A junction-by-junction feedback-based strategy with convergence analysis for dynamic traffic assignment



By considering the traffic assignment problem as a control problem, this paper develops a new realtime route guidance strategy for accurate convergence of the traffic flows to user equilibrium (UE) or system optimum (SO), in the presence of drivers’ response uncertainties. With the new guidance strategy, the drivers make routing decisions based on the route guidance information from junction to junction. Specifically, instead of total travel cost of every route from origin to destination, the travel cost of every alternative link plus the average cost to destination from the next junction corresponding to the alternative link is sent to the drivers at each specific junction. The drivers’ response to the route guidance information is directly modeled by the splitting rates at the junctions, which are simply negatively correlated with the comparison of related cost information and are able to take into account the drivers’ response uncertainties. With the proposed route guidance strategy, in the case of fixed travel demands, the accurate convergence of the traffic flows to a UE is guaranteed in the presence of drivers’ response uncertainties by using LaSalle’s invariance principle. When marginal travel cost information, instead of travel cost information, is sent to the drivers, a system optimum can be achieved under a mild condition on the marginal cost function.


本文提出了一个新的基于结点动态交通分配策略的轻松驾驶响应模型。具体地,驾驶员对路径引导信息的响应是直接有结点的分裂率表示的,它与相关成本信息进行比较,满足简单的负相关性。文中提到的动态交通分配策略,利用 LaSalle 不变性原理保证了交通流量趋于一个用户平衡。当边际出行成本信息用来代替出行成本信息时,目标函数在温和的条件下可以获得系统优化。


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  1. 1

    Xiong Z, Sheng H, Rong W G, et al. Intelligent transportation systems for smart cities: a progress review. Sci China Inf Sci, 2012, 55: 2908–2914

    Article  Google Scholar 

  2. 2

    Wardrop J G. Some theoretical aspects of road traffic research. In: Proceedings of the Institution of Civil Engineers, Part II, London, 1952. 325–378

    Google Scholar 

  3. 3

    Drew D R. Traffic Flow Theory and Control. New York: McGraw-Hill, 1968

    Google Scholar 

  4. 4

    Bar-Gera H. Traffic assignment by paired alternative segments. Transport Res Part B, 2010, 44: 1022–1046

    Article  Google Scholar 

  5. 5

    Janson B N. Dynamic traffic assignment for urban road networks. Oper Res, 1991, 25: 143–161

    Google Scholar 

  6. 6

    Papageorgiou M. Dynamic modeling, assignment and route guidance traffic networks. Transport Res Part B, 1990, 24: 471–495

    Article  Google Scholar 

  7. 7

    Peeta S, Ziliaskopoulos A K. Foundations of dynamic traffic assignment: the past, the present and the future. Netw Spat Econ, 2001, 1: 233–265

    Article  Google Scholar 

  8. 8

    Friesz T L, Bernstein D, Mehta N J, et al. Day-to-day dynamic network disequilibria and idealized traveler information systems. Oper Res, 1994, 42: 1120–1136

    MathSciNet  Article  MATH  Google Scholar 

  9. 9

    Han L, Du L. On a link-based day-to-day traffic assignment model. Transport Res Part B, 2012, 46: 72–84

    Article  Google Scholar 

  10. 10

    He X, Guo X, Liu H X. A link-based day-to-day traffic assignment model. Transport Res Part B, 2010, 44: 597–608

    Article  Google Scholar 

  11. 11

    Bellei G, Gentile G, Papola N. A within-day dynamic traffic assignment model for urban road networks. Transport Res Part B, 2005, 39: 1–29

    Article  Google Scholar 

  12. 12

    Zhang D, Nagurney A. On the local and global stability of a travel route choice adjustment process. Transport Res Part B, 1996, 30: 245–262

    Article  Google Scholar 

  13. 13

    Peeta S, Yang T H. Stability issues for dynamic traffic assignment. Automatica, 2003, 39: 21–34

    MathSciNet  Article  MATH  Google Scholar 

  14. 14

    Smith M, Mounce R. A splitting rate model of traffic re-routeing and traffic control. Transport Res Part B, 2011, 45: 1389–1409

    Article  Google Scholar 

  15. 15

    Smith M J. The stability of a dynamic model of traffic assignment: an application of a method of Lyapunov. Transport Sci, 1984, 18: 245–252

    MathSciNet  Article  Google Scholar 

  16. 16

    Kachroo P, Özbay K. Feedback Control Theory for Dynamic Traffic Assignment. Berlin: Springer, 1998

    Google Scholar 

  17. 17

    Papageorgiou M, Diakaki C, Dinopoulou V, et al. Review of road traffic control strategies. Proc IEEE, 2003, 91: 2043–2067

    Article  Google Scholar 

  18. 18

    Dafermos S C, Sparrow F T. The traffic assignment problem for a general network. J Res National Bureau Stand–B Math Sci, 1969, 73B: 91–118

    Google Scholar 

  19. 19

    Cantarella G E, Cascetta E. Dynamic processes and equilibrium in transportation networks: towards a unifying theory. Transport Sci, 1995, 29: 305–329

    Article  MATH  Google Scholar 

  20. 20

    Bie J, Lo H K. Stability and attraction domains of traffic equilibria in a day-to-day dynamical system formulation. Transport Res Part B, 2010, 44: 90–107

    Article  Google Scholar 

  21. 21

    Evans S P. Derivation and analysis of some models for combining trip distribution and assignment. Transport Res, 1976, 10: 37–57

    Article  Google Scholar 

  22. 22

    Khalil H K. Nonlinear Systems. 3rd ed. New Jersey: Prentice Hall, 2002

    Google Scholar 

  23. 23

    Friesz T L, Luque J, Tobin R L, et al. Dynamic network traffic assignment considered as a continuous time optimal control problem. Oper Res, 1989, 37: 893–901

    MathSciNet  Article  MATH  Google Scholar 

  24. 24

    Pavlis Y, Papageorgiou M. Simple decentralized feedback strategies for route guidance in traffic networks. Transport Sci, 1999, 33: 264–278

    Article  MATH  Google Scholar 

  25. 25

    Kachroo P, Özbay K. Feedback control solutions to network level system optimal real-time dynamic traffic assignment problems. Netw Spat Econ J, 2005, 5: 243–260

    Article  MATH  Google Scholar 

  26. 26

    Kachroo P, Özbay K. Modeling of network level system-optimal real-time dynamic traffic routing problem using nonlinear H8 feedback control theoretic approach. J Intell Transport Syst, 2006, 10: 159–171

    Article  MATH  Google Scholar 

  27. 27

    Xu T D, Sun L J, Peng Z R, et al. Integrated route guidance and ramp metering consistent with drivers’ en-route diversion behaviour. IET Intell Transport Syst, 2011, 5: 267–276

    Article  Google Scholar 

  28. 28

    Kachroo P, Özbay K. Solution to the user equilibrium dynamic traffic routing problem using feedback linearization. Transport Res Part B, 1998, 32: 343–360

    Article  Google Scholar 

  29. 29

    Wang F Y. Agent-based control for networked traffic management systems. IEEE Intell Syst, 2005, 20: 92–96

    Article  Google Scholar 

  30. 30

    Freeman R A, Kokotovic P V. Robust Nonlinear Control Design: State-space and Lyapunov Techniques. Boston: Birkhäuser, 1996

    Google Scholar 

  31. 31

    Papadimitratos P, de La Fortelle A, Evenssen K, et al. Vehicular communication systems: enabling technologies, applications, and future outlook on intelligent transportation. IEEE Commun Mag, 2009, 47: 84–95

    Article  Google Scholar 

  32. 32

    Christofides N. Graph Theory: An Algorithmic Approach. London: Academic Press, 1975

    Google Scholar 

  33. 33

    Aubin J P, Cellina A. Differential Inclusions. Berlin: Springer, 1984

    Google Scholar 

  34. 34

    La Salle J P. The extent of asymptotic stability. Proc Natl Acad Sci, 1960, 46: 363–365

    MathSciNet  Article  Google Scholar 

  35. 35

    La Salle J P. Some extensions of liapunovs second method. IRE Trans Circuit Theory, 1960, CT-7: 520–527

    Google Scholar 

  36. 36

    Logemann H, Ryan E P. Asymptotic behaviour of nonlinear systems. American Math Mon, 2004, 111: 864–889

    MathSciNet  Article  MATH  Google Scholar 

  37. 37

    Shevitz D, Paden B. Lyapunov stability theory of nonsmooth systems. IEEE Trans Automat Control, 1994, 39: 1910–1914

    MathSciNet  Article  MATH  Google Scholar 

  38. 38

    Sanfelice R G, Goebel R, Teel A R. Invariance principles for hybrid systems with connections to detectability and asymptotic stability. IEEE Trans Automat Control, 2007, 52: 2282–2297

    MathSciNet  Article  Google Scholar 

  39. 39

    Beckmann M, McGuire C B, Winsten C B. Studies in the Economics of Transportation. New Haven: Yale University Press, 1956

    Google Scholar 

  40. 40

    Bertsekas D P. Nonlinear Programming. Massachusetts: Athena Scientific, 1999

    Google Scholar 

  41. 41

    Hendrickson C T, Janson B N. A common network flow formulation for several civil engineering problems. Civil Eng Syst, 1984, 1: 195–203

    Article  Google Scholar 

  42. 42

    Liu T, Jiang Z P, Xin W, et al. Robust stability of a dynamic traffic assignment model with uncertainties. In: Proceedings of the 2013 American Control Conference, Washington, 2013. 4056–4061

    Google Scholar 

  43. 43

    Karafyllis I, Jiang Z P. Stability and Stabilization of Nonlinear Systems. London: Springer, 2011

    Google Scholar 

  44. 44

    Li T, Zhang J F. Sampled-data based average consensus with measurement noises: convergence analysis and uncertainty principle. Sci China Ser F-Inf Sci, 2009, 52: 2089–2103

    MathSciNet  Article  MATH  Google Scholar 

  45. 45

    Liu T, Jiang Z P, Hill D J. Decentralized output-feedback control of large-scale nonlinear systems with sensor noise. Automatica, 2012, 48: 2560–2568

    MathSciNet  Article  MATH  Google Scholar 

  46. 46

    Liu T, Jiang Z P, Hill D J. Nonlinear Control of Dynamic Networks. Boca Raton: CRC Press, 2014

    Google Scholar 

  47. 47

    Liu X M, Lin Z L. On semi-global stabilization of minimum phase nonlinear systems without vector relative degrees. Sci China Ser F-Inf Sci, 2009, 52: 2153–2162

    MathSciNet  Article  MATH  Google Scholar 

  48. 48

    Sun W J, Huang J. Output regulation for a class of uncertain nonlinear systems with nonlinear exosystems and its application. Sci China Ser F-Inf Sci, 2009, 52: 2172–2179

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Tengfei Liu or Xuesong Lu or Zhong-Ping Jiang.

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Liu, T., Lu, X. & Jiang, ZP. A junction-by-junction feedback-based strategy with convergence analysis for dynamic traffic assignment. Sci. China Inf. Sci. 59, 1–17 (2016). https://doi.org/10.1007/s11432-015-5444-1

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  • dynamic traffic assignment
  • user equilibrium (UE)
  • system optimum (SO)
  • convergence
  • LaSalle’s invariance principle
  • 010203


  • 动态交通分配
  • 用户均衡
  • 系统优化
  • 收敛性
  • LaSalle 不变原理