Formation control based on a consensus protocol under directed communications with two time delays

This paper presents an extension of the traditional consensus problem to a formation control for a group of agents driven by second order dynamics. The communication between members of the group is assumed to be directed and affected by two rationally independent time delays, one in the position and the other in the velocity information channels. These delays are considered as constant and uniform throughout the communication topology. The stability of the consensus protocol is studied first, using a simplifying factorization procedure and deploying the Cluster Treatment of Characteristic Roots (CTCR) paradigm to obtain exact stability boundaries in the domain of the delays. CTCR requires the determination of all the potential stability switching loci in the domain of the delays. For this, a surrogate domain, called the “Spectral Delay Space (SDS)” is used. As the stability of the system is studied the findings yield a paradoxical control concept of “Delay Scheduling”. It shows that the stability of the system can be regained by increasing the delays. Once the stability of the consensus protocol is guaranteed, a new control term is added to obtain a desired stable formation for the agents.