510 research outputs found
Multi-agent Systems with Compasses
This paper investigates agreement protocols over cooperative and
cooperative--antagonistic multi-agent networks with coupled continuous-time
nonlinear dynamics. To guarantee convergence for such systems, it is common in
the literature to assume that the vector field of each agent is pointing inside
the convex hull formed by the states of the agent and its neighbors, given that
the relative states between each agent and its neighbors are available. This
convexity condition is relaxed in this paper, as we show that it is enough that
the vector field belongs to a strict tangent cone based on a local supporting
hyperrectangle. The new condition has the natural physical interpretation of
requiring shared reference directions in addition to the available local
relative states. Such shared reference directions can be further interpreted as
if each agent holds a magnetic compass indicating the orientations of a global
frame. It is proven that the cooperative multi-agent system achieves
exponential state agreement if and only if the time-varying interaction graph
is uniformly jointly quasi-strongly connected. Cooperative--antagonistic
multi-agent systems are also considered. For these systems, the relation has a
negative sign for arcs corresponding to antagonistic interactions. State
agreement may not be achieved, but instead it is shown that all the agents'
states asymptotically converge, and their limits agree componentwise in
absolute values if and in general only if the time-varying interaction graph is
uniformly jointly strongly connected.Comment: SIAM Journal on Control and Optimization, In pres
Dynamics over Signed Networks
A signed network is a network with each link associated with a positive or
negative sign. Models for nodes interacting over such signed networks, where
two different types of interactions take place along the positive and negative
links, respectively, arise from various biological, social, political, and
economic systems. As modifications to the conventional DeGroot dynamics for
positive links, two basic types of negative interactions along negative links,
namely the opposing rule and the repelling rule, have been proposed and studied
in the literature. This paper reviews a few fundamental convergence results for
such dynamics over deterministic or random signed networks under a unified
algebraic-graphical method. We show that a systematic tool of studying node
state evolution over signed networks can be obtained utilizing generalized
Perron-Frobenius theory, graph theory, and elementary algebraic recursions.Comment: In press, SIAM Revie
Distributed Optimization: Convergence Conditions from a Dynamical System Perspective
This paper explores the fundamental properties of distributed minimization of
a sum of functions with each function only known to one node, and a
pre-specified level of node knowledge and computational capacity. We define the
optimization information each node receives from its objective function, the
neighboring information each node receives from its neighbors, and the
computational capacity each node can take advantage of in controlling its
state. It is proven that there exist a neighboring information way and a
control law that guarantee global optimal consensus if and only if the solution
sets of the local objective functions admit a nonempty intersection set for
fixed strongly connected graphs. Then we show that for any tolerated error, we
can find a control law that guarantees global optimal consensus within this
error for fixed, bidirectional, and connected graphs under mild conditions. For
time-varying graphs, we show that optimal consensus can always be achieved as
long as the graph is uniformly jointly strongly connected and the nonempty
intersection condition holds. The results illustrate that nonempty intersection
for the local optimal solution sets is a critical condition for successful
distributed optimization for a large class of algorithms
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