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Dynamic Routing Problem in Distributed Parameter Setting

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

Abstract

The aim of this chapter is to design control law for the DTR (Dynamic Traffic Routing) problem modeled in the distributed parameter setting. For the control design, the chapter uses the sliding mode control technique for regulating the error. Sliding mode control provides a robust method against bounded uncertainties. The price to pay for that robustness is chattering. The chapter shows methods to deal with chattering reduction in the control implementation. Control design and its software simulation using sliding mode control are presented in this chapter. The chapter also provides a simple study of discretization errors that are obtained in the numerical approximation of the distributed model for software simulation. The chapter shows the development of a simple software simulation code and its simulation to study this problem.

References

  1. 1.
    Vidyasagar M (1992) Nonlinear systems analysis, 2nd edn. Prentice-Hall IncGoogle Scholar
  2. 2.
    Filippov AF (1960) Differential equations with discontinuous right-hand side. Matematicheskii sbornik 93(1):99–128MathSciNetzbMATHGoogle Scholar
  3. 3.
    Filippov AF (1980) Differential equations with second members discontinuous on intersecting surfaces. Differ Equ 415:1292–1299zbMATHGoogle Scholar
  4. 4.
    Paden B, Sastry S (1987) A calculus for computing filippov’s differential inclusion with application to the variable structure control of robot manipulators. IEEE Trans Circuits Syst 34(1):73–82MathSciNetCrossRefGoogle Scholar
  5. 5.
    Shevitz D, Paden B (1994) Lyapunov stability theory of nonsmooth systems. IEEE Trans Autom Control 39(9):1910–1914MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kachroo P (1999) Existence of solutions to a class of nonlinear convergent chattering-free sliding mode control systems. IEEE Trans Autom Control 44(8):1620–1624MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kachroo P, Tomizuka M (1996) Chattering reduction and error convergence in the sliding-mode control of a class of nonlinear systems. IEEE Trans Autom Control 41(7):1063–1068MathSciNetCrossRefGoogle Scholar
  8. 8.
    Slotine JJE, Li W (1991) Applied nonlinear control. Prentice-HallGoogle Scholar
  9. 9.
    Slotine J-J, Sastry SS (1983) Tracking control of non-linear systems using sliding surfaces, with application to robot manipulators. Int J Control 38(2):465–492CrossRefGoogle Scholar
  10. 10.
    Slotine J-J, Coetsee JA (1986) Adaptive sliding controller synthesis for non-linear systems. Int J Control 43(6):1631–1651CrossRefGoogle Scholar
  11. 11.
    Bartolini G, Pydynowski P (1996) An improved, chattering free, VSC scheme for uncertain dynamical systems. IEEE Trans Autom Control 41(8):1220–1226CrossRefGoogle Scholar
  12. 12.
    Bartolini G, Ferrara A, Usani E (1998) Chattering avoidance by second-order sliding mode control. IEEE Trans Autom Control 43(2):241–246MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bartolini G, Pydynowski P (1993) Asymptotic linearization of uncertain nonlinear systems by means of continuous control. Int J Robust Nonlinear Control 3(2):87–103CrossRefGoogle Scholar
  14. 14.
    Kachroo P (1993) Nonlinear control strategies and vehicle traction control. University of California, BerkeleyGoogle Scholar
  15. 15.
    Kachroo P, Tomizuka M (1995) Sliding mode control with chattering reduction and error convergence for a class of discrete nonlinear systems with application to vehicle control. In: Proceedings of the international mechanical engineering congress and expo, pp 225–233Google Scholar
  16. 16.
    Francis BA, Wonham WM (1975) The internal model principle for linear multivariable regulators. Appl Math Optim 2(2):170–194MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kachroo P, Ozbay K (1997) Sliding mode for user equilibrium dynamic traffic routing control. In: IEEE conference on intelligent transportation system, ITSC’97. IEEE, pp 70–75Google Scholar
  18. 18.
    Kachroo P, Tomizuka M (1992) Integral action for chattering reduction and error convergence in sliding mode control. In: American control conference. IEEE, pp 867–870Google Scholar
  19. 19.
    Stewart J (2012) Essential calculus: Early transcendentals. Cengage LearningGoogle Scholar
  20. 20.
    Kachroo P, Sastry S (2016a) Travel time dynamics for intelligent transportation systems: theory and applications. IEEE Trans Intell Transp Syst 17(2):385–394CrossRefGoogle Scholar
  21. 21.
    Kachroo P, Sastry S (2016b) Traffic assignment using a density-based travel-time function for intelligent transportation systems. IEEE Trans Intell Transp Syst 17(5):1438–1447CrossRefGoogle Scholar
  22. 22.
    LeVeque RJ (1990) Numerical methods for conservation laws. Birkhäuser VerlagCrossRefGoogle Scholar
  23. 23.
    LeVeque RJ (2002) Finite volume methods for hyperbolic problems, vol 31. Cambridge University PressGoogle Scholar
  24. 24.
    Toro EF (2013) Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Springer Science & Business MediaGoogle 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

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