Necessary and sufficient conditions for average formation tracking of second-order multi-agent systems with multiple leaders

Average formation tracking problems for second-order multi-agent systems with multiple leaders and directed interaction topologies are studied, where the states of followers form a predefined time-varying formation while tracking the average state of the multiple leaders. An average formation tracking protocol is constructed using neighboring relative information, where only part of the followers which are named as well-informed ones are required to obtain the information of the multiple leaders. New properties of the Laplacian matrix are derived under the assumption that for each uninformed follower, there exists at least one directed path from a well-informed follower to it. Necessary and sufficient conditions for second-order multi-agent systems with multiple leaders to achieve average formation tracking are proposed by utilizing the properties of the Laplacian matrix. An approach to design the average formation tracking protocol is presented by solving an algebraic Riccati equation. The presented results can be applied to deal with the target enclosing problems, average tracking problems and consensus tracking problems for second-order multi-agent systems with one or multiple targets/leaders. An application example in multiple vehicles enclosing is provided to demonstrate the effectiveness of the theoretical results.