# Optimal Feedback Control for a Perimeter Trafﬁc Flow at an Urban Region

### Jack Haddad, Ilya Ioslovich

#### Abstract

Traffic flow control has motivated many researchers since early decades of the 19th century. Recently, the concept of a perimeter traffic control for an urban region has been strengthened by a series of works, which have shown that a perimeter controller, located at a region border, can manipulate the transfer flows across the border to maximize the total outflow of the region. The macroscopic fundamental diagram (MFD), that relates average flow with accumulation, is used to model the traffic flow dynamics in the region. Assuming that the control inputs of the cross-border flows are coupled, i.e. the border is always utilized over time for transferring flows by one of the two directions (from and towards the region), and that the urban region has two traffic flow demands generated inside the region with internal and external destinations, and a generated traffic flow outside the region with a destination to the region, the explicit formulation of the optimal feedback control policy and a proof of optimality are provided. The proof is based on the modified Krotov-Bellman sufficient conditions of optimality, where the upper and lower bounds of state variables are calculated.

#### References

- Aboudolas, K. and Geroliminis, N. (2013). Perimeter and boundary flow control in multi-reservoir heterogeneous networks. Transportation Research Part B, 55:265-281.
- Buisson, C. and Ladier, C. (2009). Exploring the impact of homogeneity of traffic measurements on the existence of macroscopic fundamental diagrams. Transportation Research Record, 2124:127-136.
- Daganzo, C. F. (2007). Urban gridlock: Macroscopic modeling and mitigation approaches. Transportation Research Part B, 41(1):49-62.
- Daganzo, C. F., Gayah, V. V., and Gonzales, E. J. (2011). Macroscopic relations of urban traffic variables: Bifurcations, multivaluedness and instability. Transportation Research Part B, 45(1):278-288.
- Gayah, V. V. and Daganzo, C. F. (2011). Clockwise hysteresis loops in the macroscopic fundamental diagram: An effect of network instability. Transportation Research Part B, 45(4):643-655.
- Geroliminis, N. and Boyaci, B. (2012). The effect of variability of urban systems characteristics in the network capacity. Transportation Research Part B, 46(10):1607-1623.
- Geroliminis, N. and Daganzo, C. F. (2008). Existence of urban-scale macroscopic fundamental diagrams: some experimental findings. Transportation Research Part B, 42(9):759-770.
- Geroliminis, N., Haddad, J., and Ramezani, M. (2013). Optimal perimeter control for two urban regions with macroscopic fundamental diagrams: A model predictive approach. IEEE Transactions on Intelligent Transportation Systems, 14(1):348-359.
- Geroliminis, N. and Sun, J. (2011a). Hysteresis phenomena of a macroscopic fundamental diagram in freeway networks. Transportation Research Part A, 45(9):966- 979.
- Geroliminis, N. and Sun, J. (2011b). Properties of a welldefined macroscopic fundamental diagram for urban traffic. Transportation Research Part B, 45(3):605- 617.
- Godfrey, J. W. (1969). The mechanism of a road network. Traffic Engineering and Control, 11(7):323-327.
- Haddad, J. and Geroliminis, N. (2012). On the stability of traffic perimeter control in two-region urban cities. Transportation Research Part B, 46(1):1159-1176.
- Haddad, J., Ramezani, M., and Geroliminis, N. (2013). Cooperative traffic control of a mixed network with two urban regions and a freeway. Transportation Research Part B, 54:17-36.
- Hajiahmadi, M., Haddad, J., Schutter, B. D., and Geroliminis, N. (2013). Optimal hybrid macroscopic traffic control for urban regions: Perimeter and switching signal plans controllers. In European Control Conference 13.
- Ji, Y., Daamen, W., Hoogendoorn, S., HoogendoornLanser, S., and Qian, X. (2010). Macroscopic fundamental diagram: Investigating its shape using simulation data. Transportation Research Record, 2161:42- 48.
- Ji, Y. and Geroliminis, N. (2012). On the spatial partitioning of urban transportation networks. Transportation Research Part B, 46(10):1639-1656.
- Keyvan-Ekbatani, M., Kouvelas, A., Papamichail, I., and Papageorgiou, M. (2012). Exploiting the fundamnetal diagram of urban networks for feedback-based gating. Transportation Research Part B, 46(10):1393-1403.
- Knoop, V., Hoogendoorn, S., and van Lint, H. (2013). The impact of traffic dynamics on the macroscopic fundamental diagram. In 92nd Annual Meeting of Transportation Research Board, Washington D.C., USA.
- Knoop, V. L., Hoogendoorn, S. P., and Van Lint, J. W. C. (2012). Routing strategies based on the macroscopic fundamental diagram. Transportation Research Record, 2315:1-10.
- Krotov, V. F. (1996). Global methods in optimal control theory. M. Dekker, NY, USA.
- Krotov, V. K., Burkreev, V. Z., and Gurman, V. I. (1971). New Variational Methods In Flight Dynamics. Coronet Books.
- Mahmassani, H., Williams, J., and Herman, R. (1987). Performance of urban traffic networks. In Gartner, N. and Wilson, N., editors, Proceedings of the 10th International Symposium on Transportation and Traffic Theory, Amsterdam, The Netherlands. Elsevier.
- Mahmassani, H. S., Saberi, M., and Zockaie, A. K. (2013). Network gridlock: Theory, characteristics, and dynamic. In Procedia - Social and Behavioral Sciences, volume 80, pages 79-98. doi: 10.1016/j.sbspro.2013.05.007 20th International Symposium on Transportation and Traffic Theory.
- Mazloumian, A., Geroliminis, N., and Helbing, D. (2010). The spatial variability of vehicle densities as determinant of urban network capacity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 368(1928):4627-4647.
- Olszewski, P., Fan, H. S. L., and Tan, Y.-W. (1995). Areawide traffic speed-flow model for the Singapore CBD. Transportation Research Part A, 29A(4):273-281.
- Saberi, M. and Mahmassani, H. (2012). Exploring properties of network-wide flow-density relations in a freeway network. In Transportation Research Board 91st Annual Meeting, Washington, D.C.
- Shraiber, A. and Haddad, J. (2014). Robust control design for a perimeter traffic flow controller at an urban region. In European Control Conference 14 (Accepted).
- Zhang, L., Garoni, T., and de Gier, J. (2013). A comparative study of macroscopic fundamental diagrams of arterial road networks governed by adaptive traffic signal systems. Transportation Research Part B, 49:1-23.

#### Paper Citation

#### in Harvard Style

Haddad J. and Ioslovich I. (2014). **Optimal Feedback Control for a Perimeter Trafﬁc Flow at an Urban Region** . In *Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,* ISBN 978-989-758-039-0, pages 14-20. DOI: 10.5220/0005009800140020

#### in Bibtex Style

@conference{icinco14,

author={Jack Haddad and Ilya Ioslovich},

title={Optimal Feedback Control for a Perimeter Trafﬁc Flow at an Urban Region},

booktitle={Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},

year={2014},

pages={14-20},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0005009800140020},

isbn={978-989-758-039-0},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,

TI - Optimal Feedback Control for a Perimeter Trafﬁc Flow at an Urban Region

SN - 978-989-758-039-0

AU - Haddad J.

AU - Ioslovich I.

PY - 2014

SP - 14

EP - 20

DO - 10.5220/0005009800140020