Dynamic Routing Problem in Distributed Parameter Setting

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.

[1]  W. Wonham,et al.  The internal model principle for linear multivariable regulators , 1975 .

[2]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[3]  Jean-Jacques E. Slotine,et al.  Tracking control of non-linear systems using sliding surfaces with application to robot manipulators , 1983 .

[4]  G. Bartolini,et al.  Chattering avoidance by second-order sliding mode control , 1998, IEEE Trans. Autom. Control..

[5]  P. Kachroo,et al.  Sliding mode for user equilibrium dynamic traffic routing control , 1997, Proceedings of Conference on Intelligent Transportation Systems.

[6]  S. Shankar Sastry,et al.  Travel Time Dynamics for Intelligent Transportation Systems: Theory and Applications , 2016, IEEE Transactions on Intelligent Transportation Systems.

[7]  Masayoshi Tomizuka,et al.  Integral Action For Chattering Reduction And Error Convergence in Sliding Mode Control , 1992, 1992 American Control Conference.

[8]  S. Shankar Sastry,et al.  Traffic Assignment Using a Density-Based Travel-Time Function for Intelligent Transportation Systems , 2016, IEEE Transactions on Intelligent Transportation Systems.

[9]  Jean-Jacques E. Slotine,et al.  Adaptive sliding controller synthesis for non-linear systems , 1986 .

[10]  B. Paden,et al.  Lyapunov stability theory of nonsmooth systems , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[11]  Pushkin Kachroo,et al.  Existence of solutions to a class of nonlinear convergent chattering-free sliding mode control systems , 1999, IEEE Trans. Autom. Control..

[12]  G. Bartolini,et al.  Asymptotic linearization of uncertain nonlinear systems by means of continuous control , 1993 .

[13]  Weiping Li,et al.  Applied Nonlinear Control , 1991 .

[14]  S. Sastry,et al.  A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulators , 1987 .

[15]  Masayoshi Tomizuka,et al.  Chattering reduction and error convergence in the sliding-mode control of a class of nonlinear systems , 1996, IEEE Trans. Autom. Control..

[16]  James Stewart,et al.  Essential Calculus: Early Transcendentals , 2006 .

[17]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[18]  Giorgio Bartolini,et al.  An improved, chattering free, VSC scheme for uncertain dynamical systems , 1996, IEEE Trans. Autom. Control..

[19]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[20]  M. Vidyasagar,et al.  Nonlinear systems analysis (2nd ed.) , 1993 .