# American Institute of Mathematical Sciences

June  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), 2348-2380. doi: 10.1137/060659569.  Google Scholar [2] A. Aw and M. Rascle, Resurrection of second order models of traffic flow, SIAM Journal on Applied Mathematics, 60 (2000), 916-938. doi: 10.1137/S0036139997332099.  Google Scholar [3] E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Communications in Partial Differential Equations, 15 (1990), 293-309. doi: 10.1080/03605309908820745.  Google Scholar [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), 749. doi: 10.3934/nhm.2007.2.751.  Google Scholar [5] A. Bayen, R. Raffard and C. Tomlin, Network congestion alleviation using adjoint hybrid control: Application to highways, Hybrid Systems: Computation and Control, (2004), 113-129. doi: 10.1007/978-3-540-24743-2_7.  Google Scholar [6] A. Ben-Tal and A. Nemirovski, Robust solutions of linear programming problems contaminated with uncertain data, Mathematical Programming, 88 (2000), 411-424. doi: 10.1007/PL00011380.  Google Scholar [7] D. Bertsimas and M. Sim, The price of robustness, Operations Research, 52 (2004), 35-53. doi: 10.1287/opre.1030.0065.  Google Scholar [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 49th IEEE Conference on, 1705-1712. IEEE, 2010. Google Scholar [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), 107-127. doi: 10.1137/090754467.  Google Scholar [10] D. Bovkovic and M. Krstic, Backstepping control of chemical tubular reactors, Computers & Chemical Engineering, 26 (2002), 1077-1085. Google Scholar [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 15th International IEEE Conference on, IEEE, 2012, 832-839. doi: 10.1109/ITSC.2012.6338639.  Google Scholar [12] E. Canepa and C. Claudel, Spoofing cyber attack detection in probe-based traffic monitoring systems using mixed integer linear programming, in Computing, Networking and Communications, 2013 International IEEE Conference on. IEEE, 2013. doi: 10.1109/ICCNC.2013.6504104.  Google Scholar [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 50th IEEE Conference on, IEEE, 2011, 5959-5964. Google Scholar [14] C. Canudas de Wit, D. Jacquet and D. Koenig, Optimal ramp metering strategy with extended LWR model, analysis and computational methods, (2005). Google Scholar [15] R. Carlson, I. Papamichail and M. Papageorgiou, Local feedback-based mainstream traffic flow control on motorways using variable speed limits, Intelligent Transportation Systems, IEEE Transactions on, 12 (2011), 1261-1276. doi: 10.1109/TITS.2011.2156792.  Google Scholar [16] C. Claudel and A. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory, Automatic Control, IEEE Transactions on, 55 (2010), 1142-1157. doi: 10.1109/TAC.2010.2041976.  Google Scholar [17] C. Claudel and A. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part II: Computational methods, Automatic Control, IEEE Transactions on, 55 (2010), 1158-1174. doi: 10.1109/TAC.2010.2045439.  Google Scholar [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), 383-402. doi: 10.1137/090778754.  Google Scholar [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). Google Scholar [20] M. Crandall and P. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8.  Google Scholar [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), 269-287. doi: 10.1016/0191-2615(94)90002-7.  Google Scholar [22] C. Daganzo, A variational formulation of kinematic waves: Basic theory and complex boundary conditions, Transporation Research B, 39 (2005), 187-196. doi: 10.1016/j.trb.2004.04.003.  Google Scholar [23] C. Daganzo, On the variational theory of traffic flow: Well-posedness, duality and applications, (2006). doi: 10.3934/nhm.2006.1.601.  Google Scholar [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, number 09-3159, (2009). Google Scholar [25] H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, SIAM Journal on Control and Optimization, 31 (1993), 257-272. doi: 10.1137/0331016.  Google Scholar [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), 245-265. doi: 10.1007/s00607-009-0038-7.  Google Scholar [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), 1155-1176. doi: 10.1137/040605503.  Google Scholar [28] V. Gabrel, C. Murat and N. Remli, Best and worst optimum for linear programs with interval right hand sides, in Modelling, Computation and Optimization in Information Systems and Management Sciences, Springer, 2008, 126-134. doi: 10.1007/978-3-540-87477-5_14.  Google Scholar [29] V. Gabrel, C. Murat and N. Remli, Linear programming with interval right hand sides, International Transactions in Operational Research, 17 (2010), 397-408. doi: 10.1111/j.1475-3995.2009.00737.x.  Google Scholar [30] R. J. Gibbens and F. P. Kelly, An investigation of proportionally fair ramp metering, in Intelligent Transportation Systems (ITSC), 2011 14th International IEEE Conference on, IEEE, 2011, 490-495. doi: 10.1109/ITSC.2011.6082812.  Google Scholar [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), 589-616. doi: 10.1007/s10957-005-5499-z.  Google Scholar [32] M. Herty and A. Klar, Simplified dynamics and optimization of large scale traffic networks, Mathematical Models and Methods in Applied Sciences, 14 (2004), 579-601. doi: 10.1142/S0218202504003362.  Google Scholar [33] M. Hladík, Interval linear programming: A survey, Linear Programming-New Frontiers in Theory and Applications, (2010), 85-120. Google Scholar [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), 317-345. doi: 10.1098/rspa.1955.0089.  Google Scholar [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), 375-382. Google Scholar [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), 1727-1748. Google Scholar [37] M. Minoux, Robust LP with right-handside uncertainty, duality and applications, (2007). Google Scholar [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), 43-44. Google Scholar [39] G. Newell, A simplified theory of kinematic waves in highway traffic, part I, II and III, Transportation Research Part B: Methodological, 27 (1993), 281-287. Google Scholar [40] M. Papageorgiou, H. Hadj-Salem and J. Blosseville, Alinea: A local feedback control law for on-ramp metering, Transportation Research Record, 1320 (1991), 58-64. Google Scholar [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 IEEE 51st Annual Conference on, IEEE, 2012, 6969-6974. doi: 10.1109/CDC.2012.6426749.  Google Scholar [42] P. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar [43] S. Ulbrich, Optimal control of nonlinear hyperbolic conservation laws with source terms, Technische Universitaet Muenchen, (2001). Google Scholar [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), 1-35. doi: 10.1093/amrx/abq002.  Google Scholar [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, Sensing and Control (ICNSC), 2011 IEEE International Conference on, IEEE, 2011, 121-126. doi: 10.1109/ICNSC.2011.5874888.  Google Scholar

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), 2348-2380. doi: 10.1137/060659569.  Google Scholar [2] A. Aw and M. Rascle, Resurrection of second order models of traffic flow, SIAM Journal on Applied Mathematics, 60 (2000), 916-938. doi: 10.1137/S0036139997332099.  Google Scholar [3] E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Communications in Partial Differential Equations, 15 (1990), 293-309. doi: 10.1080/03605309908820745.  Google Scholar [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), 749. doi: 10.3934/nhm.2007.2.751.  Google Scholar [5] A. Bayen, R. Raffard and C. Tomlin, Network congestion alleviation using adjoint hybrid control: Application to highways, Hybrid Systems: Computation and Control, (2004), 113-129. doi: 10.1007/978-3-540-24743-2_7.  Google Scholar [6] A. Ben-Tal and A. Nemirovski, Robust solutions of linear programming problems contaminated with uncertain data, Mathematical Programming, 88 (2000), 411-424. doi: 10.1007/PL00011380.  Google Scholar [7] D. Bertsimas and M. Sim, The price of robustness, Operations Research, 52 (2004), 35-53. doi: 10.1287/opre.1030.0065.  Google Scholar [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 49th IEEE Conference on, 1705-1712. IEEE, 2010. Google Scholar [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), 107-127. doi: 10.1137/090754467.  Google Scholar [10] D. Bovkovic and M. Krstic, Backstepping control of chemical tubular reactors, Computers & Chemical Engineering, 26 (2002), 1077-1085. Google Scholar [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 15th International IEEE Conference on, IEEE, 2012, 832-839. doi: 10.1109/ITSC.2012.6338639.  Google Scholar [12] E. Canepa and C. Claudel, Spoofing cyber attack detection in probe-based traffic monitoring systems using mixed integer linear programming, in Computing, Networking and Communications, 2013 International IEEE Conference on. IEEE, 2013. doi: 10.1109/ICCNC.2013.6504104.  Google Scholar [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 50th IEEE Conference on, IEEE, 2011, 5959-5964. Google Scholar [14] C. Canudas de Wit, D. Jacquet and D. Koenig, Optimal ramp metering strategy with extended LWR model, analysis and computational methods, (2005). Google Scholar [15] R. Carlson, I. Papamichail and M. Papageorgiou, Local feedback-based mainstream traffic flow control on motorways using variable speed limits, Intelligent Transportation Systems, IEEE Transactions on, 12 (2011), 1261-1276. doi: 10.1109/TITS.2011.2156792.  Google Scholar [16] C. Claudel and A. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory, Automatic Control, IEEE Transactions on, 55 (2010), 1142-1157. doi: 10.1109/TAC.2010.2041976.  Google Scholar [17] C. Claudel and A. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part II: Computational methods, Automatic Control, IEEE Transactions on, 55 (2010), 1158-1174. doi: 10.1109/TAC.2010.2045439.  Google Scholar [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), 383-402. doi: 10.1137/090778754.  Google Scholar [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). Google Scholar [20] M. Crandall and P. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8.  Google Scholar [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), 269-287. doi: 10.1016/0191-2615(94)90002-7.  Google Scholar [22] C. Daganzo, A variational formulation of kinematic waves: Basic theory and complex boundary conditions, Transporation Research B, 39 (2005), 187-196. doi: 10.1016/j.trb.2004.04.003.  Google Scholar [23] C. Daganzo, On the variational theory of traffic flow: Well-posedness, duality and applications, (2006). doi: 10.3934/nhm.2006.1.601.  Google Scholar [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, number 09-3159, (2009). Google Scholar [25] H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, SIAM Journal on Control and Optimization, 31 (1993), 257-272. doi: 10.1137/0331016.  Google Scholar [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), 245-265. doi: 10.1007/s00607-009-0038-7.  Google Scholar [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), 1155-1176. doi: 10.1137/040605503.  Google Scholar [28] V. Gabrel, C. Murat and N. Remli, Best and worst optimum for linear programs with interval right hand sides, in Modelling, Computation and Optimization in Information Systems and Management Sciences, Springer, 2008, 126-134. doi: 10.1007/978-3-540-87477-5_14.  Google Scholar [29] V. Gabrel, C. Murat and N. Remli, Linear programming with interval right hand sides, International Transactions in Operational Research, 17 (2010), 397-408. doi: 10.1111/j.1475-3995.2009.00737.x.  Google Scholar [30] R. J. Gibbens and F. P. Kelly, An investigation of proportionally fair ramp metering, in Intelligent Transportation Systems (ITSC), 2011 14th International IEEE Conference on, IEEE, 2011, 490-495. doi: 10.1109/ITSC.2011.6082812.  Google Scholar [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), 589-616. doi: 10.1007/s10957-005-5499-z.  Google Scholar [32] M. Herty and A. Klar, Simplified dynamics and optimization of large scale traffic networks, Mathematical Models and Methods in Applied Sciences, 14 (2004), 579-601. doi: 10.1142/S0218202504003362.  Google Scholar [33] M. Hladík, Interval linear programming: A survey, Linear Programming-New Frontiers in Theory and Applications, (2010), 85-120. Google Scholar [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), 317-345. doi: 10.1098/rspa.1955.0089.  Google Scholar [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), 375-382. Google Scholar [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), 1727-1748. Google Scholar [37] M. Minoux, Robust LP with right-handside uncertainty, duality and applications, (2007). Google Scholar [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), 43-44. Google Scholar [39] G. Newell, A simplified theory of kinematic waves in highway traffic, part I, II and III, Transportation Research Part B: Methodological, 27 (1993), 281-287. Google Scholar [40] M. Papageorgiou, H. Hadj-Salem and J. Blosseville, Alinea: A local feedback control law for on-ramp metering, Transportation Research Record, 1320 (1991), 58-64. Google Scholar [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 IEEE 51st Annual Conference on, IEEE, 2012, 6969-6974. doi: 10.1109/CDC.2012.6426749.  Google Scholar [42] P. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.  Google Scholar [43] S. Ulbrich, Optimal control of nonlinear hyperbolic conservation laws with source terms, Technische Universitaet Muenchen, (2001). Google Scholar [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), 1-35. doi: 10.1093/amrx/abq002.  Google Scholar [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, Sensing and Control (ICNSC), 2011 IEEE International Conference on, IEEE, 2011, 121-126. doi: 10.1109/ICNSC.2011.5874888.  Google Scholar
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