On the Consensus of Homogeneous Multiagent Systems With Positivity Constraints

This paper investigates the consensus problem for multiagent systems, under the assumptions that the agents are homogeneous and described by a single-input positive state-space model, the mutual interactions are cooperative, and the static state-feedback law that each agent adopts to achieve consensus preserves the positivity of the overall system. Necessary conditions for the problem solvability, which allow us to address only the special case when the state matrix is irreducible, are provided. Under the irreducibility assumption, equivalent sets of sufficient conditions are derived. Special conditions either on the system description or on the Laplacian of the communication graph allow us to obtain necessary and sufficient conditions for the problem solvability. Finally, by exploiting some results about robust stability either of positive systems or of polynomials, further sufficient conditions for the problem solvability are derived. Numerical examples illustrate the proposed results.

[1]  B. Mohar.  THE LAPLACIAN SPECTRUM OF GRAPHS y , 1991 .

[2]  Ιωαννησ Τσιτσικλησ,et al.  PROBLEMS IN DECENTRALIZED DECISION MAKING AND COMPUTATION , 1984 .

[3]  Ettore Fornasini,et al.  Linear Copositive Lyapunov Functions for Continuous-Time Positive Switched Systems , 2010, IEEE Transactions on Automatic Control.

[4]  Thomas L. Markham,et al.  On the eigenvalues and diagonal entries of a Hermitian matrix , 1993 .

[5]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[6]  Snezana Pejic.  Algebraic graph theory in the analysis of frequency assignment problems , 2008 .

[7]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[8]  Randal W. Beard,et al.  Consensus seeking in multiagent systems under dynamically changing interaction topologies , 2005, IEEE Transactions on Automatic Control.

[9]  B. Ross Barmish,et al.  New Tools for Robustness of Linear Systems , 1993 .

[10]  Jie Lin,et al.  The multi-agent rendezvous problem , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[11]  Hal L. Smith,et al.  Monotone Dynamical Systems: An Introduction To The Theory Of Competitive And Cooperative Systems (Mathematical Surveys And Monographs) By Hal L. Smith , 1995 .

[12]  Karl Henrik Johansson,et al.  Structural Balance and Opinion Separation in Trust–Mistrust Social Networks , 2016, IEEE Transactions on Control of Network Systems.

[13]  F. Allgöwer,et al.  On consensus in multi-agent systems with linear high-order agents , 2008 .

[14]  Wei Ren.  On Consensus Algorithms for Double-Integrator Dynamics , 2008, IEEE Trans. Autom. Control..

[15]  Xiwei Liu.  Distributed nonlinear control algorithms for network consensus , 2010 .

[16]  Anders Rantzer,et al.  Distributed control of positive systems , 2011, IEEE Conference on Decision and Control and European Control Conference.

[17]  Corentin Briat.  Robust stability and stabilization of uncertain linear positive systems via Integral Linear Constraints : L 1-and L ∞-gains characterization , 2014 .

[18]  F. Harary,et al.  A survey of the theory of hypercube graphs , 1988 .

[19]  M. H. Schultz,et al.  Topological properties of hypercubes , 1988, IEEE Trans. Computers.

[20]  Dimitri Peaucelle,et al.  Efficient Convergence Rate Analysis of Multi-Agent Positive Systems Under Formation Control , 2014 .

[21]  M. Degroot.  Reaching a Consensus , 1974 .

[22]  William Watkins,et al.  The laplacian matrix of a graph: unimodular congruence , 1990 .

[23]  Alexey S. Matveev,et al.  Opinion Dynamics in Social Networks With Hostile Camps: Consensus vs. Polarization , 2015, IEEE Transactions on Automatic Control.

[24]  Corentin Briat,et al.  Robust stability and stabilization of uncertain linear positive systems via integral linear constraints: L1‐gain and L∞‐gain characterization , 2012, ArXiv.

[25]  Maria Elena Valcher,et al.  On the consensus problem with positivity constraints , 2016, 2016 American Control Conference (ACC).

[26]  M. Lewin.  On nonnegative matrices , 1971 .

[27]  Dimitri Peaucelle,et al.  Stability and persistence analysis of large scale interconnected positive systems , 2013, 2013 European Control Conference (ECC).

[28]  Maria Elena Valcher,et al.  On the Stabilizability and Consensus of Positive Homogeneous Multi-Agent Dynamical Systems , 2014, IEEE Transactions on Automatic Control.

[29]  Maria Elena Valcher,et al.  New results on the solution of the positive consensus problem , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[30]  F. Tadeo,et al.  Controller Synthesis for Positive Linear Systems With Bounded Controls , 2007, IEEE Transactions on Circuits and Systems II: Express Briefs.

[31]  Frank Allgöwer,et al.  On topology and dynamics of consensus among linear high-order agents , 2011, Int. J. Syst. Sci..

[32]  S. Rinaldi,et al.  Positive Linear Systems: Theory and Applications , 2000 .

[33]  Laxman M. Waghmare,et al.  APPLICATION OF WIRELESS SENSOR NETWORKS FOR GREENHOUSE PARAMETER CONTROL IN PRECISION AGRICULTURE , 2011 .

[34]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[35]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[36]  Xianping Liu,et al.  Further Results on Consensus of Second-Order Multi-Agent Systems With Exogenous Disturbance , 2013, IEEE Transactions on Circuits and Systems I: Regular Papers.

[37]  Ella M. Atkins,et al.  Distributed multi‐vehicle coordinated control via local information exchange , 2007 .

[38]  Jie Huang,et al.  Stability of a Class of Linear Switching Systems with Applications to Two Consensus Problems , 2011, IEEE Transactions on Automatic Control.

[39]  R. Merris.  Laplacian matrices of graphs: a survey , 1994 .

[40]  Robert Shorten,et al.  On Linear Copositive Lyapunov Functions and the Stability of Switched Positive Linear Systems , 2007, IEEE Transactions on Automatic Control.

[41]  Rodolphe Sepulchre,et al.  Synchronization in networks of identical linear systems , 2009, Autom..

[42]  Ravindra B. Bapat,et al.  On algebraic connectivity of graphs with at most two points of articulation in each block , 2012 .

[43]  Robert Shorten,et al.  A result on implicit consensus with application to emissions control , 2011, IEEE Conference on Decision and Control and European Control Conference.

[44]  D. Hinrichsen,et al.  Robust Stability of positive continuous time systems , 1996 .

[45]  Richard M. Murray,et al.  INFORMATION FLOW AND COOPERATIVE CONTROL OF VEHICLE FORMATIONS , 2002 .

[46]  M. Fiedler.  Algebraic connectivity of graphs , 1973 .

[47]  Diederich Hinrichsen,et al.  Robust stability and transient behaviour of positive linear systems , 2007 .

[48]  Alessandro Astolfi,et al.  Design of Positive Linear Observers for Positive Linear Systems via Coordinate Transformations and Positive Realizations , 2008, SIAM J. Control. Optim..

[49]  Robert Shorten,et al.  On linear co-positive Lyapunov functions for sets of linear positive systems , 2009, Autom..

[50]  Wei Ren,et al.  Information consensus in multivehicle cooperative control , 2007, IEEE Control Systems.

[51]  Kinkar Ch. Das,et al.  An improved upper bound for Laplacian graph eigenvalues , 2003 .

[52]  Yuguang Fang,et al.  Sufficient conditions for the stability of interval matrices , 1993 .

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