Sampling control on collaborative flocking motion of discrete-time system with time-delays
Introduction
Flocking, as a core problem in the complexity science, is essentially a kind of bionics method. Without centralized control, flocking motion produces macroscopic synchronous effect by mutual perception, which enhances their abilities of searching for food and avoiding predators. In the nature, flocking behavior exists widely, which is collaborative group behavior in a certain gathering way. In the military, flocking can combat replacing the army in order to reduce casualties.
With the development of biology, computer, control science, and artificial intelligence, flocking problem has drawn increasing attention of many scholars recently. In 1987, a computer model, imitating biological aggregation behavior, which contain gather, separation and adjustment three rules, is proposed by Reynolds [1]. Tanner et al. [2], [3] firstly provide a theoretical explanation for the computer model, for the fixed and dynamic topology, an effective control protocol is proposed by Tanner. The local distributed protocol, usable the multi-agent systems to achieve flocking motion without collisions, is designed by Olfati-Saber [4], [5]. As a kind of flocking motion, consensus of multi-agent systems has been studied deeply. Consensus of a heterogeneous multi-agent system with input saturation is studied in [6]. Reference [7] researches bounded consensus algorithms for multi-agent systems in directed networks. Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics is presented in [8], distributed adaptive control for synchronization in directed complex networks is studied in [9]. Consensus of high-order multi-agent systems is analyzed in switching topologies [10], the observer-based consensus for nonlinear multi-agent systems is discussed with intermittent communication [11]. Dynamical consensus seeking of second-order multi-agent systems with communication delays is studied [12], [13]. Flocking control algorithms are taken into account for multi-agent systems with communication time-delays and switching topology [14], [15], [16].
Up to now, most researches focus on the continuous system. The study of discrete system is more and more important with the rapid development of digital communication technology. In this aspect, Vicsek et al. [17] have proposed a discrete system model, where the velocity of agents can asymptotically converge to consistent state. For the communication delay problem, basing on frequency analysis and time analysis method, consistency of discrete-time systems with time delays is researched [18], [19], [20].
Research on multi-rate sampling begins 1950s. Using Kranc operator and sampler decomposition method, Kranc [21] solved the different sample frequency problem in the input and output sample. But when the sampling rates are more than two kinds, the complexity of the mathematical model makes traditional transfer function restricted in the application. Araki and Yamamoto [22] presented a complete state space description of the multi-rate sampling system. Using two different lifting technology, Yanamoto [23] put forward to the function space model of the multi-rate sampling system. However, up to now, the results of multi-rate sampling control of networked system with communication time-delays have not been reported.
The main objective of this paper is to research multi-rate sampling control of collaborative flocking motion with time delays. The rest of the paper is organized as follows. In second part, problem description and preliminaries are given. The main results are discussed in third part. In fourth part, simulation examples are used to illustrate the correct of theoretical results. Conclusions are finally drawn in the fifth part.
Section snippets
Algebraic graph theory
Assume that the system consist of n agents. Let be a graph consisting of with n agents and . is a adjacency matrix of the graph G. The relationships among agents are described by the graph without self-loops. Moreover, the adjacency element when , otherwise, . The set of neighbors of is denoted by . The Laplacian matrix of the graph G is defined as , where . Suppose that n
Main results
Theorem 1 Consider discrete-time multi-agent systems with communication time-delays. The dynamic behaviors of the agents are described by Eq. (1), (2). Then, control protocol (4), (5) can solve the collaborative flocking motion problem. Moreover, all agents can asymptotically achieve consistent velocities, and the distances will remain stable between any two agents without collision. Proof Define the following variable: By substituting Eqs. (1), (2) into Eq. (6), we can
Simulation examples
Supposing that there are four identical agents and a leader in the topology (Fig. 1) and only the fourth agent can receive information of the leader. We choose control gain k=1 and in the distributed control protocol. c=2 and d=2.8 in the energy function equation. In the 2 dimensional space, the leader and four agents initial positions and the initial velocities are q[0]=[7.6602 6.3923 7.1707 8.7727 8.9256 9.1127 6.6863 6.592 6.0024 9.6622], p[0]=[0.07 0.5964 0.7335 0.5188 0.8958 0.8959
Conclusions
For the discrete-time multi-agent systems, a distributed control protocol, which ensures the collaborative flocking motion of the multi-agent systems, is presented in the paper. Moreover, if the two-rate sampling periods are smaller than single sampling period, there are more fluctuations and larger adjustment time of the velocities with similar position curves. If the two-rate sampling periods are larger than single sampling period, the fluctuation number is smaller with shorter adjustment
Acknowledgments
This research is supported by the National Natural Science Foundation of China (under grant 61273152, 61304052, 51407088), the Science Foundation of Education Office of Shandong Province of China (under grant ZR2011FM07, BS2015DX018).
Han Fujun received his Ph.D. degree in Precision Instruments and Machinery from Beijing University of Aeronautics and Astronautics in 2009. He is an Associate Professor in School of Information and Electrical Engineering, Ludong University. His main research direction is Control theory and control engineering.
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Han Fujun received his Ph.D. degree in Precision Instruments and Machinery from Beijing University of Aeronautics and Astronautics in 2009. He is an Associate Professor in School of Information and Electrical Engineering, Ludong University. His main research direction is Control theory and control engineering.
Gao Lei received his Master degree in Instrument from Harbin Institute of Technology in 2011. He is a Senior Engineer in 513 Institute of China Aerospace Science and Technology Corporation. His main research direction is data management of space vehicle and test equipments research.
Yang Hongyong received his Ph.D. degree in control theory and control engineering from Southeast University in 2005. He is a Professor in School of Information and Electrical Engineering, Ludong University. His research interest covers complex network, multi-agent systems, Intelligence control.