# American Institute of Mathematical Sciences

2014, 7(3): 525-542. doi: 10.3934/dcdss.2014.7.525

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

 1 Department of Mechanical Engineering, Ibn Sina Building, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Jeddah, Saudi Arabia 2 Department of Electrical Engineering, Ibn Sina Building, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Jeddah, Saudi Arabia 3 Department of Electrical Engineering, Office 3275, Ibn Sina Building, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Jeddah, Saudi Arabia

Received  June 2013 Revised  September 2013 Published  January 2014

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.
Citation: Yanning Li, Edward Canepa, Christian Claudel. Efficient robust control of first order scalar conservation laws using semi-analytical solutions. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 525-542. doi: 10.3934/dcdss.2014.7.525
##### References:
 [1] J. P. Aubin, A. M Bayen and P. Saint-Pierre, Dirichlet problems for some Hamilton-Jacobi equations with inequality constraints,, SIAM Journal on Control and Optimization, 47 (2008), 2348. doi: 10.1137/060659569. [2] A. Aw and M. Rascle, Resurrection of second order models of traffic flow,, SIAM Journal on Applied Mathematics, 60 (2000), 916. doi: 10.1137/S0036139997332099. [3] E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians,, Communications in Partial Differential Equations, 15 (1990), 293. doi: 10.1080/03605309908820745. [4] G. Bastin, B. Haut, J. M. Coron and B. d'Andréa Novel, Lyapunov stability analysis of networks of scalar conservation laws,, Networks and Heterogeneous Media, 2 (2007). doi: 10.3934/nhm.2007.2.751. [5] A. Bayen, R. Raffard and C. Tomlin, Network congestion alleviation using adjoint hybrid control: Application to highways,, Hybrid Systems: Computation and Control, (2004), 113. doi: 10.1007/978-3-540-24743-2_7. [6] A. Ben-Tal and A. Nemirovski, Robust solutions of linear programming problems contaminated with uncertain data,, Mathematical Programming, 88 (2000), 411. doi: 10.1007/PL00011380. [7] D. Bertsimas and M. Sim, The price of robustness,, Operations Research, 52 (2004), 35. doi: 10.1287/opre.1030.0065. [8] S. Blandin, X. Litrico and A. Bayen, Boundary stabilization of the inviscid burgers equation using a Lyapunov method,, in Decision and Control (CDC), (2010), 1705. [9] S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic,, SIAM Journal on Applied Mathematics, 71 (2011), 107. doi: 10.1137/090754467. [10] D. Bovkovic and M. Krstic, Backstepping control of chemical tubular reactors,, Computers & Chemical Engineering, 26 (2002), 1077. [11] E. Canepa and C. Claudel, Exact solutions to traffic density estimation problems involving the Lighthill-Whitham-Richards traffic flow model using mixed integer programming,, in Intelligent Transportation Systems (ITSC), (2012), 832. doi: 10.1109/ITSC.2012.6338639. [12] E. Canepa and C. Claudel, Spoofing cyber attack detection in probe-based traffic monitoring systems using mixed integer linear programming,, in Computing, (2013). doi: 10.1109/ICCNC.2013.6504104. [13] C. Canudas de Wit, Best-effort highway traffic congestion control via variable speed limits,, in Decision and Control and European Control Conference (CDC-ECC), (2011), 5959. [14] C. Canudas de Wit, D. Jacquet and D. Koenig, Optimal ramp metering strategy with extended LWR model, analysis and computational methods,, (2005)., (2005). [15] R. Carlson, I. Papamichail and M. Papageorgiou, Local feedback-based mainstream traffic flow control on motorways using variable speed limits,, Intelligent Transportation Systems, 12 (2011), 1261. doi: 10.1109/TITS.2011.2156792. [16] C. Claudel and A. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory,, Automatic Control, 55 (2010), 1142. doi: 10.1109/TAC.2010.2041976. [17] C. Claudel and A. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part II: Computational methods,, Automatic Control, 55 (2010), 1158. doi: 10.1109/TAC.2010.2045439. [18] C. Claudel and A. Bayen, Convex formulations of data assimilation problems for a class of Hamilton-Jacobi equations,, SIAM Journal on Control and Optimization, 49 (2011), 383. doi: 10.1137/090778754. [19] C. Claudel, T. Chamoin and A. Bayen, Solutions to estimation problems for Hamilton-Jacobi equations using linear programming,, to appear in IEEE Transactions on Control Sytems Technology, (2013). [20] M. Crandall and P. Lions, Viscosity solutions of Hamilton-Jacobi equations,, Trans. Amer. Math. Soc., 277 (1983), 1. doi: 10.1090/S0002-9947-1983-0690039-8. [21] C. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory,, Transportation Research Part B: Methodological, 28 (1994), 269. doi: 10.1016/0191-2615(94)90002-7. [22] C. Daganzo, A variational formulation of kinematic waves: Basic theory and complex boundary conditions,, Transporation Research B, 39 (2005), 187. doi: 10.1016/j.trb.2004.04.003. [23] C. Daganzo, On the variational theory of traffic flow: Well-posedness, duality and applications,, (2006)., (2006). doi: 10.3934/nhm.2006.1.601. [24] G. Dervisoglu, G. Gomes, J. Kwon, and P. Horowitz and R. Varaiya, Automatic calibration of the fundamental diagram and empirical observations on capacity,, in Transportation Research Board 88th Annual Meeting, (2009), 09. [25] H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations,, SIAM Journal on Control and Optimization, 31 (1993), 257. doi: 10.1137/0331016. [26] A. Fügenschuh, S. Göttlich, M. Herty, C. Kirchner and A. Martin, Efficient reformulation and solution of a nonlinear PDE-controlled flow network model,, Computing, 85 (2009), 245. doi: 10.1007/s00607-009-0038-7. [27] A. Fügenschuh, M. Herty, A. Klar and A. Martin, Combinatorial and continuous models for the optimization of traffic flows on networks,, SIAM Journal on Optimization, 16 (2006), 1155. doi: 10.1137/040605503. [28] V. Gabrel, C. Murat and N. Remli, Best and worst optimum for linear programs with interval right hand sides,, in Modelling, (2008), 126. doi: 10.1007/978-3-540-87477-5_14. [29] V. Gabrel, C. Murat and N. Remli, Linear programming with interval right hand sides,, International Transactions in Operational Research, 17 (2010), 397. doi: 10.1111/j.1475-3995.2009.00737.x. [30] R. J. Gibbens and F. P. Kelly, An investigation of proportionally fair ramp metering,, in Intelligent Transportation Systems (ITSC), (2011), 490. doi: 10.1109/ITSC.2011.6082812. [31] M. Gugat, A. Herty, M.and Klar and G. Leugering, Optimal control for traffic flow networks,, Journal of Optimization Theory and Applications, 126 (2005), 589. doi: 10.1007/s10957-005-5499-z. [32] M. Herty and A. Klar, Simplified dynamics and optimization of large scale traffic networks,, Mathematical Models and Methods in Applied Sciences, 14 (2004), 579. doi: 10.1142/S0218202504003362. [33] M. Hladík, Interval linear programming: A survey,, Linear Programming-New Frontiers in Theory and Applications, (2010), 85. [34] M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 229 (1999), 317. doi: 10.1098/rspa.1955.0089. [35] X. Lu, P. P. Varaiya and R. Horowitz, An equivalent second order model with application to traffic control,, in Control in Transportation Systems, (2009), 375. [36] P. Mazaré, A. Dehwah, C. Claudel and A. Bayen, Analytical and grid-free solutions to the Lighthill-Whitham-Richards traffic flow model,, Transportation Research Part B: Methodological, 45 (2011), 1727. [37] M. Minoux, Robust LP with right-handside uncertainty, duality and applications,, (2007)., (2007). [38] K. Moskowitz, Discussion of freeway level of service as influenced by volume and capacity characteristics by DR Drew and CJ Keese,, Highway Research Record, 99 (1965), 43. [39] G. Newell, A simplified theory of kinematic waves in highway traffic, part I, II and III,, Transportation Research Part B: Methodological, 27 (1993), 281. [40] M. Papageorgiou, H. Hadj-Salem and J. Blosseville, Alinea: A local feedback control law for on-ramp metering,, Transportation Research Record, 1320 (1991), 58. [41] D. Pisarski and C. Canudas de Wit, Optimal balancing of road traffic density distributions for the cell transmission model,, in Decision and Control (CDC), (2012), 6969. doi: 10.1109/CDC.2012.6426749. [42] P. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42. doi: 10.1287/opre.4.1.42. [43] S. Ulbrich, Optimal control of nonlinear hyperbolic conservation laws with source terms,, Technische Universitaet Muenchen, (2001). [44] D. Work, S. Blandin, O. Tossavainen, B. Piccoli and A. Bayen, A traffic model for velocity data assimilation,, Applied Mathematics Research eXpress, 2010 (2010), 1. doi: 10.1093/amrx/abq002. [45] Y. Yuan, J. W. C. Van Lint, S. P. Hoogendoorn, J. L. M. Vrancken and T. Schreiter, Freeway traffic state estimation using extended Kalman filter for first-order traffic model in Lagrangian coordinates,, in Networking, (2011), 121. doi: 10.1109/ICNSC.2011.5874888.

show all references

##### References:
 [1] J. P. Aubin, A. M Bayen and P. Saint-Pierre, Dirichlet problems for some Hamilton-Jacobi equations with inequality constraints,, SIAM Journal on Control and Optimization, 47 (2008), 2348. doi: 10.1137/060659569. [2] A. Aw and M. Rascle, Resurrection of second order models of traffic flow,, SIAM Journal on Applied Mathematics, 60 (2000), 916. doi: 10.1137/S0036139997332099. [3] E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians,, Communications in Partial Differential Equations, 15 (1990), 293. doi: 10.1080/03605309908820745. [4] G. Bastin, B. Haut, J. M. Coron and B. d'Andréa Novel, Lyapunov stability analysis of networks of scalar conservation laws,, Networks and Heterogeneous Media, 2 (2007). doi: 10.3934/nhm.2007.2.751. [5] A. Bayen, R. Raffard and C. Tomlin, Network congestion alleviation using adjoint hybrid control: Application to highways,, Hybrid Systems: Computation and Control, (2004), 113. doi: 10.1007/978-3-540-24743-2_7. [6] A. Ben-Tal and A. Nemirovski, Robust solutions of linear programming problems contaminated with uncertain data,, Mathematical Programming, 88 (2000), 411. doi: 10.1007/PL00011380. [7] D. Bertsimas and M. Sim, The price of robustness,, Operations Research, 52 (2004), 35. doi: 10.1287/opre.1030.0065. [8] S. Blandin, X. Litrico and A. Bayen, Boundary stabilization of the inviscid burgers equation using a Lyapunov method,, in Decision and Control (CDC), (2010), 1705. [9] S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic,, SIAM Journal on Applied Mathematics, 71 (2011), 107. doi: 10.1137/090754467. [10] D. Bovkovic and M. Krstic, Backstepping control of chemical tubular reactors,, Computers & Chemical Engineering, 26 (2002), 1077. [11] E. Canepa and C. Claudel, Exact solutions to traffic density estimation problems involving the Lighthill-Whitham-Richards traffic flow model using mixed integer programming,, in Intelligent Transportation Systems (ITSC), (2012), 832. doi: 10.1109/ITSC.2012.6338639. [12] E. Canepa and C. Claudel, Spoofing cyber attack detection in probe-based traffic monitoring systems using mixed integer linear programming,, in Computing, (2013). doi: 10.1109/ICCNC.2013.6504104. [13] C. Canudas de Wit, Best-effort highway traffic congestion control via variable speed limits,, in Decision and Control and European Control Conference (CDC-ECC), (2011), 5959. [14] C. Canudas de Wit, D. Jacquet and D. Koenig, Optimal ramp metering strategy with extended LWR model, analysis and computational methods,, (2005)., (2005). [15] R. Carlson, I. Papamichail and M. Papageorgiou, Local feedback-based mainstream traffic flow control on motorways using variable speed limits,, Intelligent Transportation Systems, 12 (2011), 1261. doi: 10.1109/TITS.2011.2156792. [16] C. Claudel and A. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory,, Automatic Control, 55 (2010), 1142. doi: 10.1109/TAC.2010.2041976. [17] C. Claudel and A. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part II: Computational methods,, Automatic Control, 55 (2010), 1158. doi: 10.1109/TAC.2010.2045439. [18] C. Claudel and A. Bayen, Convex formulations of data assimilation problems for a class of Hamilton-Jacobi equations,, SIAM Journal on Control and Optimization, 49 (2011), 383. doi: 10.1137/090778754. [19] C. Claudel, T. Chamoin and A. Bayen, Solutions to estimation problems for Hamilton-Jacobi equations using linear programming,, to appear in IEEE Transactions on Control Sytems Technology, (2013). [20] M. Crandall and P. Lions, Viscosity solutions of Hamilton-Jacobi equations,, Trans. Amer. Math. Soc., 277 (1983), 1. doi: 10.1090/S0002-9947-1983-0690039-8. [21] C. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory,, Transportation Research Part B: Methodological, 28 (1994), 269. doi: 10.1016/0191-2615(94)90002-7. [22] C. Daganzo, A variational formulation of kinematic waves: Basic theory and complex boundary conditions,, Transporation Research B, 39 (2005), 187. doi: 10.1016/j.trb.2004.04.003. [23] C. Daganzo, On the variational theory of traffic flow: Well-posedness, duality and applications,, (2006)., (2006). doi: 10.3934/nhm.2006.1.601. [24] G. Dervisoglu, G. Gomes, J. Kwon, and P. Horowitz and R. Varaiya, Automatic calibration of the fundamental diagram and empirical observations on capacity,, in Transportation Research Board 88th Annual Meeting, (2009), 09. [25] H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations,, SIAM Journal on Control and Optimization, 31 (1993), 257. doi: 10.1137/0331016. [26] A. Fügenschuh, S. Göttlich, M. Herty, C. Kirchner and A. Martin, Efficient reformulation and solution of a nonlinear PDE-controlled flow network model,, Computing, 85 (2009), 245. doi: 10.1007/s00607-009-0038-7. [27] A. Fügenschuh, M. Herty, A. Klar and A. Martin, Combinatorial and continuous models for the optimization of traffic flows on networks,, SIAM Journal on Optimization, 16 (2006), 1155. doi: 10.1137/040605503. [28] V. Gabrel, C. Murat and N. Remli, Best and worst optimum for linear programs with interval right hand sides,, in Modelling, (2008), 126. doi: 10.1007/978-3-540-87477-5_14. [29] V. Gabrel, C. Murat and N. Remli, Linear programming with interval right hand sides,, International Transactions in Operational Research, 17 (2010), 397. doi: 10.1111/j.1475-3995.2009.00737.x. [30] R. J. Gibbens and F. P. Kelly, An investigation of proportionally fair ramp metering,, in Intelligent Transportation Systems (ITSC), (2011), 490. doi: 10.1109/ITSC.2011.6082812. [31] M. Gugat, A. Herty, M.and Klar and G. Leugering, Optimal control for traffic flow networks,, Journal of Optimization Theory and Applications, 126 (2005), 589. doi: 10.1007/s10957-005-5499-z. [32] M. Herty and A. Klar, Simplified dynamics and optimization of large scale traffic networks,, Mathematical Models and Methods in Applied Sciences, 14 (2004), 579. doi: 10.1142/S0218202504003362. [33] M. Hladík, Interval linear programming: A survey,, Linear Programming-New Frontiers in Theory and Applications, (2010), 85. [34] M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 229 (1999), 317. doi: 10.1098/rspa.1955.0089. [35] X. Lu, P. P. Varaiya and R. Horowitz, An equivalent second order model with application to traffic control,, in Control in Transportation Systems, (2009), 375. [36] P. Mazaré, A. Dehwah, C. Claudel and A. Bayen, Analytical and grid-free solutions to the Lighthill-Whitham-Richards traffic flow model,, Transportation Research Part B: Methodological, 45 (2011), 1727. [37] M. Minoux, Robust LP with right-handside uncertainty, duality and applications,, (2007)., (2007). [38] K. Moskowitz, Discussion of freeway level of service as influenced by volume and capacity characteristics by DR Drew and CJ Keese,, Highway Research Record, 99 (1965), 43. [39] G. Newell, A simplified theory of kinematic waves in highway traffic, part I, II and III,, Transportation Research Part B: Methodological, 27 (1993), 281. [40] M. Papageorgiou, H. Hadj-Salem and J. Blosseville, Alinea: A local feedback control law for on-ramp metering,, Transportation Research Record, 1320 (1991), 58. [41] D. Pisarski and C. Canudas de Wit, Optimal balancing of road traffic density distributions for the cell transmission model,, in Decision and Control (CDC), (2012), 6969. doi: 10.1109/CDC.2012.6426749. [42] P. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42. doi: 10.1287/opre.4.1.42. [43] S. Ulbrich, Optimal control of nonlinear hyperbolic conservation laws with source terms,, Technische Universitaet Muenchen, (2001). [44] D. Work, S. Blandin, O. Tossavainen, B. Piccoli and A. Bayen, A traffic model for velocity data assimilation,, Applied Mathematics Research eXpress, 2010 (2010), 1. doi: 10.1093/amrx/abq002. [45] Y. Yuan, J. W. C. Van Lint, S. P. Hoogendoorn, J. L. M. Vrancken and T. Schreiter, Freeway traffic state estimation using extended Kalman filter for first-order traffic model in Lagrangian coordinates,, in Networking, (2011), 121. doi: 10.1109/ICNSC.2011.5874888.
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