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A two-dimensional multi-class traffic flow model
Periodic consensus in network systems with general distributed processing delays
1. | College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, 410073, P. R. China |
2. | College of Mathematics and Statistics, Changsha University of Science Technology, Changsha, 410076, P. R. China |
How to understand the dynamical consensus patterns in network systems is of particular significance in both theories and applications. In this paper, we are interested in investigating the influences of distributed processing delay on the consensus patterns in a network model. As new observations, we show that the desired network model undergoes both weak consensus and periodic consensus behaviors when the parameters reach a threshold value and the connectedness of the network system may be absent. In results, some criterions of weak consensus and periodic consensus with exponential convergent rate are established by the standard functional differential equations analysis. An analytic formula is given to calculate the asymptotic periodic consensus in terms of model parameters and the initial time interval. Also, we post the threshold values for some typical distributions included uniform distribution and Gamma distribution. Finally, we give the numerical simulation and analyse the influences of different delays on the consensus.
References:
[1] |
M. H. DeGroot, Reaching a consensus, J. Am. Stat. Assoc., 69 (1974), 118-121. Google Scholar |
[2] |
J. A. Fax and R. M. Murray,
Information flow and cooperative control of vehicle formations, IEEE Trans. Autom. Control, 49 (2004), 1465-1476.
doi: 10.1109/TAC.2004.834433. |
[3] |
Q. Feng, S. K. Nguang and A. Seuret,
Stability analysis of linear coupled differential–difference systems with general distributed delays, IEEE Transactions on Automatic Control, 65 (2020), 1356-1363.
doi: 10.1109/TAC.2019.2928145. |
[4] |
J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[5] |
J. Jost, F. M. Atay and W. Lu,
Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays, Networks and Heterogeneous Media, 6 (2011), 329-349.
doi: 10.3934/nhm.2011.6.329. |
[6] |
F. A. Khasawneh and B. P. Mann,
A spectral element approach for the stability of delay systems, Int. J. Numer. Meth. Engng., 87 (2011), 566-592.
|
[7] |
N. A. Lynch, Distributed Algorithms, San Francisco, CA: Morgan Kaufmann, 1996.
doi: 10.1108/IMDS-01-2014-0013. |
[8] |
F. Mazenc, M. Malisoff and H. $\ddot{O}$zbay,
Stability and robustness analysis for switched systems with time-varying delays, SIAM J. Control Optim., 56 (2018), 158-182.
doi: 10.1137/16M1104895. |
[9] |
M. Michiels, I. C. Morărescu and S.-I. Niculescu,
Consensus problems with distributed delays, with application to traffic flow models, SIAM J. Control Optim., 48 (2009), 77-101.
doi: 10.1137/060671425. |
[10] |
I. C. Morarescu, W. Michiels and M. Jungers, Synchronization of coupled nonlinear oscillatiors with gamma-distributed delays, in American Control Conference, ACC 2013, (2013), Washington, United States. Google Scholar |
[11] |
R. Olfati-Saber and R. M. Murray,
Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control, 49 (2004), 1520-1533.
doi: 10.1109/TAC.2004.834113. |
[12] |
A. V. Proskurnikov, Average consensus in networks with nonlinearly delayed couplings and switching topology, Automatica, 49 (2013) 2928–2932.
doi: 10.1016/j.automatica.2013.06.007. |
[13] |
D. Serre, Matrices, Graduate Texts in Mathematics, 216 (2010), Springer.
doi: 10.1007/978-1-4419-7683-3. |
[14] |
J. W. H. So, X. Tang and X. Zou,
Stability in a linear delay system without instantaneous negative feedback, SIAM J. Math. Anal., 33 (2002), 1297-1304.
doi: 10.1137/S0036141001389263. |
[15] |
O. Solomon and E. Fridman,
New stability conditions for systems with distributed delays, Automatica, 49 (2013), 3467-3475.
doi: 10.1016/j.automatica.2013.08.025. |
[16] |
T. Vicsek, A Czirók, E. Ben-Jacob, I. Cohen and O. Shochet,
Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.
doi: 10.1103/PhysRevLett.75.1226. |
show all references
References:
[1] |
M. H. DeGroot, Reaching a consensus, J. Am. Stat. Assoc., 69 (1974), 118-121. Google Scholar |
[2] |
J. A. Fax and R. M. Murray,
Information flow and cooperative control of vehicle formations, IEEE Trans. Autom. Control, 49 (2004), 1465-1476.
doi: 10.1109/TAC.2004.834433. |
[3] |
Q. Feng, S. K. Nguang and A. Seuret,
Stability analysis of linear coupled differential–difference systems with general distributed delays, IEEE Transactions on Automatic Control, 65 (2020), 1356-1363.
doi: 10.1109/TAC.2019.2928145. |
[4] |
J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[5] |
J. Jost, F. M. Atay and W. Lu,
Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays, Networks and Heterogeneous Media, 6 (2011), 329-349.
doi: 10.3934/nhm.2011.6.329. |
[6] |
F. A. Khasawneh and B. P. Mann,
A spectral element approach for the stability of delay systems, Int. J. Numer. Meth. Engng., 87 (2011), 566-592.
|
[7] |
N. A. Lynch, Distributed Algorithms, San Francisco, CA: Morgan Kaufmann, 1996.
doi: 10.1108/IMDS-01-2014-0013. |
[8] |
F. Mazenc, M. Malisoff and H. $\ddot{O}$zbay,
Stability and robustness analysis for switched systems with time-varying delays, SIAM J. Control Optim., 56 (2018), 158-182.
doi: 10.1137/16M1104895. |
[9] |
M. Michiels, I. C. Morărescu and S.-I. Niculescu,
Consensus problems with distributed delays, with application to traffic flow models, SIAM J. Control Optim., 48 (2009), 77-101.
doi: 10.1137/060671425. |
[10] |
I. C. Morarescu, W. Michiels and M. Jungers, Synchronization of coupled nonlinear oscillatiors with gamma-distributed delays, in American Control Conference, ACC 2013, (2013), Washington, United States. Google Scholar |
[11] |
R. Olfati-Saber and R. M. Murray,
Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control, 49 (2004), 1520-1533.
doi: 10.1109/TAC.2004.834113. |
[12] |
A. V. Proskurnikov, Average consensus in networks with nonlinearly delayed couplings and switching topology, Automatica, 49 (2013) 2928–2932.
doi: 10.1016/j.automatica.2013.06.007. |
[13] |
D. Serre, Matrices, Graduate Texts in Mathematics, 216 (2010), Springer.
doi: 10.1007/978-1-4419-7683-3. |
[14] |
J. W. H. So, X. Tang and X. Zou,
Stability in a linear delay system without instantaneous negative feedback, SIAM J. Math. Anal., 33 (2002), 1297-1304.
doi: 10.1137/S0036141001389263. |
[15] |
O. Solomon and E. Fridman,
New stability conditions for systems with distributed delays, Automatica, 49 (2013), 3467-3475.
doi: 10.1016/j.automatica.2013.08.025. |
[16] |
T. Vicsek, A Czirók, E. Ben-Jacob, I. Cohen and O. Shochet,
Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.
doi: 10.1103/PhysRevLett.75.1226. |










Cases | Descriptions | ||
Uniform distribution | |||
116.7278 | 16.8680 | Exponential distribution | |
3.8152 | 2.8801 | Special |
|
2.7019 | 2.3530 | Special |
|
Bernoulli distribution |
Cases | Descriptions | ||
Uniform distribution | |||
116.7278 | 16.8680 | Exponential distribution | |
3.8152 | 2.8801 | Special |
|
2.7019 | 2.3530 | Special |
|
Bernoulli distribution |
7.0605 | 0.3183 | 2.7692 | 0.4617 | 0.9713 |
8.2346 | 6.9483 | 3.1710 | 9.5022 | 0.3445 |
where the numbers are randomly selected in interval (0, 10). |
7.0605 | 0.3183 | 2.7692 | 0.4617 | 0.9713 |
8.2346 | 6.9483 | 3.1710 | 9.5022 | 0.3445 |
where the numbers are randomly selected in interval (0, 10). |
Cases | Results | ||
Uniform distribution (Fig. 1) | consensus | ||
periodic consensus | |||
Exponential distribution(Fig. 2) | consensus | ||
periodic consensus | |||
Special |
consensus | ||
periodic consensus | |||
Special |
consensus | ||
periodic consensus | |||
Bernoulli distribution(Fig. 5) | consensus | ||
periodic consensus |
Cases | Results | ||
Uniform distribution (Fig. 1) | consensus | ||
periodic consensus | |||
Exponential distribution(Fig. 2) | consensus | ||
periodic consensus | |||
Special |
consensus | ||
periodic consensus | |||
Special |
consensus | ||
periodic consensus | |||
Bernoulli distribution(Fig. 5) | consensus | ||
periodic consensus |
Distribution cases | Group 1(blue) | Group 2(red) | ||
Uniform (Fig. 6) | consensus | periodic consensus | ||
periodic consensus | divergence | |||
Exponential (Fig. 7) | consensus | periodic consensus | ||
periodic consensus | divergence | |||
Gamma 1(Fig. 8) | consensus | periodic consensus | ||
periodic consensus | divergence | |||
Gamma 2(Fig. 9) | consensus | periodic consensus | ||
periodic consensus | divergence | |||
Bernoulli(Fig. 10) | consensus | periodic consensus | ||
periodic consensus | divergence |
Distribution cases | Group 1(blue) | Group 2(red) | ||
Uniform (Fig. 6) | consensus | periodic consensus | ||
periodic consensus | divergence | |||
Exponential (Fig. 7) | consensus | periodic consensus | ||
periodic consensus | divergence | |||
Gamma 1(Fig. 8) | consensus | periodic consensus | ||
periodic consensus | divergence | |||
Gamma 2(Fig. 9) | consensus | periodic consensus | ||
periodic consensus | divergence | |||
Bernoulli(Fig. 10) | consensus | periodic consensus | ||
periodic consensus | divergence |
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