Solving a modified consensus problem of linear multi-agent systems

A distributed protocol is proposed for a modified consensus problem of a network of agents that have the same continuous-time linear dynamics. Each agent estimates its own state using its output information and then sends the estimated state to its neighbor agents for the purpose of reaching a consensus. The modified consensus problem requires the group decision value to be a linear function of initial states and initial estimated states of all agents in the network, and the transformation matrix associated with this linear function not to be a zero matrix. It is proved that under the proposed control protocol, the modified consensus problem can be solved if and only if the system matrices of the agent’s dynamics are stabilizable and detectable, the input matrix is not a zero matrix, and the communication topology graph has a spanning tree. The proposed protocol can also be extended to multi-agent systems where agents are described by discrete-time linear dynamics. The corresponding necessary and sufficient conditions are provided as well.