Multi-Agent Systems with Reciprocal Interaction Laws

In this thesis, we investigate a special class of multi-agent systems, which we call reciprocal multi-agent (RMA) systems. The evolution of agents in a RMA system is governed by interactions between pairs of agents. Each interaction is reciprocal, and the magnitude of attraction/repulsion depends only on distances between agents. We investigate the class of RMA systems from four perspectives, these are two basic properties of the dynamical system, one formula for computing the Morse indices/co-indices of critical formations, and one formation control model as a variation of the class of RMA systems. An important aspect about RMA systems is that there is an equivariant potential function associated with each RMA system so that the equations of motion of agents are actually a gradient flow. The two basic properties about this class of gradient systems we will investigate are about the convergence of the gradient flow, and about the question whether the associated potential function is generically an equivariant Morse function. We develop systematic approaches for studying these two problems, and establish important results. A RMA system often has multiple critical formations and in general, these are hard to locate. So in this thesis, we consider a special class of RMA systems whereby there is a geometric characterization for each critical formation. A formula associated with the characterization is developed for computing the Morse index/co-index of each critical formation. This formula has a potential impact on the design and control of RMA systems. In this thesis, we also consider a formation control model whereby the control of formation is achieved by varying interactions between selected pairs of agents. This model can be interpreted in different ways in terms of patterns of information flow, and we establish results about the controllability of this control system for both centralized and decentralized problems.

[1]  Tianguang Chu,et al.  Self-organized motion in anisotropic swarms , 2003 .

[2]  Mohamed-Ali Belabbas,et al.  On global feedback stabilization of decentralized formation control , 2011, IEEE Conference on Decision and Control and European Control Conference.

[3]  A. Stephen Morse,et al.  Operations on Rigid Formations of Autonomous Agents , 2003, Commun. Inf. Syst..

[4]  H. Sussmann,et al.  Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[5]  R. Murray,et al.  Consensus protocols for networks of dynamic agents , 2003, Proceedings of the 2003 American Control Conference, 2003..

[6]  Reza Olfati-Saber,et al.  Flocking for multi-agent dynamic systems: algorithms and theory , 2006, IEEE Transactions on Automatic Control.

[7]  John Baillieul,et al.  Information patterns and Hedging Brockett's theorem in controlling vehicle formations , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[8]  K. Passino,et al.  A class of attractions/repulsion functions for stable swarm aggregations , 2004 .

[9]  M. Field.  Stratifications of equivariant varieties , 1977, Bulletin of the Australian Mathematical Society.

[10]  Jorge Cortés,et al.  Finite-time convergent gradient flows with applications to network consensus , 2006, Autom..

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

[12]  Panos J. Antsaklis,et al.  Control and Communication Challenges in Networked Real-Time Systems , 2007, Proceedings of the IEEE.

[13]  J.A. Marshall,et al.  On Invariance of Cyclic Group Symmetries in Multiagent Formations , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[14]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[15]  Brian D. O. Anderson,et al.  Control of a three-coleader formation in the plane , 2007, Syst. Control. Lett..

[16]  Brian D. O. Anderson,et al.  Equivariant Morse theory and formation control , 2013, 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[17]  Brian D. O. Anderson,et al.  Controlling a triangular formation of mobile autonomous agents , 2007, 2007 46th IEEE Conference on Decision and Control.

[18]  M. Golubitsky,et al.  Stable mappings and their singularities , 1973 .

[19]  Brian D. O. Anderson,et al.  Counting Critical Formations on a Line , 2014, SIAM J. Control. Optim..

[20]  J. Sylvester.  XIX. A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares , 1852 .

[21]  Richard M. Murray,et al.  DISTRIBUTED COOPERATIVE CONTROL OF MULTIPLE VEHICLE FORMATIONS USING STRUCTURAL POTENTIAL FUNCTIONS , 2002 .

[22]  R. Pignoni.  Density and stability of Morse functions on a stratified space , 1979 .

[23]  Roger W. Brockett,et al.  Centralized and decentralized formation control with controllable interaction laws , 2014, 53rd IEEE Conference on Decision and Control.

[24]  B. Anderson.  Morse theory and formation control , 2011, 2011 19th Mediterranean Conference on Control & Automation (MED).

[25]  Florian Dörfler,et al.  Geometric Analysis of the Formation Problem for Autonomous Robots , 2010, IEEE Transactions on Automatic Control.

[26]  J. Mather,et al.  Stratifications and Mappings , 1973 .

[27]  Mireille E. Broucke,et al.  Stabilisation of infinitesimally rigid formations of multi-robot networks , 2009, Int. J. Control.

[28]  B. Anderson,et al.  Controlling Four Agent Formations , 2010 .

[29]  R. Brockett.  On the control of a flock by a leader , 2010 .

[30]  Jeff Moehlis,et al.  Equivariant dynamical systems , 2007, Scholarpedia.

[31]  Michael Field.  Symmetry breaking for compact Lie groups , 1996 .

[32]  Mohamed-Ali Belabbas.  Mathematical aspects of decentralized control of formations in the plane , 2011, ArXiv.