American Institute of Mathematical Sciences

Advanced
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, (2009), 31.  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, (2010), 161.  doi: 10.1145/1755952.1755976.  Google Scholar

[3]

J.-P. Aubin, "Viability Theory,", Systems and Control: Foundations and Applications, (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.  doi: 10.1137/060659569.  Google Scholar

[5]

M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of {Hamilton-Jacobi-Bellman} Equations,", Birkhäuser, (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.  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, (2012), 832.   Google Scholar

[8]

P. D. Christofides, "Nonlinear and Robust Control of Partial Differential Equation Systems: Methods and Applications to Transport-Reaction Processes,", Birkhäuser, (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.  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.  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.  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.  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.  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.  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.  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.  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.  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, (2008), 15.  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.  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.  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.   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.   Google Scholar

[23]

R. C. Smith and M. A. Demetriou, "Research Directions in Distributed Parameter Systems,", SIAM, (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.  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.   Google Scholar

[26]

, , ().   Google Scholar

[27]

, , ().   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, (2009), 31.  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, (2010), 161.  doi: 10.1145/1755952.1755976.  Google Scholar

[3]

J.-P. Aubin, "Viability Theory,", Systems and Control: Foundations and Applications, (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.  doi: 10.1137/060659569.  Google Scholar

[5]

M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of {Hamilton-Jacobi-Bellman} Equations,", Birkhäuser, (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.  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, (2012), 832.   Google Scholar

[8]

P. D. Christofides, "Nonlinear and Robust Control of Partial Differential Equation Systems: Methods and Applications to Transport-Reaction Processes,", Birkhäuser, (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.  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.  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.  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.  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.  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.  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.  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.  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.  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, (2008), 15.  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.  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.  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.   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.   Google Scholar

[23]

R. C. Smith and M. A. Demetriou, "Research Directions in Distributed Parameter Systems,", SIAM, (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.  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.   Google Scholar

[26]

, , ().   Google Scholar

[27]

, , ().   Google Scholar

[1]

Elham Mardaneh, Ryan Loxton, Qun Lin, Phil Schmidli. A mixed-integer linear programming model for optimal vessel scheduling in offshore oil and gas operations. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1601-1623. doi: 10.3934/jimo.2017009
[2]

Mahdi Roozbeh, Saman Babaie–Kafaki, Zohre Aminifard. Two penalized mixed–integer nonlinear programming approaches to tackle multicollinearity and outliers effects in linear regression models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020128
[3]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053
[4]

Ye Tian, Cheng Lu. Nonconvex quadratic reformulations and solvable conditions for mixed integer quadratic programming problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1027-1039. doi: 10.3934/jimo.2011.7.1027
[5]

Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks & Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014
[6]

Louis Caccetta, Syarifah Z. Nordin. Mixed integer programming model for scheduling in unrelated parallel processor system with priority consideration. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 115-132. doi: 10.3934/naco.2014.4.115
[7]

René Henrion, Christian Küchler, Werner Römisch. Discrepancy distances and scenario reduction in two-stage stochastic mixed-integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 363-384. doi: 10.3934/jimo.2008.4.363
[8]

Zhiguo Feng, Ka-Fai Cedric Yiu. Manifold relaxations for integer programming. Journal of Industrial & Management Optimization, 2014, 10 (2) : 557-566. doi: 10.3934/jimo.2014.10.557
[9]

Jiahui Zhu, Zdzisław Brzeźniak. Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3269-3299. doi: 10.3934/dcdsb.2016097
[10]

Mahmoud Ameri, Armin Jarrahi. An executive model for network-level pavement maintenance and rehabilitation planning based on linear integer programming. Journal of Industrial & Management Optimization, 2020, 16 (2) : 795-811. doi: 10.3934/jimo.2018179
[11]

Fanwen Meng, Kiok Liang Teow, Kelvin Wee Sheng Teo, Chee Kheong Ooi, Seow Yian Tay. Predicting 72-hour reattendance in emergency departments using discriminant analysis via mixed integer programming with electronic medical records. Journal of Industrial & Management Optimization, 2019, 15 (2) : 947-962. doi: 10.3934/jimo.2018079
[12]

Wan Nor Ashikin Wan Ahmad Fatthi, Adibah Shuib, Rosma Mohd Dom. A mixed integer programming model for solving real-time truck-to-door assignment and scheduling problem at cross docking warehouse. Journal of Industrial & Management Optimization, 2016, 12 (2) : 431-447. doi: 10.3934/jimo.2016.12.431
[13]

Sigurdur Hafstein, Skuli Gudmundsson, Peter Giesl, Enrico Scalas. Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 939-956. doi: 10.3934/dcdsb.2018049
[14]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[15]

Frank Pörner, Daniel Wachsmuth. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Mathematical Control & Related Fields, 2018, 8 (1) : 315-335. doi: 10.3934/mcrf.2018013
[16]

Yacine Chitour, Jean-Michel Coron, Mauro Garavello. On conditions that prevent steady-state controllability of certain linear partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 643-672. doi: 10.3934/dcds.2006.14.643
[17]

Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure & Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211
[18]

Xiaoyu Fu. Stabilization of hyperbolic equations with mixed boundary conditions. Mathematical Control & Related Fields, 2015, 5 (4) : 761-780. doi: 10.3934/mcrf.2015.5.761
[19]

T. Candan, R.S. Dahiya. Oscillation of mixed neutral differential equations with forcing term. Conference Publications, 2003, 2003 (Special) : 167-172. doi: 10.3934/proc.2003.2003.167
[20]

Alain Bensoussan, Shaokuan Chen, Suresh P. Sethi. Linear quadratic differential games with mixed leadership: The open-loop solution. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 95-108. doi: 10.3934/naco.2013.3.95

2019 Impact Factor: 1.053

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences