability conditions for state synchronization with homogeneous discrete-time multiagent systems with a directed and weighted communication network under partial- or full-state coupling. Our solvability conditions reveal that the synchronization problem is solvable for all possible, a priori given, set of graphs associated with a communication network only under the condition that the agents are at most weakly unstable (ie, agents have all eigenvalues in the closed unit disc). However, if an upper bound on the eigenvalues inside the unit disc of the row stochastic matrices associated with any graph in a given set of graphs is known, then one can achieve synchronization for a class of unstable agents. We provide protocol design for at most weakly unstable agents based on a direct eigenstructure assignment method and a standard H₂ discrete-time algebraic Riccati equation. We also provide protocol design for strictly unstable agents (ie, agents have at least one eigenvalue outside the unit disc) based on the standard H₂ discrete-time algebraic Riccati equation.