`a`
Discrete and Continuous Dynamical Systems - Series S (DCDS-S)
 

Efficient robust control of first order scalar conservation laws using semi-analytical solutions

Pages: 525 - 542, Volume 7, Issue 3, June 2014      doi:10.3934/dcdss.2014.7.525

 
       Abstract        References        Full Text (4378.1K)       Related Articles       

Yanning Li - Department of Mechanical Engineering, Ibn Sina Building, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Jeddah, Saudi Arabia (email)
Edward Canepa - Department of Electrical Engineering, Ibn Sina Building, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Jeddah, Saudi Arabia (email)
Christian Claudel - Department of Electrical Engineering, Office 3275, Ibn Sina Building, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Jeddah, Saudi Arabia (email)

Abstract: This article presents a new robust control framework for transportation problems in which the state is modeled by a first order scalar conservation law. Using an equivalent formulation based on a Hamilton-Jacobi equation, we pose the problem of controlling the state of the system on a network link, using initial density control and boundary flow control, as a Linear Program. We then show that this framework can be extended to arbitrary control problems involving the control of subsets of the initial and boundary conditions. Unlike many previously investigated transportation control schemes, this method yields a globally optimal solution and is capable of handling shocks (i.e. discontinuities in the state of the system). We also demonstrate that the same framework can handle robust control problems, in which the uncontrollable components of the initial and boundary conditions are encoded in intervals on the right hand side of inequalities in the linear program. The lower bound of the interval which defines the smallest feasible solution set is used to solve the robust LP/MILP. Since this framework leverages the intrinsic properties of the Hamilton-Jacobi equation used to model the state of the system, it is extremely fast. Several examples are given to demonstrate the performance of the robust control solution and the trade-off between the robustness and the optimality.

Keywords:  Traffic control, distributed parameter systems, optimal control, robust control, interval linear programming.
Mathematics Subject Classification:  90C08.

Received: June 2013;      Revised: September 2013;      Available Online: January 2014.

 References