Traffic Regulation via Controlled Speed Limit

We study an optimal control problem for traffic regulation via variable speed limit. The traffic flow dynamics is described with the Lighthill-Whitham-Richards (LWR) model with Newell-Daganzo flux function. We aim at minimizing the $L^2$ quadratic error to a desired outflow, given an inflow on a single road. We first provide existence of a minimizer and compute analytically the cost functional variations due to needle-like variation in the control policy. Then, we compare three strategies: instantaneous policy; random exploration of control space; steepest descent using numerical expression of gradient. We show that the gradient technique is able to achieve a cost within 10% of random exploration minimum with better computational performances.

[1]  Antonella Ferrara,et al.  Nonlinear optimization for freeway control using variable-speed signaling , 1999 .

[2]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

[3]  Stefan Ulbrich,et al.  Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws , 2003, Syst. Control. Lett..

[4]  Antonella Ferrara,et al.  Optimal control of freeways via speed signalling and ramp metering , 1997 .

[5]  Stefan Ulbrich,et al.  A Sensitivity and Adjoint Calculus for Discontinuous Solutions of Hyperbolic Conservation Laws with Source Terms , 2002, SIAM J. Control. Optim..

[6]  M. Krstić,et al.  Optimal control of scalar one-dimensional conservation laws , 2006, 2006 American Control Conference.

[7]  A. Hegyi,et al.  Optimal Coordination of Variable Speed Limits to Suppress Shock Waves , 2002, IEEE Transactions on Intelligent Transportation Systems.

[8]  P. I. Richards.  Shock Waves on the Highway , 1956 .

[9]  Richard Bellman,et al.  Introduction to the mathematical theory of control processes , 1967 .

[10]  Stefan Ulbrich,et al.  Convergence of Linearized and Adjoint Approximations for Discontinuous Solutions of Conservation Laws. Part 2: Adjoint Approximations and Extensions , 2010, SIAM J. Numer. Anal..

[11]  Markos Papageorgiou,et al.  Optimal Motorway Traffic Flow Control Involving Variable Speed Limits and Ramp Metering , 2010, Transp. Sci..

[12]  M J Lighthill,et al.  ON KINEMATIC WAVES.. , 1955 .

[13]  Rinaldo M. Colombo,et al.  A well posed conservation law with a variable unilateral constraint , 2007 .

[14]  Jian-Xin Xu,et al.  Freeway Traffic Control Using Iterative Learning Control-Based Ramp Metering and Speed Signaling , 2007, IEEE Transactions on Vehicular Technology.

[15]  Eduardo F. Camacho,et al.  Global Versus Local MPC Algorithms in Freeway Traffic Control With Ramp Metering and Variable Speed Limits , 2012, IEEE Transactions on Intelligent Transportation Systems.

[16]  István Varga,et al.  Freeway shockwave control using ramp metering and variable speed limits , 2013, 21st Mediterranean Conference on Control and Automation.

[17]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[18]  Alexandre M. Bayen,et al.  Adjoint-Based Optimization on a Network of Discretized Scalar Conservation Laws with Applications to Coordinated Ramp Metering , 2015, J. Optim. Theory Appl..

[19]  Nan Zou,et al.  Optimal Variable Speed Limit Control for Real-time Freeway Congestions , 2013 .

[20]  Michael Herty,et al.  Adjoint IMEX-based schemes for control problems governed by hyperbolic conservation laws , 2012, Comput. Optim. Appl..

[21]  A. Marson,et al.  Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws , 2007 .

[22]  Antonella Ferrara,et al.  OPTIMAL CONTROL OF FREEWAYS VIA SPEED SIGNALLING AND RAMP METERING , 1997 .

[23]  M J Lighthill,et al.  On kinematic waves II. A theory of traffic flow on long crowded roads , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[24]  Mauro Garavello,et al.  Traffic Flow on Networks , 2006 .

[25]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[26]  Grégoire Allaire,et al.  Numerical analysis and optimization , 2007 .

[27]  M. Herty,et al.  Optimal Control for Traffic Flow Networks , 2005 .

[28]  Carlos Canudas de Wit,et al.  Best-effort highway traffic congestion control via variable speed limits , 2011, IEEE Conference on Decision and Control and European Control Conference.

[29]  Andreas Hegyi,et al.  Dynamic speed limit control to resolve shock waves on freeways - Field test results of the SPECIALIST algorithm , 2010, 13th International IEEE Conference on Intelligent Transportation Systems.

[30]  J. Nédélec,et al.  First order quasilinear equations with boundary conditions , 1979 .

[31]  Andreas Hegyi,et al.  SPECIALIST: A dynamic speed limit control algorithm based on shock wave theory , 2008, 2008 11th International IEEE Conference on Intelligent Transportation Systems.

[32]  A. Bressan.  Hyperbolic systems of conservation laws : the one-dimensional Cauchy problem , 2000 .

[33]  Massimo Fornasier,et al.  Mean-field sparse optimal control , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[34]  Armin Fügenschuh,et al.  Combinatorial and Continuous Models for the Optimization of Traffic Flows on Networks , 2006, SIAM J. Optim..

[35]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[36]  C. Daganzo.  THE CELL TRANSMISSION MODEL.. , 1994 .

[37]  Bart De Schutter,et al.  Model predictive control for optimal coordination of ramp metering and variable speed limits , 2005 .

[38]  D. Wishart.  Introduction to the Mathematical Theory of Control Processes. Volume 1—Linear Equations and Quadratic Criteria , 1969 .

[39]  Simone Göttlich,et al.  Speed limit and ramp meter control for traffic flow networks , 2016 .

[40]  Andreas Hegyi,et al.  The expected effectivity of the dynamic speed limit algorithm SPECIALIST - A field data evaluation method , 2009, 2009 European Control Conference (ECC).