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We investigate questions related to the time evolution of discrete graph dynamical systems where each node has a state from {0,1}. The configuration of a system at any time instant is a Boolean vector that specifies the state of each node at that instant. We say that two configurations are similar if the Hamming distance between them is small. Also, a predecessor of a configuration B is a configuration A such that B can be reached in one step from A. We study problems related to the similarity of predecessor configurations from which two similar configurations can be reached in one time step. We address these problems both analytically and experimentally. Our analytical results point out that the level of similarity between predecessors of two similar configurations depends on the local functions of the dynamical system. Our experimental results, which consider random graphs as well as small world networks, rely on the fact that the problem of finding predecessors can be reduced to the Boolean Satisfiability problem (SAT).
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Controllability of Formation Systems on Special Orthogonal Groups over Directed Graphs
Gradient flows provide a means for a networked formation system to reach and stabilize at a target configuration. However, the decentralization constraints and the geometry of the state space makes the appearance of stable but undesired configurations inevitable. The presence of these undesired stable configurations precludes global convergence to the target configuration. In this paper, we address the issue by considering a controlled formation system on special orthogonal groups over a directed graph. Agents of the system are tasked with stabilizing from others at target relative attitudes. The nominal dynamics of the agents are gradient flows of certain potential functions. These functions are parameter dependent, pretuned by the controller. To prevent the formation system from being trapped at an undesired configuration, we formulate and address the problem of whether the controller can steer the system from any configuration to any other configuration by retuning, on the fly, the parameters of the potential functions. We show that the answer is affirmative provided that the underlying graph is rooted with a single root node being fully actuated.
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- Award ID(s):
- 1809315
- PAR ID:
- 10289277
- Date Published:
- Journal Name:
- IEEE Transactions on Control of Network Systems
- ISSN:
- 2372-2533
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TCNS.2021.3050125
