3. Riccati Design for Synchronization of Continuous-Time Systems

This chapter studies cooperative tracking control of multi-agent dynamical systems interconnected by a fixed communication graph topology. Each agent or node is mathematically modeled by identical continuous linear time-invariant (LTI) systems, which includes the single-integrator and double-integrator as special cases. The communication network among the agents is described by a directed graph. A command generator or leader node generates the desired tracking trajectory to which all agents should synchronize. Only a few nodes are aware of information from the leader node. A locally optimal Riccati design approach is introduced here to synthesize the distributed cooperative control protocols. A framework for cooperative tracking control is proposed, including full state feedback control protocols, observer design, and dynamic output regulator control. The classical system theory notion of duality is extended to networked cooperative systems on graphs. It is shown that the local Riccati design method guarantees synchronization of multi-agent systems regardless of graph topology, as long as certain connectivity properties hold. This is formalized through the notion of synchronization region. It is shown that the Riccati design method yields unbounded synchronization regions and so achieves synchronization on arbitrary digraphs containing a spanning tree.