CooperativeControl - A Study of Synchronization and Group...

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A Study of Synchronization and Group Cooperation Using Partial Contraction Theory Jean-Jacques E. Slotine 1 , 2 and Wei Wang 1 , 3 1 Nonlinear Systems Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA 2 3 1 Introduction Synchronization, collective behavior, and group cooperation have been the object of extensive recent research. A fundamental understanding of aggregate motions in the natural world, such as bird flocks, Fsh schools, animal herds, or bee swarms, for instance, would greatly help in achieving desired collective behaviors of artiFcial multi-agent systems, such as vehicles with distributed cooperative control rules. In [38], Reynolds published his well-known computer model of “boids,” successfully forming an animation flock using three local rules: collision avoidance , velocity matching ,and fock centering . Motivated by the growth of colonies of bacteria, Viscek et al. [55] proposed a similar discrete-time model which realizes heading matching using information only from neighbors. Viscek’s model was later analyzed analytically [16, 52, 53]. Models in continuous-time [1, 22, 32, 33, 62] and combinations of Reynolds’ three rules [21, 34, 35, 49, 50] were also studied. Related questions can also be found e.g. in [3, 18, 20, 42], in oscillator synchronization [48], as well as in physics in the study of lasers [39] or of Bose-Einstein condensation [17]. This article provides a theoretical analysis tool, partial contraction theory [62], for the study of group cooperation and especially group agreement and synchronization. Partial contraction (or meta-contraction) theory is a straight- forward but very general application of contraction theory, a recent nonlinear system analysis tool based on studying convergence between two arbitrary system trajectories [26, 27, 45, 46]. Actually, partial contraction extends con- traction theory in that, while the latter is concerned with convergence to a unique trajectory, the former can describe convergence to particular proper- ties or manifolds [46, 62]. In particular, partial contraction theory can be used to easily derive sufficient conditions for coupled nonlinear networks to reach group agreement or synchronize. The article is organized as follows. Section 2 briefly reviews contraction theory and two important combination properties of contracting systems, and
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2 Jean-Jacques E. Slotine and Wei Wang introduces partial contraction theory. The collective behavior of coupled net- works of identical dynamic elements is studied in Section 3. Synchronization conditions for general diFusion-coupled networks are derived, and then are extended to networks with switching topologies or including group leaders. Adaptive versions are also derived. Section 4 studies a simpli±ed model of flocking in continuous-time. Concluding remarks are oFered in Section 5.
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CooperativeControl - A Study of Synchronization and Group...

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