# Modeling nonlinear road traffic networks for junction control

• Volume: 22, Issue: 3, page 723-732
• ISSN: 1641-876X

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## Abstract

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The paper introduces a method of mathematical modeling of high scale road traffic networks, where a new special hypermatrix structure is intended to be used. The structure describes the inner-inner, inner-outer and outer-outer relations, and laws of a network area. The research examines the nonlinear equation system. The analysed model can be applied to the testing and planning of large-scale road traffic networks and the regulation of traffic systems. The elaborated model is in state space form, where the states are vehicle densities on a particular lane and the dynamics are described by a nonlinear state constrained positive system. This model can be used directly for simulation and analysis and as a starting point for investigating various control strategies. The stability of the traffic over the network can be analyzed by constructing a linear Lyapunov function and the associated theory. The model points out that in intersection control one must take the traffic density values of both the input and the output sections into account. Generally, the control of any domain has to take the density of input and output sections into consideration.

## How to cite

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Tamás Péter. "Modeling nonlinear road traffic networks for junction control." International Journal of Applied Mathematics and Computer Science 22.3 (2012): 723-732. <http://eudml.org/doc/244066>.

@article{TamásPéter2012,
abstract = {The paper introduces a method of mathematical modeling of high scale road traffic networks, where a new special hypermatrix structure is intended to be used. The structure describes the inner-inner, inner-outer and outer-outer relations, and laws of a network area. The research examines the nonlinear equation system. The analysed model can be applied to the testing and planning of large-scale road traffic networks and the regulation of traffic systems. The elaborated model is in state space form, where the states are vehicle densities on a particular lane and the dynamics are described by a nonlinear state constrained positive system. This model can be used directly for simulation and analysis and as a starting point for investigating various control strategies. The stability of the traffic over the network can be analyzed by constructing a linear Lyapunov function and the associated theory. The model points out that in intersection control one must take the traffic density values of both the input and the output sections into account. Generally, the control of any domain has to take the density of input and output sections into consideration.},
author = {Tamás Péter},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {mathematical modeling and control; traffic networks; nonlinear positive system; linear Lyapunov function},
language = {eng},
number = {3},
pages = {723-732},
title = {Modeling nonlinear road traffic networks for junction control},
url = {http://eudml.org/doc/244066},
volume = {22},
year = {2012},
}

TY - JOUR
AU - Tamás Péter
TI - Modeling nonlinear road traffic networks for junction control
JO - International Journal of Applied Mathematics and Computer Science
PY - 2012
VL - 22
IS - 3
SP - 723
EP - 732
AB - The paper introduces a method of mathematical modeling of high scale road traffic networks, where a new special hypermatrix structure is intended to be used. The structure describes the inner-inner, inner-outer and outer-outer relations, and laws of a network area. The research examines the nonlinear equation system. The analysed model can be applied to the testing and planning of large-scale road traffic networks and the regulation of traffic systems. The elaborated model is in state space form, where the states are vehicle densities on a particular lane and the dynamics are described by a nonlinear state constrained positive system. This model can be used directly for simulation and analysis and as a starting point for investigating various control strategies. The stability of the traffic over the network can be analyzed by constructing a linear Lyapunov function and the associated theory. The model points out that in intersection control one must take the traffic density values of both the input and the output sections into account. Generally, the control of any domain has to take the density of input and output sections into consideration.
LA - eng
KW - mathematical modeling and control; traffic networks; nonlinear positive system; linear Lyapunov function
UR - http://eudml.org/doc/244066
ER -

## References

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1. Arneson, H. and Langbort, C. (2009). Linear programming based routing design for a class of positive systems with integral and capacity constraints, Proceedings of the 1st IFAC Workshop on Estimation and Control of Networked Systems, Venice, Italy, pp. 352-357.
2. Bacciotti, A. (1983). On the positive orthant controllability of two-dimensional bilinear systems, System Control Letters 3(1): 53-55. Zbl0532.93005
3. Bastin, G. (1999). Issues in Modeling and Control of Massbalanced Systems, in D. Aeyels, F. Lamnabhilagerrigve and A. van der Schaft (Eds.), Stability and Stabilization of Nonlinear Systems, Lecture Notes in Control and Information Sciences, Vol. 246, Springer, London, pp. 53-74. Zbl0934.93011
4. Bertsimas, D. and Patterson, S. (1998). The air traffic flow management problem with en route capacities, Operations Research 46(3): 406-422. Zbl0996.90010
5. Boothby, W.M. (1982). Some comments on positive orthant controllability of bilinear systems, SIAM Journal of Control and Optimization 20(5): 634-644. Zbl0488.93009
6. Caccetta, L. and Rumchev, V. (2000). A survey of reachability and controllability for positive linear systems, Annals of Operations Research 98(1): 101-122. Zbl0972.93003
7. Cantoni, M., Weyer, E., Li, Y., Ooi, S., Mareels, I. and Ryan, M. (2007). Control of large-scale irrigation networks, Proceedings of the IEEE 95(1): 75-91.
8. Chang, T. and Lin, J. (2000). Optimal signal timing for an oversaturated intersection, Transportation Research B 34(6): 471-491.
9. Coxson, P. and Shapiro, H. (1987). Positive input reachability and controllability of positive systems, Linear Algebra and Its Applications 94: 35-53. Zbl0633.93008
10. Diakaki, C., Dinopoulou, V., Aboudolas, K., Papageorgiou, M., Ben-Shabat, E., Seider, E., and Leibov, A. (2003). Extensions and new applications of the traffic control strategy TUC, TRB 2003 Annual Meeting, Washington, DC, USA.
11. Diakaki, C., Papageorgiou, M. and McLean, T. (1998). Integrated traffic-responsive urban control strategy, IN-TUC: Application and Evaluation in Glasgow. DACCORD Workshop on Advanced Motorway Traffic Control, Lancaster UK, pp. 46-55.
12. Drew, D.R. (1968). Traffic Flow Theory and Control, McGrawHill Book Company, New York, NY.
13. Farina, L. and Rinaldi, S. (2000). Positive Linear Systems: Theory and Applications, Wiley, New York, NY. Zbl0988.93002
14. Földesi, P. and Botzheim, J. (2010). Modeling of loss aversion in solving fuzzy road transport traveling salesman problem using eugenic bacterial memetic algorithm, Memetic Computing 2(4): 259-271.
15. Fu, Y., Wang, H., Lu, C. and Chandra, R.S. (2006). Distributed utilization control for real-time clusters with load balancing, Proceedings of the 27th IEEE International Real-Time Systems Symposium, Rio de Janeiro, Brazil, pp. 137-146.
16. Gazis, D. (1964). Optimum control of a system of oversaturated intersections, Operations Research 12(6): 815-831. Zbl0131.18903
17. Gazis, D.C. (1976). Optimal control of oversaturated store-andforward transportation networks, Transportation Science 10(1): 1-9.
18. Gazis, D. and Potts, R. (1963). The oversaturated intersection, Proceedings of the 2nd International Symposium on Theory of Traffic Flow, London, UK, pp. 221-237.
19. Greenberg, H. (1959). An analysis of traffic flow, Operations Research 7(1): 79-85.
20. Greenshields, B. (1934). A study of traffic capacity, Proceedings of the Highway Research Board 14: 448-477.
21. Guardabassi, G., Locatelli, A. and Papageorgiou, M. (1984). A note on the optimal control of an oversaturated intersection, Transportation Research B 18(2): 111-113.
22. Haddad, J., De Schutter, B., Mahalel, D., Ioslovich, I. and Gutman, P. (2010). Optimal steady-state control for isolated traffic intersections, IEEE Transactions on Automatic Control 55(11): 2612-2617.
23. Harmati, I., Orbán, G. and Várlaki, P. (2007). Takagi-Sugeno fuzzy control models for large scale logistics systems, Proceedings of the IEEE International Symposium on Computational Intelligence and Intelligent Informatics (ISCIII'07), Agadir, Morocco, pp. 199-203.
24. Harmati, I., Rövid, A. and Várlaki, P. (2010). Approximation of force and energy in vehicle crash using LPV type description, WSEAS Transactions on Systems 9(7): 734-743.
25. Improta, G. and Cantarella, G. (1984). Control systems design for an individual signalised junction, Transportation Research B 18(2): 147-167.
26. Kachroo, P. and Ozbay, K. (1999). Feedback Control Theory for Dynamic Traffic Assignment, Springer, London. Zbl1043.93500
27. Kashani, H. and Saridis, G. (1983). Intelligent control for urban traffic systems, Automatica 19(2): 191-197. Zbl0503.90045
28. Krozel, J. Jakobovits, R. and Penny, S. (2006). An algorithmic approach for airspace how programs, Air Traffic Control Quartelly 14(3): 203-229.
29. Kulcsar, B., Varga, I. and Bokor, J. (2005). Constrained split rate estimation by moving horizon, 16th IFAC World Congress, Prague, Czech Republic, Vol. 16, Part I, p. 2035.
30. Luenberger, D.G. (1979). Introduction to Dynamical Systems. Theory, Models and Applications, John Wiley and Sons, New York, NY. Zbl0458.93001
31. Mazloumi, E. (2008). A new delay function for signalised intersections, Road Transportation Research 17(3): 3-12.
32. Michalopoulos, P. and Stephanopoulos, G. (1978). Optimal control of oversaturated intersections: Theoretical and practical considerations, Traffic Engineering Control 19(5): 216-221.
33. Papageorgiu, M. (1991). Concise Encyclopedia of Traffic and Transportation Systems, Pergamon Press, Oxford.
34. Peter, T. and Basset, M. (2009). Application of new traffic models for determine optimal trajectories, International Forum on Strategic Technologies (IFOST 2009), Ho Chi Minh, Vietnam, pp. 89-94.
35. Peter, T. and Bokor, J. (2010). Modeling road traffic networks for control, Annual International Conference on Network Technologies and Communications, NTC2010, Phuket Beach Resort, Thailand, pp. 18-22.
36. Rouphail, N. and Akcelik, R. (1992). Oversaturation delay estimates with consideration of peaking, Transportation Research Rec 1365: 71-81.
37. Sachkov, Y.L. (1997). On positive orthant controllability of bilinear systems in small codimensions, SIAM Journal of Control and Optimization 35(1): 29-35. Zbl0876.93013
38. Talmor, I. and Mahalel, D. (2007). Signal design for an isolated intersection during congestion, Operation Research Soc 58(4): 454-466. Zbl1143.90319
39. Tettamanti, T., Varga, I., Kulcsár, B and Bokor, J. (2008). Model predictive control in urban traffic network management, 16th Mediterranean Conference on Control and Automation, Ajaccio, France, pp. 1538-1543.
40. Valcher, M. (1996). Controllability and reachability criteria for discrete-time positive systems, International Journal of Control 65(3): 511-536. Zbl0873.93009
41. Webster, F. (1958). Traffic signal settings, Technical report, Great Britain Road Research Lab., London.

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