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Effective Algorithms for a Class of Discrete Valued Optimal Control Problems

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Optimization and Cooperative Control Strategies

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 381))

Abstract

We consider a general class of optimal control problems where the control functions are assumed piecewise constant and only take on values from a finite discrete set. The aim in such a problem is to find a sequence of discrete control values and a corresponding set of exact switching times (i.e., times where control should switch between the discrete values) such that a given functional representing cost or risk is minimized. Such problems arise in a range of applications. One application is the ‘Transit Path Problem’ where an object such as a robot or vehicle (air, naval, space or land) needs to traverse a specified region (discrete or continuous) between two points in a prescribed time so as to avoid detection. The objective is to find a path for the object which satisfies the time constraints and which minimizes the total risk of detection. The risk function is not simple and depends on a range of factors such as the environment, the types of sensors, the speed, direction and position of the vehicle. The main difficulty with these problems is that the range of some of the controls is discrete and hence not convex. Since the gradients with respect to the switching time parameters are discontinuous, ordinary gradients based solution methods perform poorly. An additional difficulty is to determine exactly how many switching times are involved in an optimal solution. We address the first difficulty by using the Control Parameterization Enhancing Transform (CPET) and the second difficulty by solving a sequence of problems which are transformed via CPET. With respect to the transit path problem our strategy involves a two stage approach. The first stage involves a discretization of the problem and the solution of a constrained path problem in a network. The second stage involves the use of an optimal control model and a solution procedure that utilizes the solution obtained from the first stage.

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References

  1. Ahuja, R.L., Magnanti, J.L., Orlin, J.B.: Network Flows: Theory, algorithms and applications. Prentice-Hall, Englewood Cliffs (1993)

    Google Scholar 

  2. Aneja, Y.P., Aggarwal, V., Nair, K.P.K.: Shortest chain subject to side constraints. Networks 13(2), 295–302 (1983)

    Article  MathSciNet  Google Scholar 

  3. Beasley, J.E., Christofides, N.: An algorithm for the resource constrained shortest path Problem. Networks 19(4), 379–394 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  4. Boland, N., Dethridge, J., Dumitrescu, I.: Accelerated label setting algorithms for the elementary resource constrained shortest path problem. Operations Research Letters 34, 379–394 (1989)

    MathSciNet  Google Scholar 

  5. Caccetta, L.: Branch and cut methods for mixed integer programming problems. Progess in Optimization II. Applied Optimization Series, vol. 39, pp. 21–44. Kluwer Publishers, Dordrecht (2000)

    Google Scholar 

  6. Caccetta, L., Loosen, I., Rehbock, V.: Optimal transit path problem for submarines. Modelling Supplementary Volume of the DCDIS, Series B, Applications & Algorithms, pp. 874–880 (2005)

    Google Scholar 

  7. Carlyle, W.M., Wood, R.K.: Lagrangian relaxation and enumeration for solving constrained shortest-path problems. In: Proceedings of the 38th Annual ORSNZ Conference, University of Waikato, pp. 3–12 (2003)

    Google Scholar 

  8. Carlyle, W.M., Royset, J.O., Wood, R.K.: Routing military aircraft with a constrained shortest-path algorithm (in review, 2007)

    Google Scholar 

  9. Chan, Y.K., Foddy, M.: Real time optimal flight path generation by storage of massive data bases. In: Proceedings of the IEEE NEACON 1985, Institute of Electrical and Electronics Engineerings, New York, pp. 516–521 (1985)

    Google Scholar 

  10. Chaudhry, A., Misovec, K., D’Andrea, R.: Low observability path planning for an unmanned air vehicle using mixed integer linear programming. In: 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, December 14–17, pp. 14–17 (2004)

    Google Scholar 

  11. Desrochers, M., Soumis, F.: A generalized permanent labelling algorithm for the shortest path problem with time windows. INFOR 26, 191–212 (1988)

    MATH  Google Scholar 

  12. Dijkstra, E.W.: A note of two problems in connection with graphs. Numerische Mathematik 1, 269–271 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dogan, A.: Probabilistic approach in path planning for UAVs. In: Proceedings of the 18th IEEE International Symposium on Intelligent Control, Houston, Texas, October 5–8, pp. 608–613 (2003)

    Google Scholar 

  14. Dumitrescu, I., Boland, N.: Improved preprocessing, labeling and scaling algorithms for the weight constrained shortest path problem. Networks 42(3), 135–153 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Garey, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of NP-completeness. W.H. Freeman and Co., San Francisco (1979)

    MATH  Google Scholar 

  16. Gill, P.E., Murray, W., Saunders, M.A., Wright, M.H.: User’s guide for NPSOL (version 4.0): A fortran package for nonlinear programming. Report SOL 86–2, Department of Operations Research, Stanford University (1998)

    Google Scholar 

  17. Grignon, L., Prasetio, Y., Toktas, B., Yen, J., Zabinsky, Z.: A basis space-time network formulation for aircraft rerouting with airspace closures. Technical Report, Department of Industrial Engineering, University of Washington, Seattle, Washington (2002)

    Google Scholar 

  18. Hallam, C.L.: Hierarchical path generation: An application to submarine transit paths. Honours Dissertation. Murdoch University, Western Australia (1997)

    Google Scholar 

  19. Hallam, C.L., Harrison, K.J., Ward, J.A.: A multiobjective optimal path algorithm. Digital Signal Processing 11(2), 133–143 (2001)

    Article  Google Scholar 

  20. Handler, G., Zang, I.: A dual algorithm for the constrained shortest path problem. Networks 10(4), 293–310 (1980)

    Article  MathSciNet  Google Scholar 

  21. Hassin, R.: Approximation schemes for the restricted shortest path problem. Mathematics of Operations Research 17, 36–42 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hebert, J., Jacques, D., Novy, M., Pachter, M.: Cooperative control of UAVs. In: AIAA Guidance, Navigation and Control Conference and Exhibit, Montreal, Canada (2001)

    Google Scholar 

  23. Hespanha, J.P., Kizilocak, H.H., Ateskan, Y.S.: Probalistic map building for aircraft-tracking radars. In: Proceedings of the American Control Conference, Arlington, Virginia, vol. 6, pp. 4381–4386 (2001)

    Google Scholar 

  24. Howlett, P., Pudney, P., Benjamin, B.: Determination of optimal driving strategies for the control of a train. In: Computational Techniques and Applications: CTAC 1991, Computational Mathematics Group, Australian Mathematical Society, pp. 241–248 (1991)

    Google Scholar 

  25. Howlett, P.: Optimal strategies for the control of a train. Automatica 32, 519–532 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. ILOG CPLEX 9.0: User’s Manual, ILOG, S.A (2003), http://www.ilog.com/

  27. Inanc, T., Misovec, K., Murray, R.M.: Nonlinear trajectory generation for unmanned air vehicles with multiple radars. In: Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, December 14–17, pp. 3817–3822 (2004)

    Google Scholar 

  28. Jaumard, B., Semet, F., Vovor, T.: A two-phase resource constrained shortest path algorithm for acyclic graphs. Les Cahier du GERAD, G-96–48 (1996)

    Google Scholar 

  29. Jennings, L.S., Fisher, M.E., Teo, K.L., Goh, C.J.: MISER3: Solving optimal control problems - an update. Advanced Engineering Software 13, 190–196 (1991)

    Article  MATH  Google Scholar 

  30. Jennings, L.S., Teo, K.L., Fisher, M.E., Goh, C.J.: MISER3 version 2, optimal control software, theory and user manual Department of Mathematics, University of Western Australia (1997), http://cado.maths.uwa.edu.au/miser/manual.html

  31. Joksch, H.C.: The shortest route problem with constraints. Journal of Mathematical Analysis and Application 14, 191–197 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  32. Jun, M., D’Andrea, R.: Path planning for unmanned aerial vehicles in uncertain and adversarial environments. Cooperative Control: Models, Applications and Algorithms, 99–110 (2003)

    Google Scholar 

  33. Kim, J., Hespanha, J.P.: Discrete approximation to continuous shortest-path: Application to minimum-risk path planning for groups of UAV’s. In: 42nd IEEE Conference on Decision and Control, Maui, Hawaii, December 9–12 (2003)

    Google Scholar 

  34. Latourell, J.L., Wallet, B.C., Copeland, B.: Genetic algorithm to solve constrained routing problems with applications for cruise missile routing. In: Proceedings of SPIE, Applications and Science of Computational Intelligence, vol. 3390, pp. 490–500 (1998)

    Google Scholar 

  35. Lakshmanan, R., Stephanopoulos, G.: Synthesis of operating procedures for complete chemical plants-II: A nonlinear planning methodology. Computer and Chemical Engineering 12, 1003–1021 (1988)

    Article  Google Scholar 

  36. Leary, J.: Search for a stealthy flight path through a hostile radar defence network. Master’s Thesis, Naval Postgraduate School, Monterey, California (1995)

    Google Scholar 

  37. Lee, S.H.K.: Route optimization model for strike aircraft. Master’s Thesis, Naval Postgraduate School, Monterey, California (1995)

    Google Scholar 

  38. Lee, H.W.J., Teo, K.L., Jennings, L.S., Rehbock, V.: Control parametrization enhancing technique for time optimal control problems. Dynamic Systems and Applications 6(2), 243–262 (1997)

    MATH  MathSciNet  Google Scholar 

  39. Lee, H.W.J., Teo, K.L., Rehbock, V., Jennings, L.S.: Control parametrization enhancing technique for discrete-valued control problems. Automatica 35(8), 1401–1407 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  40. Lewis, D.H.: Optimal three-dimensional path planning using visibility constaints. Master’s Thesis, Naval Postgraduate School, Monterey, California (1988)

    Google Scholar 

  41. Loosen, I.: Optimization techniques for constrained path problems. Ph.D Thesis, Department of Mathematics and Statistiics, Curtin University of Technology (2007)

    Google Scholar 

  42. Lorenz, D.H., Raz, D.: A simple efficient approximation scheme for the restricted shortest path problem. Operations Research Letters 28, 213–219 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  43. McFarland, M.B., Acjeru, R.A., Taylor, B.L.: Motion planning for reduced observability of autonomous aerial vehicles. In: IEEE International Conference on Control Applications (1999)

    Google Scholar 

  44. Milam, M.: Optimal trajectory generation for constrained dynamical systems. Ph.D Thesis, Control and Dynamical Systems, Caltech (2003)

    Google Scholar 

  45. Misovec, K., Inanc, T., Wohletz, J., Murray, R.M.: Low-observable nonlinear trajectory generation for unmanned air vehicles. In: 42nd IEEE Conference on Decision and Control, Maui, Hawaii, December 9–12 (2003)

    Google Scholar 

  46. Murphey, R., Zabarankin, M., Uryasev, S.: Trajectory optimization in a threat environment. Research Report 2003–9, Department of Industrial and System Engineering, University of Florida, Gainesville, Florida (2003)

    Google Scholar 

  47. Norsell, M.: Radar cross section constraints in flight path optimization. In: 41st Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 6–9 (2003)

    Google Scholar 

  48. Novy, M.C., Jacques, D.R., Pachter, M.: Air vehicle optimal trajectories between two radars. In: Proceedings of the American Control Conference, Anchorage, Alaska, May 8–10 (2002)

    Google Scholar 

  49. Pachter, M., Hebert, F.: Optimal aircraft trajectories for radar exposure minimization. In: Proceedings of the American Control Conference, Arlington, Virginia, June 25–27 (2001)

    Google Scholar 

  50. Pfeiffer, B., Batta, R., Klamroth, K., Nagi, R.: Path planning for UAVs in the presence of threat zones using probabilistic modeling (in review, 2005)

    Google Scholar 

  51. Rehbock, V., Teo, K.L., Jennings, L.S., Lee, H.W.J.: A survey of the control parametrization and control parametrization enhancing methods for constrained optimal control problems. In: Progress in Optimization: Contributions from Australia, pp. 247–275. Kluwer Academic Press, Dordrecht (1999)

    Google Scholar 

  52. Rehbock, V., Caccetta, L., Hallam, C.L., O’Dowd, R.: Optimal submarine transit paths through sonar fields. Research Report, Department of Mathematics and Statistics, Curtin University of Technology, Western Australia (2000)

    Google Scholar 

  53. Rehbock, V., Caccetta, L.: Two defence applications involving discrete valued optimal control. Australian & New Zealand Industrial and Applied Mathematics Journal 44, 33–54 (2002)

    MathSciNet  Google Scholar 

  54. Ribeiro, C., Minoux, M.: Solving hard constrained shortest path problems by lagrangian relaxation and branch-and-bound algorithms. Methods of Operations Research 53, 303–316 (1986)

    MATH  MathSciNet  Google Scholar 

  55. Savelsbergh, M.W.P.: Preprocessing and probing techniques for mixed integer programming problems. ORSA Journal on Computing 6(4), 445–454 (1994)

    MATH  MathSciNet  Google Scholar 

  56. Schittkowski, K.: NLPQL: A Fortran subroutine for solving constrained nonlinear programming problems. Annals of Operations Research 5, 485–500 (1985/1986)

    MathSciNet  Google Scholar 

  57. Shapiro, J.F.: A survey of lagrangian techniques for discrete optimization. Annals of Discrete Mathematics 5, 113–138 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  58. Teo, K.L., Goh, C.J., Wong, K.H.: A unified computational approach to optimal control problems. Longman Scientific and Technical, London (1991)

    MATH  Google Scholar 

  59. Vian, J.L., Moore, J.R.: Trajectory optimization with risk minimization for military aircraft. AIAA Journal of Guidance, Control and Dynamics 12(3), 311–317 (1989)

    Article  Google Scholar 

  60. Yen, J.Y.: Finding the k-shortest, loopless paths in a network. Management Science 17, 711–715 (1971)

    Article  Google Scholar 

  61. Zabarankin, M., Uryasev, S., Pardalos, P.: Optimal risk path algorithms. In: Murphey, R., Pardalos, P. (eds.) Cooperative Control and Optimization, vol. 66, pp. 271–303. Kluwer Academic, Dordrecht (2002)

    Chapter  Google Scholar 

  62. Zabarankin, M., Uryasev, S., Murphey, R.: Aircraft routing under the risk of detection. Naval Research Logistics 53, 728–747 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Western Australian Centre of Excellence in Industrial Optimisation (WACEIO) Department of Mathematics and Statistics, Curtin University of Technology, Kent Street, Bentley, WA 6102, Australia

    Louis Caccetta, Ian van Loosen & Volker Rehbock

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  1. Louis Caccetta
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  2. Ian van Loosen
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  3. Volker Rehbock
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Editors and Affiliations

  1. Raytheon, Inc, St. Petersburg, FL, USA

    Michael J. Hirsch

  2. Munitions Directorate, Eglin AFB, USA

    Clayton W. Commander  & Robert Murphey  & 

  3. University of Florida, Gainesville, FL, USA

    Panos M. Pardalos

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Caccetta, L., van Loosen, I., Rehbock, V. (2009). Effective Algorithms for a Class of Discrete Valued Optimal Control Problems. In: Hirsch, M.J., Commander, C.W., Pardalos, P.M., Murphey, R. (eds) Optimization and Cooperative Control Strategies. Lecture Notes in Control and Information Sciences, vol 381. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88063-9_1

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  • DOI: https://doi.org/10.1007/978-3-540-88063-9_1

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