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
jjs@mit.edu
3
wangwei@mit.edu
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