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Decentralized Formation Control Part II: Algebraic aspects of information flow and singularities

Given an ensemble of autonomous agents and a task to achieve cooperatively, how much do the agents need to know about the state of the ensemble and about the task in order to achieve it? We introduce new methods to understand these aspects of decentralized control. Precisely, we introduce a framework to capture what agents with partial information can achieve by cooperating and illustrate its use by deriving results about global stabilization of directed formations. This framework underscores the need to differentiate the knowledge an agent has about the task to accomplish from the knowledge an agent has about the current state of the system. The control of directed formations has proven to be more difficult than initially thought, as is exemplified by the lack of global result for formations with n \geq 4 agents. We established in part I that the space of planar formations has a non-trivial global topology. We propose here an extension of the notion of global stability which, because it acknowledges this non-trivial topology, can be applied to the study of formation control. We then develop a framework that reduces the question of whether feedback with partial information can stabilize the system to whether two sets of functions intersect. We apply this framework to the study of a directed formation with n = 4 agents and show that the agents do not have enough information to implement locally stabilizing feedback laws. Additionally, we show that feedback laws that respect the information flow cannot stabilize a target configuration without stabilizing other, unwanted configurations.