Abstract
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.
Highlights
In the last decade, many scientists paid their attentions to design different cooperative control strategies of swarm systems with different application backgrounds, such as explanations for animal flocking phenomena [1, 2], information consensus analysis and design [3,4,5,6,7,8,9], analysis for different source data [10], and formation structure design and maintenance [10,11,12,13]
The whole dynamics of the leader-following swarm system is separated as the dynamics of the leader and the relative dynamics among all agents by constructing the specific nonsingular transformation matrix, and design and analysis criteria for time-varying formation tracking with the limited energy are proposed, respectively, where the relationship matrix between the matrix variable and the limited energy is associated with the Laplacian matrix of a star graph with the leader being the central vertex
For high-order linear time-invariant swarm systems with the limited energy and fixed topologies, a new formation tracking protocol with an energy integral term was proposed to realize formation tracking under the condition that the whole energy consumption is less than or equal to the limited energy supply
Summary
Many scientists paid their attentions to design different cooperative control strategies of swarm systems with different application backgrounds, such as explanations for animal flocking phenomena [1, 2], information consensus analysis and design [3,4,5,6,7,8,9], analysis for different source data [10], and formation structure design and maintenance [10,11,12,13]. To the best of our knowledge, the design and analysis problems of time-varying formation tracking for high-order linear time-variant swarm systems with the limited energy and fixed topologies are still open and are not comprehensively discussed. The whole dynamics of the leader-following swarm system is separated as the dynamics of the leader and the relative dynamics among all agents by constructing the specific nonsingular transformation matrix, and design and analysis criteria for time-varying formation tracking with the limited energy are proposed, respectively, where the relationship matrix between the matrix variable and the limited energy is associated with the Laplacian matrix of a star graph with the leader being the central vertex. Notations. e symbols Rn and Rn×n denote the n-dimensional real column vector and the n-dimensional real matrix space, respectively. e symbol 1N represents the Ndimensional column vector with all components 1. e notation 0 denotes the zero vector or zero matrix with compatible dimensions, and the matrix Ir means an identity matrix with dimension r. e symbol ⊗ represents the Kronecker product. e notation QT Q > 0 shows that the matrix Q is symmetric and positive definite
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