From Global, Finite-Time, Linear Computations to Local, Edge-Based Interaction Rules

A network of locally interacting agents can be thought of as performing a distributed computation. But not all computations can be faithfully distributed. This technical note investigates which global, linear transformations can be computed in finite time using local rules with time varying weights, i.e., rules which rely solely on information from adjacent nodes in a network. The main result states that a linear transformation is computable in finite time using local rules if and only if the transformation has positive determinant. An optimal control problem is solved for finding the local interaction rules, and simulations are performed to elucidate how optimal solutions can be obtained.

[1]  Tucker R. Balch,et al.  Behavior-based formation control for multirobot teams , 1998, IEEE Trans. Robotics Autom..

[2]  Francesco Bullo,et al.  Coordination and Geometric Optimization via Distributed Dynamical Systems , 2003, SIAM J. Control. Optim..

[3]  Amit Bhaya,et al.  Real Matrices with Positive Determinant are Homotopic to the Identity , 1998, SIAM Rev..

[4]  R. Brockett Finite Dimensional Linear Systems , 2015 .

[5]  Randal W. Beard,et al.  Distributed Consensus in Multi-vehicle Cooperative Control - Theory and Applications , 2007, Communications and Control Engineering.

[6]  Santiago Grijalva,et al.  Prosumer-based control architecture for the future electricity grid , 2011, 2011 IEEE International Conference on Control Applications (CCA).

[7]  W. Press,et al.  Numerical Recipes: The Art of Scientific Computing , 1987 .

[8]  Kay Römer,et al.  The design space of wireless sensor networks , 2004, IEEE Wireless Communications.

[9]  Sanjay Lall,et al.  A Characterization of Convex Problems in Decentralized Control$^ast$ , 2005, IEEE Transactions on Automatic Control.

[10]  Jie Lin,et al.  Coordination of groups of mobile autonomous agents using nearest neighbor rules , 2003, IEEE Trans. Autom. Control..

[11]  S. Sastry Nonlinear Systems: Analysis, Stability, and Control , 1999 .

[12]  Uwe Helmke,et al.  Controllability of Bilinear Interconnected Systems , 2014 .

[13]  R. Olfati-Saber,et al.  Consensus Filters for Sensor Networks and Distributed Sensor Fusion , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[14]  G. Strang Introduction to Linear Algebra , 1993 .

[15]  J. Trinkle,et al.  Controlling Shapes of Ensembles of Robots of Finite Size with Nonholonomic Constraints , 2009 .

[16]  Jorge Cortes,et al.  Distributed Control of Robotic Networks: A Mathematical Approach to Motion Coordination Algorithms , 2009 .

[17]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[18]  Sanjay Lall,et al.  A graph-theoretic approach to distributed control over networks , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[19]  George J. Pappas,et al.  Stable flocking of mobile agents part I: dynamic topology , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[20]  Magnus Egerstedt,et al.  Distributed Power Allocation in Prosumer Networks , 2012 .

[21]  A.L. Dimeas,et al.  Operation of a multiagent system for microgrid control , 2005, IEEE Transactions on Power Systems.

[22]  Raphaël M. Jungers,et al.  Graph diameter, eigenvalues, and minimum-time consensus , 2012, Autom..

[23]  Naomi Ehrich Leonard,et al.  Coordinated patterns of unit speed particles on a closed curve , 2007, Syst. Control. Lett..

[24]  Asuman E. Ozdaglar,et al.  Constrained Consensus and Optimization in Multi-Agent Networks , 2008, IEEE Transactions on Automatic Control.

[25]  R. Brockett System Theory on Group Manifolds and Coset Spaces , 1972 .

[26]  Magnus Egerstedt,et al.  Graph Theoretic Methods in Multiagent Networks , 2010, Princeton Series in Applied Mathematics.

[27]  Shreyas Sundaram,et al.  Distributed function calculation and consensus using linear iterative strategies , 2008, IEEE Journal on Selected Areas in Communications.

[28]  M. Egerstedt,et al.  From Global Linear Computations to Local Interaction Rules , 2013, 1311.5959.

[29]  Magnus Egerstedt,et al.  Distributed Coordination Control of Multiagent Systems While Preserving Connectedness , 2007, IEEE Transactions on Robotics.

[30]  Zamora,et al.  Electronic textiles: a platform for pervasive computing , 2003, Proceedings of the IEEE.