American Institute of Mathematical Sciences

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September  2013, 8(3): 783-802. doi: 10.3934/nhm.2013.8.783

Spoofing cyber attack detection in probe-based traffic monitoring systems using mixed integer linear programming

Edward S. Canepa 1, , Alexandre M. Bayen 2, and  Christian G. Claudel 1,

1. 

King Abdullah University of Science and Technology, Electrical Engineering Department, Thuwal, Makkah 23955, KSA, Saudi Arabia, Saudi Arabia
2. 

University of California at Berkely, Electrical Engineering and Computer Sciences, Berkeley CA 94720-170

Received  September 2012 Revised  March 2013 Published  October 2013

Traffic sensing systems rely more and more on user generated (insecure) data, which can pose a security risk whenever the data is used for traffic flow control. In this article, we propose a new formulation for detecting malicious data injection in traffic flow monitoring systems by using the underlying traffic flow model. The state of traffic is modeled by the Lighthill-Whitham-Richards traffic flow model, which is a first order scalar conservation law with concave flux function. Given a set of traffic flow data generated by multiple sensors of different types, we show that the constraints resulting from this partial differential equation are mixed integer linear inequalities for a specific decision variable. We use this fact to pose the problem of detecting spoofing cyber attacks in probe-based traffic flow information systems as mixed integer linear feasibility problem. The resulting framework can be used to detect spoofing attacks in real time, or to evaluate the worst-case effects of an attack offline. A numerical implementation is performed on a cyber attack scenario involving experimental data from the Mobile Century experiment and the Mobile Millennium system currently operational in Northern California.
Keywords: hyperbolic partial differential equations, cybersecurity., Mixed integer linear programming.
Mathematics Subject Classification: 68U35, 90C11, 90C9.
Citation: Edward S. Canepa, Alexandre M. Bayen, Christian G. Claudel. Spoofing cyber attack detection in probe-based traffic monitoring systems using mixed integer linear programming. Networks & Heterogeneous Media, 2013, 8 (3) : 783-802. doi: 10.3934/nhm.2013.8.783
References:
[1]

S. Amin, A. Cardenas and S. Sastry, Safe and secure networked control systems under denial-of-service attacks, in "Proceedings of the 12th International Conference on Hybrid Systems: Computation and Control," Springer-Verlag, (2009), 31-45. doi: 10.1007/978-3-642-00602-9_3.  Google Scholar

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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

[5]

M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of {Hamilton-Jacobi-Bellman} Equations," Birkhäuser, Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[6]

E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Communications in Partial Differential Equations, 15 (1990), 1713-1742. doi: 10.1080/03605309908820745.  Google Scholar

[7]

E. S. Canepa and C. G. Claudel, Exact solutions to traffic density estimation problems involving the Lighthill-Whitman-Richards traffic flow model using Mixed Integer Linear Programing, In "Proceedings of the 15th International IEEE Conference on Intelligent Transportation Systems", (2012), 832-839. Google Scholar

[8]

P. D. Christofides, "Nonlinear and Robust Control of Partial Differential Equation Systems: Methods and Applications to Transport-Reaction Processes," Birkhäuser, Boston, MA, 2001. doi: 10.1007/978-1-4612-0185-4.  Google Scholar

[9]

C. G. Claudel and A. M. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory, IEEE Transactions on Automatic Control, 55 (2010), 1142-1157. doi: 10.1109/TAC.2010.2041976.  Google Scholar

[10]

C. G. Claudel and A. M. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part {II: Computational methods}, IEEE Transactions on Automatic Control, 55 (2010), 1158-1174. doi: 10.1109/TAC.2010.2045439.  Google Scholar

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C. G. Claudel and A. M 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

[12]

M. G. Crandall and P.-L. Lions, Viscosity solutions of {Hamilton-Jacobi equations}, Transactions of the American Mathematical Society, 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8.  Google Scholar

[13]

C. Daganzo, The cell transmission model: a dynamic representation of highway traffic consistent with the hydrodynamic theory, Transportation Research, 28B (1994), 269-287. doi: 10.1016/0191-2615(94)90002-7.  Google Scholar

[14]

C. F. Daganzo, A variational formulation of kinematic waves: basic theory and complex boundary conditions, Transportation Research B, 39B (2005), 187-196. doi: 10.1016/j.trb.2004.04.003.  Google Scholar

[15]

C. F. Daganzo, On the variational theory of traffic flow: well-posedness, duality and applications, Networks and Heterogeneous Media, 1 (2006), 601-619. doi: 10.3934/nhm.2006.1.601.  Google Scholar

[16]

H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, SIAM Journal of Control and Optimization, 31 (1993), 257-272. doi: 10.1137/0331016.  Google Scholar

[17]

J. C. Herrera, D. B. Work, R. Herring, X. J. Ban, Q. Jacobson and A. M. Bayen, Evaluation of traffic data obtained via GPS-enabled mobile phones: The Mobile Century field experiment, Transportation Research Part C: Emerging Technologies, 18 (2010), 568-583. doi: 10.1016/j.trc.2009.10.006.  Google Scholar

[18]

B. Hoh, M. Gruteser, R. Herring, J. Ban, D. Work, J. C. Herrera, A. M. Bayen, M. Annavaram and Q. Jacobson, Virtual trip lines for distributed privacy-preserving traffic monitoring, in "Proceedings of the 6th International Conference on Mobile Systems, Applications, and Services," ACM, (2008), 15-28. doi: 10.1145/1378600.1378604.  Google Scholar

[19]

M. Krstic and A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic pdes and application to systems with actuator and sensor delays, Systems & Control Letters, 57 (2008), 750-758. doi: 10.1016/j.sysconle.2008.02.005.  Google Scholar

[20]

P. E. Mazare, A. Dehwah, C. G. Claudel and A. M. Bayen, Analytical and grid-free solutions to the lighthill-whitham-richards traffic flow model, Transportation Research Part B: Methodological, 45 (2011), 1727-1748. doi: 10.1016/j.trb.2011.07.004.  Google Scholar

[21]

K. Moskowitz, Discussion of "freeway level of service as influenced by volume and capacity characteristics' by D.R. Drew and C. J. Keese, Highway Research Record, 99 (1965), 43-44. Google Scholar

[22]

G. F. Newell, A simplified theory of kinematic waves in highway traffic, Part (I), (II) and (III). Transporation Research B, 27B (1993), 281-313. Google Scholar

[23]

R. C. Smith and M. A. Demetriou, "Research Directions in Distributed Parameter Systems," SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898717525.  Google Scholar

[24]

I. S. Strub and A. M. Bayen, Weak formulation of boundary conditions for scalar conservation laws, International Journal of Robust and Nonlinear Control, 16 (2006), 733-748. doi: 10.1002/rnc.1099.  Google Scholar

[25]

D. Work, S. Blandin, O. Tossavainen, B. Piccoli and A. Bayen, A distributed highway velocity model for traffic state reconstruction, Applied Research Mathematics eXpress (ARMX), 1 (2010), 1-35. Google Scholar

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, , ().   Google Scholar

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, , ().   Google Scholar

show all references

References:
[1]

S. Amin, A. Cardenas and S. Sastry, Safe and secure networked control systems under denial-of-service attacks, in "Proceedings of the 12th International Conference on Hybrid Systems: Computation and Control," Springer-Verlag, (2009), 31-45. doi: 10.1007/978-3-642-00602-9_3.  Google Scholar

[2]

S. Amin, X. Litrico, S. Sastry and A. Bayen, Stealthy deception attacks on water scada systems, In "Proceedings of the 13th ACM International Conference on Hybrid Systems: Computation and Control," ACM, (2010),161-170. doi: 10.1145/1755952.1755976.  Google Scholar

[3]

J.-P. Aubin, "Viability Theory," Systems and Control: Foundations and Applications, Birkhäuser, Boston, MA, 1991.  Google Scholar

[4]

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

[5]

M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of {Hamilton-Jacobi-Bellman} Equations," Birkhäuser, Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[6]

E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Communications in Partial Differential Equations, 15 (1990), 1713-1742. doi: 10.1080/03605309908820745.  Google Scholar

[7]

E. S. Canepa and C. G. Claudel, Exact solutions to traffic density estimation problems involving the Lighthill-Whitman-Richards traffic flow model using Mixed Integer Linear Programing, In "Proceedings of the 15th International IEEE Conference on Intelligent Transportation Systems", (2012), 832-839. Google Scholar

[8]

P. D. Christofides, "Nonlinear and Robust Control of Partial Differential Equation Systems: Methods and Applications to Transport-Reaction Processes," Birkhäuser, Boston, MA, 2001. doi: 10.1007/978-1-4612-0185-4.  Google Scholar

[9]

C. G. Claudel and A. M. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory, IEEE Transactions on Automatic Control, 55 (2010), 1142-1157. doi: 10.1109/TAC.2010.2041976.  Google Scholar

[10]

C. G. Claudel and A. M. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part {II: Computational methods}, IEEE Transactions on Automatic Control, 55 (2010), 1158-1174. doi: 10.1109/TAC.2010.2045439.  Google Scholar

[11]

C. G. Claudel and A. M 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

[12]

M. G. Crandall and P.-L. Lions, Viscosity solutions of {Hamilton-Jacobi equations}, Transactions of the American Mathematical Society, 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8.  Google Scholar

[13]

C. Daganzo, The cell transmission model: a dynamic representation of highway traffic consistent with the hydrodynamic theory, Transportation Research, 28B (1994), 269-287. doi: 10.1016/0191-2615(94)90002-7.  Google Scholar

[14]

C. F. Daganzo, A variational formulation of kinematic waves: basic theory and complex boundary conditions, Transportation Research B, 39B (2005), 187-196. doi: 10.1016/j.trb.2004.04.003.  Google Scholar

[15]

C. F. Daganzo, On the variational theory of traffic flow: well-posedness, duality and applications, Networks and Heterogeneous Media, 1 (2006), 601-619. doi: 10.3934/nhm.2006.1.601.  Google Scholar

[16]

H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, SIAM Journal of Control and Optimization, 31 (1993), 257-272. doi: 10.1137/0331016.  Google Scholar

[17]

J. C. Herrera, D. B. Work, R. Herring, X. J. Ban, Q. Jacobson and A. M. Bayen, Evaluation of traffic data obtained via GPS-enabled mobile phones: The Mobile Century field experiment, Transportation Research Part C: Emerging Technologies, 18 (2010), 568-583. doi: 10.1016/j.trc.2009.10.006.  Google Scholar

[18]

B. Hoh, M. Gruteser, R. Herring, J. Ban, D. Work, J. C. Herrera, A. M. Bayen, M. Annavaram and Q. Jacobson, Virtual trip lines for distributed privacy-preserving traffic monitoring, in "Proceedings of the 6th International Conference on Mobile Systems, Applications, and Services," ACM, (2008), 15-28. doi: 10.1145/1378600.1378604.  Google Scholar

[19]

M. Krstic and A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic pdes and application to systems with actuator and sensor delays, Systems & Control Letters, 57 (2008), 750-758. doi: 10.1016/j.sysconle.2008.02.005.  Google Scholar

[20]

P. E. Mazare, A. Dehwah, C. G. Claudel and A. M. Bayen, Analytical and grid-free solutions to the lighthill-whitham-richards traffic flow model, Transportation Research Part B: Methodological, 45 (2011), 1727-1748. doi: 10.1016/j.trb.2011.07.004.  Google Scholar

[21]

K. Moskowitz, Discussion of "freeway level of service as influenced by volume and capacity characteristics' by D.R. Drew and C. J. Keese, Highway Research Record, 99 (1965), 43-44. Google Scholar

[22]

G. F. Newell, A simplified theory of kinematic waves in highway traffic, Part (I), (II) and (III). Transporation Research B, 27B (1993), 281-313. Google Scholar

[23]

R. C. Smith and M. A. Demetriou, "Research Directions in Distributed Parameter Systems," SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898717525.  Google Scholar

[24]

I. S. Strub and A. M. Bayen, Weak formulation of boundary conditions for scalar conservation laws, International Journal of Robust and Nonlinear Control, 16 (2006), 733-748. doi: 10.1002/rnc.1099.  Google Scholar

[25]

D. Work, S. Blandin, O. Tossavainen, B. Piccoli and A. Bayen, A distributed highway velocity model for traffic state reconstruction, Applied Research Mathematics eXpress (ARMX), 1 (2010), 1-35. Google Scholar

[26]

, , ().   Google Scholar

[27]

, , ().   Google Scholar

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