Modeling and Problem Formulation

  • Pushkin Kachroo
  • Kaan M. A. Özbay
Part of the Advances in Industrial Control book series (AIC)


This chapter provides the system dynamics model for control design in continuous and discrete time and space variables for traffic flow and also develops the basic framework for the control design. The problem is formulated as a feedback control design for the traffic as a distributed parameter system, i.e., in terms of Partial Differential Equations (PDE). DTR formulation for two alternate routes and then generalized for n routes is developed. Space discretization of the model results in Ordinary Differential Equation (ODE) representation, and its further time discretization produces difference equations. System dynamic equations for sample problems are developed, and a simple control law is also shown with computer simulation results.


  1. 1.
    Lighthill MJ, Whitham GB (1955) On kinematic waves II. A theory of traffic flow on long crowded roads. Proc Royal Soci London. Ser A. Math Phys Sci 229(1178):317–345. Scholar
  2. 2.
    Papgeorgiou M, Messmer A (1991) Dynamic network traffic assignment and route guidance via feedback regulation. Trans Res Record 1306:49–58Google Scholar
  3. 3.
    Greenshields BD, Channing W, Miller H (1935) A study of traffic capacity. In: Highway research board proceedings, vol 1935. National Research Council (USA), Highway Research BoardGoogle Scholar
  4. 4.
    Haberman R (1977) Mathematical models. Prentice-Hall, New JerseyzbMATHGoogle Scholar
  5. 5.
    Kachroo P (2009) Pedestrian dynamics: Mathematical theory and evacuation control. CRC PressGoogle Scholar
  6. 6.
    Fletcher CAJ (1982) Burgers equation: a model for all reasons. Numerical solutions of partial differential equations. North-Holland Pub. Co, Amsterdam, HollandGoogle Scholar
  7. 7.
    Musha T, Higuchi H (1978) Traffic current fluctuation and the burgers equation. Jpn J Appl Phys 17(5):811CrossRefGoogle Scholar
  8. 8.
    Burns JA, Kang S (1991) A control problem for burgers’ equation with bounded input/output. Nonlinear Dyn 2(4):235–262CrossRefGoogle Scholar
  9. 9.
    Burns JA, Kang S (1991) A stabilization problem for burgers equation with unbounded control and observation. In: Estimation and control of distributed parameter systems. Springer, pp 51–72Google Scholar
  10. 10.
    Burgers JM (1995) Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. In: Selected Papers of JM Burgers. Springer, pp 281–334CrossRefGoogle Scholar
  11. 11.
    Burgers JM (1948) A mathematical model illustrating the theory of turbulence. Adv Appl Mech 1:171–199MathSciNetCrossRefGoogle Scholar
  12. 12.
    Burgers JM (1972) Statistical problems connected with asymptotic solutions of the one-dimensional nonlinear diffusion equation. Springer. In: Statistical models and turbulence. Springer, pp 41–60CrossRefGoogle Scholar
  13. 13.
    Papageorgiou M, Blosseville J-M, Hadj-Salem H (1989) Macroscopic modelling of traffic flow on the boulevard périphérique in paris. Transp Res Part B: Methodol 23(1):29–47CrossRefGoogle Scholar
  14. 14.
    Cole JD (1951) On a quasi-linear parabolic equation occurring in aerodynamics. Quart Appl Math 9(3):225–236MathSciNetCrossRefGoogle Scholar
  15. 15.
    Glimm J, Lax PD (1970) Decay of solutions of systems of nonlinear hyperbolic conservation laws. American Mathematical SocCrossRefGoogle Scholar
  16. 16.
    Maslov VP (1987) A new approach to generalized solutions of nonlinear systems. In: Soviet Math. Dokl, vol 1, pp 29–33Google Scholar
  17. 17.
    Curtain RF (1984) Stability of semilinear evolution equations in hilbert space. J Math Pures et Appl 63:121–128zbMATHGoogle Scholar
  18. 18.
    Kielhöfer H (1974) Stability and semilinear evolution equations in hilbert space. Arch Rat Mech Anal 57(2):150–165MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ran B, Boyce DE, LeBlanc LJ (1993) A new class of instantaneous dynamic user-optimal traffic assignment models. Oper Res 41(1):192–202CrossRefGoogle Scholar
  20. 20.
    Friesz TL, Luque J, Tobin RL, Wie B-W (1989) Dynamic network traffic assignment considered as a continuous time optimal control problem. Oper Res 37(6):893–901MathSciNetCrossRefGoogle Scholar
  21. 21.
    Papageorgiou M (1983) Applications of automatic control concepts to traffic flow modeling and control. Springer (Lecture Notes in Control and Information Sciences)Google Scholar
  22. 22.
    Kuo BC (1981) Automatic control systems. Prentice Hall PTRGoogle Scholar
  23. 23.
    Mosca E (1995) Optimal, predictive, and adaptive control, vol 151Google Scholar
  24. 24.
    Isidori A (1995) Nonlinear control systems. SpringerCrossRefGoogle Scholar
  25. 25.
    Slotine JJE, Li W (1991) Applied nonlinear control. Prentice-HallGoogle Scholar
  26. 26.
    Richards PI (1956) Shock waves on the highway. Oper Res 4(1):42–51MathSciNetCrossRefGoogle Scholar
  27. 27.
    Berger CR, Shaw L (1978) Discrete-event simulation of freeway traffic. In: Simulation Council Proc., Series 7, pp 85–93Google Scholar
  28. 28.
    Payne HJ (1971) Models of freeway traffic and control. Simulation Councils, IncorporatedGoogle Scholar
  29. 29.
    Nijmeijer H, Van der Schaft A (1990) Nonlinear dynamical control systems. Springer, New YorkCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringUniversity of NevadaLas VegasUSA
  2. 2.Department of Civil and Urban EngineeringNew York UniversityBrooklynUSA

Personalised recommendations