Convex Formulations of Air Traffic Flow Optimization Problems

Proceedings of the IEEE

Volumn 96, Issue 12, 2008, Pages 2096-2112

  Work, Daniel B ^a   Bayen, Alexandre M ^a  

^a Berkeley   (United States)

Author keywords

Convex optimization; finite differences; partial differential equations

Indexed keywords

EID: 85008581847     PISSN: 00189219     EISSN: 15582256     Source Type: Journal    
DOI: 10.1109/JPROC.2008.2006150     Document Type: Article

References (43)

1. S. Amin, A. BAYEN, L. El Ghaoui, and S. Sastry, “Robust feasibility for control of water flow in a reservoir-canal system”, in Proc. 46th IEEE Conf. Decision Contr. Conf., New Orleans, LA, Dec. 2007, pp. 1571–1577.
2. G. Bastin and V. Guffen, “Congestion control in compartmental network systems”, Syst. Contr. Lett., vol. 55, no. 8, pp. 689–696, 2006.
3. A. Bayen, P. Grieder, G. Meyer, and C. Tomlin, “Lagrangian delay predictive model for sector-based air traffic flow”, AIAA J. Guidance, Contr., Dyn., vol. 28, no. 5, pp. 1015–1026, 2005.
4. A. Bayen, R. Raffard, and C. Tomlin, “Adjoint-based control of a new Eulerian network model of air traffic flow”, IEEE Trans. Contr. Syst. Technol., vol. 14, no. 5, pp. 804–818, 2006.
5. A. Bennett, Inverse Methods in Physical Oceanography. Cambridge, U.K.: Cambridge Univ. Press, 1992.
6. D. Bertsimas and S. S. Patterson, “The air traffic flow management problem with enroute capacities”, Oper. Res., vol. 46, no. 3, pp. 406–422, 1998.
7. K. Bilimoria, B. Sridhar, G. Chatterji, K. Sheth, and S. Grabbe, “FACET: Future ATM concepts evaluation tool”, in Proc. 3rd USA/Eur. ATM 2001 R&D Seminar, Naples, Italy, Jun. 2001.
8. F. Borrelli, Constrained Optimal Control of Linear & Hybrid Systems, vol. 290, Lecture Notes in Control and Information Sciences. Berlin, Germany: Springer-Verlag, 2003.
9. S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004.
10. J. Coron, B. d’Andrea-Novel, and G. Bastin, “A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws”, IEEE Trans. Autom. Control, vol. 52, no. 1, pp. 2–11, 2007.
11. R. Courant, K. Friedrichs, and H. Lewy, “Uber die partiellen Differenzengleichungen der mathematischen Physik”, Math. Ann., vol. 100, pp. 32–74, 1928.
12. C. Daganzo, “The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory”, Transp. Res. B, vol. 28, no. 4, pp. 269–287, 1994.
13. C. Daganzo, “The cell transmission model, Part II: Network traffic”, Transp. Res. B, vol. 29, no. 2, pp. 79–93, 1995.
14. Federal Aviation Administration, FAA Aerospace Forecasts: Fiscal Years 2007–2020, 2007.
15. C. Hirsch, Numerical Computation of Internal & External Flows: Fundamentals of Numerical Discretization. New York: Wiley, 1988.
16. A. Jameson, L. Martinelli, and N. Pierce, “Optimum aerodynamic design using the Navier-Stokes equations”, Theor. Comput. Fluid Dyn., vol. 10, no. 1, pp. 213–237, 1998.
17. C. Kirchner, M. Herty, S. Gottlich, and A. Klar, “Optimal control for continuous supply network models”, Netw. Heterogen. Media, vol. 1, no. 4, pp. 675–688, 2006.
18. O. Koroleva, M. Krstic, and G. Schmid-Schonbein, “Decentralized and adaptive control of nonlinear fluid flow networks”, Int. J. Contr., vol. 16, pp. 801–818, 2006.
19. M. Lighthill and G. Whitham, “On kinematic waves. II. A theory of traffic flow on long crowded roads”, Proc. Roy. Soc. London
20. Series A, Math. Phys. Sci., vol. 229, no. 1178, pp. 317–345, 1955.
21. X. Litrico and V. Fromion, “Frequency modeling of open-channel flow”, J. Hydraulic Eng., vol. 130, no. 8, pp. 806–815, 2004.
22. P. Malaterre, “PILOTE: Linear quadratic optimal controller for irrigation canals”, J. Irrig. Drain. Eng., vol. 124, no. 4, pp. 187–194, 1998.
23. P. Malaterre and M. Khammash, “Controller design for a high-order 5-pool irrigation canal system”, J. Dyn. Syst., Measure., Contr., vol. 125, p. 639, 2004.
24. P. Menon, G. Sweriduk, and K. Bilimoria, “New approach for modeling, analysis, and control of air traffic flow”, AIAA J. Guidance, Contr., Dyn., vol. 27, no. 5, pp. 737–744, 2004.
25. P. Menon, G. Sweriduk, T. Lam, G. Diaz, and K. Bilimoria, “Computer-aided Eulerian air traffic flow modeling and predictive control,” AIAA J. Guidance, Contr., Dyn., vol. 29, no. 1, pp. 12–19, 2006.
26. C. Meyer, Matrix Analysis and Applied Linear Algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2000.
27. K. Mueller, D. Schleicher, and K. D. Bilimoria, “Conflict detection and resolution with traffic flow constraints”, presented at the AIAA Conf. Guidance, Navig., Contr., Monterey, CA, 2002, AIAA Paper 2002–4445.
28. M. Nolan, Fundamentals of Air Traffic Control. 4th ed. Reading, MA: Brooks Cole, 2003.
29. B. Piccoli and M. Garavello, Traffic Flow on Networks. Springfield, MO: American Institute of Mathematical Sciences on Applied Mathematics, 2006.
30. R. Raffard, K. Amonlirdviman, J. Axelrod, and C. Tomlin, “An adjoint-based parameter
31. identification algorithm applied to planar cell polarity signaling”, IEEE Trans. Autom. Control (Special Issue on Systems Biology), vol. 53, pp. 109–121, Jan. 2008.
32. R. Raffard, J. Hu, and C. Tomlin, Adjoint-Based Optimal Control of the Expected Exit Time for Stochastic Hybrid Systems, M. Morari and Thiele, Eds. Berlin, Germany: Springer-Verlag, 2005, pp. 557–572.
33. R. Raffard, S. Waslander, and C. Tomlin, “Toward efficient and equitable distributed air traffic flow control”, in Proc. 2006 Amer. Contr. Conf., Minneapolis, MN, Jun. 2006, pp. 5189–5194.
34. R. L. Raffard and C. J. Tomlin, “Second order adjoint-based optimization of ordinary and partial differential equations with application to air traffic flow”, in Proc. Amer. Contr. Conf., Portland, OR, 2005, pp. 798–803.
35. P. I. Richards, “Shockwaves on the highway”, Oper. Res., vol. 4, no. 1, pp. 42–51, 1956.
36. S. Roy, B. Sridhar, and G. C. Verghese, “An aggregate dynamic stochastic model for air traffic control”, in Proc. 5th USA/Eur. ATM 2003 R&D Seminar, Budapest, Hungary, Jun. 2003.
37. B. Sanders and N. Katopodes, “Control of canal flow by adjoint sensitivity method”, J. Irrig. Drain. Eng., vol. 125, no. 5, pp. 287–297, 1999.
38. B. Sridhar, T. Soni, K. Sheth, and G. Chatterji, “An aggregate flow model for air traffic management”, presented at the AIAA Conf. Guidance, Navig., Contr., Providence, RI, Aug. 2004, AIAA Paper 2004–5316.
39. J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations. Philadelphia, PA: SIAM, 2004.
40. I. Strub and A. Bayen, “Weak formulation of boundary conditions for scalar conservation laws: An application to highway modeling”, Int. J. Robust Nonlinear Contr., vol. 16, pp. 733–748, 2006.
41. D. Sun and A. Bayen, “A new Eulerian-Lagrangian large-capacity cell transmission model for en route traffic,” AIAA J. Guidance, Contr. Dyn., vol. 31, no. 3, pp. 616–628, 2008.
42. D. Sun, I. Strub, and A. Bayen, “Comparison of the performance of four Eulerian network flow models for strategic air traffic flow management”, Netw. Hetero. Media, vol. 2, no. 4, pp. 569–594, 2007.
43. C. Zhu, R. Byrd, and J. Nocedal, “L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization”, ACM Trans. Math. Software, vol. 23, no. 4, pp. 550–560, 1997.