The current paper investigates design and analysis problems of formation tracking for high-order linear time-invariant swarm systems, where communication topologies among agents have leader-following structures and the whole energy supply is limited. Firstly, the communication topology of a swarm system is depicted by a directed graph with a spanning tree, where the communication channels from the leader to followers are directional and the communication channels among followers are bidirectional, and a new formation tracking protocol with an energy integral term and a given upper bound is proposed to achieve formation tracking with the limited energy. Then, sufficient conditions for time-varying formation tracking design and analysis with the limited energy are presented, respectively, which include two/three linear matrix inequality constraints associated with the maximum and minimum nonzero eigenvalues of the Laplacian matrix of a communication topology. Especially, time-invariant formation tracking criteria are further deduced. Finally, two numerical examples are revealed to verify main theoretical conclusions.

The minimum‐energy formation tracking design and analysis problems for interconnected networks with distributed tracking protocols are addressed, where the effects of the total energy restriction and the interaction silence on the formation tracking are discussed, respectively. The key characteristic of this article is that the value of the energy restriction is required to be minimum, while the guaranteed‐cost tracking strategy and the energy‐constraint tracking strategy cannot ensure the minimum property of the total energy restriction. For interconnected networks without the interaction silence, sufficient conditions for minimum‐energy formation tracking are presented, whose effectiveness can be checked by the optimization procedure associated with the generalized eigenvalue method. Then, for interconnected networks with the interaction silence, by constructing the specific constraint associated with the maximum silent rate and two inhibition parameters, sufficient conditions for minimum‐energy formation tracking are proposed. Finally, numerical examples are shown to illustrate the efficacy of main results.

This paper discusses limited-budget time-varying formation design and analysis problems for a high-order linear swarm system with a fixed communication topology. Firstly, the communication topology among agents is modeled as an undirected and connected graph, and a new formation control protocol with an energy integral term is proposed to realize formation control and to guarantee the practical energy assumption is less than the limited energy budget. Then, by the matrix inequality tool, sufficient conditions for limited-budget formation design and analysis are proposed, respectively, which are scalable and checkable since they are independent of the number of agents of a swarm system and can be transformed into linear matrix inequality constraints. Moreover, an explicit expression of the formation center function is given, which contains the formation function part and the cooperative state part and is not associated with the derivatives of the formation functions. Finally, a numerical simulation is shown to demonstrate the effectiveness of theoretical results.