Abstract
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.
Highlights
Formation control of a distributed swarm system is inspired by the biological formation behaviors without the superintend node, where it is needed that some specific geometric structure is achieved and maintained by local interactions among all animals
For high-order linear swarm systems with fixed communication topologies, a new formation control protocol with an energy integral term was proposed by using the state errors and the formation function errors among neighboring agents, which can guarantee that the practical energy assumption is less than the whole energy budget
By an orthonormal transformation associated with the Laplacian matrix of the fixed communication topology, limited-budget formation control problems were converted into asymptotic stability ones, and sufficient conditions for swam systems to achieve limitedbudget formation were proposed, where the relationship between the matrix variable and the limited budget was constructed on the basis of the specific structure feature of the orthonormal transformation matrix
Summary
Many researchers from system and control fields paid their attentions to the distributed coordination of swarm systems, which have potential applications in many aspects, such as flocking algorithm and theory [1, 2], synchronization analysis and design [3,4,5,6,7,8,9,10,11], formation control [12,13,14], and distributed parallel computation [15,16,17,18,19]. In [27,28,29], formation control design and analysis problems for swarm systems with different topology structures were investigated, where the influences of the limited energy supply were not dealt with, which is critically important in practical applications. The whole energy supply is usually limited in practical swarm systems and it is critically important to discuss the influences of the whole energy supply on time-varying formation design and analysis problems. Is paper deals with time-varying formation design and analysis for high-order linear swarm systems with the limited energy budget. By constructing the relationship between the matrix variable and the energy budget, limited-budget formation design and analysis criteria are proposed, respectively, where the scalability property and the checkable property are discussed in detail. Symbols Rn and Rn×n, respectively, represent the n-dimensional real column vector and the n-dimensional real matrix space. 1N denotes the N-dimensional column vector with all components 1. 0 represents the zero vector or zero matrix with compatible dimensions. 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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