Distributed adaptive formation with state constraints for multi-agent systems: NE and RNE searching in aggregative games

A distributed adaptive formation problem with state constraints for multi-agent systems is studied in this article. It can be transformed into an aggregative game, where the local cost function for each agent relies on several aggregate items. Thus, auxiliary variables are employed to estimate these system-wide aggregate items under a distributed communication topology, and a distributed continuous-time Nash equilibrium (NE) seeking strategy is proposed for the multi-agent system to search for an NE point in the aggregative game. Furthermore, the uniqueness of the NE point is analyzed, which depends on the common state constraint set. Considering an extra property of the NE point, a concept of robust NE (RNE) point is first introduced in this paper. The multi-agent system located at the RNE point has the ability to be maintained within the common state constraint set as much as possible under disturbances. For general convex state constraints, an RNE searching algorithm is proposed with a distributed proximal continuous-time strategy. The strategy deals with a max-min fair allocation problem with a mixture of discrete and continuous variables and relaxes the condition that the cost function is monotone. Finally, we provide several simulations to demonstrate the theoretical results.