Consensus Problems with Distributed Delays, with Application to Traffic Flow Models

This paper focuses on consensus problems for a class of linear systems with distributed delay that are encountered in modeling traffic flow dynamics. In the application problems the distributed delay, whose kernel is a $\gamma$-distribution with a gap, represents the human drivers' behavior in the average. The aim of the paper is to give a characterization of the regions in the corresponding delay parameter space, where a consensus is reached for all initial conditions. The structure and properties of the system are fully exploited, which leads to explicit and computationally tractable expressions. As a by-product a stability theory for distributed delay systems with a $\gamma$-distribution kernel is developed. Also explicit expressions for the consensus function(al) of time-delay systems with constant and distributed delays that solve a consensus problem are provided. Several illustrative examples complete the presentation.

[1]  Wassim M. Haddad,et al.  On System State Equipartitioning and Semistability in Network Dynamical Systems with Arbitrary Time-Delays , 2006, CDC.

[2]  D. Helbing.  Traffic and related self-driven many-particle systems , 2000, cond-mat/0012229.

[3]  Wim Michiels,et al.  Stability and Stabilization of Time-Delay Systems (Advances in Design & Control) (Advances in Design and Control) , 2007 .

[4]  Wassim M. Haddad,et al.  On System State Equipartitioning and Semistability in Network Dynamical Systems with Arbitrary Time-Delays , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[5]  Dirk Roose,et al.  Equations with distributed delays: bifurcation analysis using computational tools for discrete delay equations , 2004 .

[6]  G. Samaey,et al.  DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations , 2001 .

[7]  Marc Green,et al.  "How Long Does It Take to Stop?" Methodological Analysis of Driver Perception-Brake Times , 2000 .

[8]  Robert Herman,et al.  Traffic Dynamics: Analysis of Stability in Car Following , 1959 .

[9]  G. Stépán,et al.  Subcritical Hopf bifurcations in a car-following model with reaction-time delay , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[11]  J. Hale.  Theory of Functional Differential Equations , 1977 .

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

[13]  Vladimir L. Kharitonov,et al.  Stability of Time-Delay Systems , 2003, Control Engineering.

[14]  E. Montroll,et al.  Traffic Dynamics: Studies in Car Following , 1958 .

[15]  L. C. Davis,et al.  Modifications of the optimal velocity traffic model to include delay due to driver reaction time , 2003 .

[16]  Silviu-Iulian Niculescu,et al.  Stability of Traffic Flow Behavior with Distributed Delays Modeling the Memory Effects of the Drivers , 2007, SIAM J. Appl. Math..

[17]  Wim Michiels,et al.  Asymptotic stability of some distributed delay systems: an algebraic approach , 2006 .

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

[19]  Rifat Sipahi,et al.  Analytical stability study of a deterministic car following model under multiple delay interactions , 2006 .

[20]  S. Lunel,et al.  Delay Equations. Functional-, Complex-, and Nonlinear Analysis , 1995 .

[21]  K. Hasebe,et al.  Analysis of optimal velocity model with explicit delay , 1998, patt-sol/9805002.

[22]  Chaouki T. Abdallah,et al.  Delay effects on static output feedback stabilization , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

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

[24]  S. Niculescu,et al.  Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach , 2007 .

[25]  D. D. Perlmutter,et al.  Stability of time‐delay systems , 1972 .

[26]  R. E. Wilson,et al.  Bifurcations and multiple traffic jams in a car-following model with reaction-time delay , 2005 .

[27]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.