Journal of Global Optimization

, Volume 61, Issue 4, pp 721–743 | Cite as

Efficient upper and lower bounds for global mixed-integer optimal control

  • Sebastian Sager
  • Mathieu Claeys
  • Frédéric Messine
Article

Abstract

We present a control problem for an electrical vehicle. Its motor can be operated in two discrete modes, leading either to acceleration and energy consumption, or to a recharging of the battery. Mathematically, this leads to a mixed-integer optimal control problem (MIOCP) with a discrete feasible set for the controls taking into account the electrical and mechanical dynamic equations. The combination of nonlinear dynamics and discrete decisions poses a challenge to established optimization and control methods, especially if global optimality is an issue. Probably for the first time, we present a complete analysis of the optimal solution of such a MIOCP: solution of the integer-relaxed problem both with a direct and an indirect approach, determination of integer controls by means of the sum up rounding strategy, and calculation of global lower bounds by means of the method of moments. As we decrease the control discretization grid and increase the relaxation order, the obtained series of upper and lower bounds converge for the electrical car problem, proving the asymptotic global optimality of the calculated chattering behavior. We stress that these bounds hold for the optimal control problem in function space, and not on an a priori given (typically coarse) control discretization grid, as in other approaches from the literature. This approach is generic and is an alternative to global optimal control based on probabilistic or branch-and-bound based techniques. The main advantage is a drastic reduction of computational time. The disadvantage is that only local solutions and certified lower bounds are provided with no possibility to reduce these gaps. For the instances of the electrical car problem, though, these gaps are very small. The main contribution of the paper is a survey and new combination of state-of-the-art methods for global mixed-integer optimal control and the in-depth analysis of an important, prototypical control problem. Despite the comparatively low dimension of the problem, the optimal solution structure of the relaxed problem exhibits a series of bang-bang, path-constrained, and sensitivity-seeking arcs.

Keywords

Mixed-integer optimal control Global optimal control  Maximum principle Method of moments Energy consumption Electrical car 

References

  1. 1.
    Ascher, U., Mattheij, R., Russell, R.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM, Philadelphia (1995)CrossRefMATHGoogle Scholar
  2. 2.
    Bemporad, A., Borodani, P., Mannelli, M.: Hybrid control of an automotive robotized gearbox for reduction of consumptions and emissions. In: Hybrid Systems: Computation and Control, Springer Lecture Notes in Computer Science, vol. 2623/2003, pp. 81–96. Springer, Heidelberg (2003). doi:10.1007/3-540-36580-X_9
  3. 3.
    Betts, J.: Practical Methods for Optimal Control Using Nonlinear Programming. SIAM, Philadelphia (2001)MATHGoogle Scholar
  4. 4.
    Biegler, L.: Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation. Comput. Chem. Eng. 8, 243–248 (1984)CrossRefGoogle Scholar
  5. 5.
    Biegler, L.: Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes. Series on Optimization. SIAM, Philadelphia (2010)CrossRefGoogle Scholar
  6. 6.
    Bock, H., Longman, R.: Optimal control of velocity profiles for minimization of energy consumption in the New York subway system. In: Proceedings of the Second IFAC Workshop on Control Applications of Nonlinear Programming and Optimization, pp. 34–43. International Federation of Automatic, Control (1980)Google Scholar
  7. 7.
    Bock, H., Longman, R.: Computation of optimal controls on disjoint control sets for minimum energy subway operation. In: Proceedings of the American Astronomical Society. Symposium on Engineering Science and Mechanics. Taiwan (1982)Google Scholar
  8. 8.
    Bock, H., Plitt, K.: A multiple shooting algorithm for direct solution of optimal control problems. In: Proceedings of the 9th IFAC World Congress, pp. 242–247. Pergamon Press, Budapest (1984)Google Scholar
  9. 9.
    Borrelli, F., Bemporad, A., Fodor, M., Hrovat, D.: An MPC/hybrid system approach to traction control. IEEE Trans. Control Syst. Technol. 14(3), 541–552 (2006). doi:10.1109/TCST.2005.860527 CrossRefGoogle Scholar
  10. 10.
    Borrelli, F., Falcone, P., Keviczky, T., Asgari, J., Hrovat, D.: MPC-based approach to active steering for autonomous vehicle systems. Int. J. Veh. Auton. Syst. 3(2–4), 265–291 (2005)CrossRefGoogle Scholar
  11. 11.
    Bulirsch, R.: Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung. Tech. rep, Carl-Cranz-Gesellschaft, Oberpfaffenhofen (1971)Google Scholar
  12. 12.
    Chachuat, B., Singer, A., Barton, P.: Global methods for dynamic optimization and mixed-integer dynamic optimization. Ind. Eng. Chem. Res. 45(25), 8373–8392 (2006)CrossRefGoogle Scholar
  13. 13.
    De Jager, B., Van Keulen, T., Kessels, J.: Optimal Control of Hybrid Vehicles. Advances in Industrial Control. Springer, London (2013)Google Scholar
  14. 14.
    Gamkrelidze, R.: Principles of Optimal Control Theory. Plenum Press, Berlin (1978)CrossRefMATHGoogle Scholar
  15. 15.
    Gerdts, M.: Solving mixed-integer optimal control problems by branch&bound: a case study from automobile test-driving with gear shift. Optim. Control Appl. Methods 26, 1–18 (2005)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Gerdts, M.: A variable time transformation method for mixed-integer optimal control problems. Optim. Control Appl. Methods 27(3), 169–182 (2006)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Gerdts, M., Sager, S.: Mixed-integer DAE Optimal control problems: necessary conditions and bounds. In: Biegler, L., Campbell, S., Mehrmann, V. (eds.) Control and Optimization with Differential-Algebraic Constraints, pp. 189–212. SIAM, Philadelphia (2012)CrossRefGoogle Scholar
  18. 18.
    Hellström, E., Ivarsson, M., Aslund, J., Nielsen, L.: Look-ahead control for heavy trucks to minimize trip time and fuel consumption. Control Eng. Pract. 17, 245–254 (2009)CrossRefGoogle Scholar
  19. 19.
    Henrion, D., Daafouz, J., Claeys, M.: Optimal switching control design for polynomial systems: an LMI approach. In: 2013 Conference on Decision and Control. Florence, Italy (2013). Accepted for publicationGoogle Scholar
  20. 20.
    Henrion, D., Lasserre, J.B., Löfberg, J.: Gloptipoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24(4–5), 761–779 (2009)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Jansson, C.: VSDP: a Matlab software package for verified semidefinite programming. Nonlinear Theory and its Applications pp. 327–330 (2006)Google Scholar
  22. 22.
    Jung, M., Reinelt, G., Sager, S.: The Lagrangian Relaxation for the Combinatorial Integral Approximation Problem. Optimization Methods and Software (2014). (Accepted)Google Scholar
  23. 23.
    Kim, T.S., Manzie, C., Sharma, R.: Model predictive control of velocity and torque split in a parallel hybrid vehicle. In: Proceedings of the 2009 IEEE International Conference on Systems. Man and Cybernetics, SMC’09, pp. 2014–2019. IEEE Press, Piscataway, NJ (2009)Google Scholar
  24. 24.
    Kirches, C., Bock, H., Schlöder, J., Sager, S.: Mixed-integer NMPC for predictive cruise control of heavy-duty trucks. In: European Control Conference, pp. 4118–4123. Zurich, Switzerland (2013)Google Scholar
  25. 25.
    Kirches, C., Sager, S., Bock, H., Schlöder, J.: Time-optimal control of automobile test drives with gear shifts. Optim. Control Appl. Methods 31(2), 137–153 (2010)CrossRefMATHGoogle Scholar
  26. 26.
    Lasserre, J.B.: Positive Polynomials and Their Applications. Imperial College Press, London (2009)Google Scholar
  27. 27.
    Lasserre, J.B., Henrion, D., Prieur, C., Trélat, E.: Nonlinear optimal control via occupation measures and LMI-relaxations. SIAM J. Control Optim. 47(4), 1643–1666 (2008)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Lee, H., Teo, K., Jennings, L., Rehbock, V.: Control parametrization enhancing technique for optimal discrete-valued control problems. Automatica 35(8), 1401–1407 (1999)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Leineweber, D.: Efficient reduced SQP methods for the optimization of chemical processes described by large sparse DAE models, Fortschritt-Berichte VDI Reihe 3, Verfahrenstechnik, vol. 613. VDI Verlag, Düsseldorf (1999)Google Scholar
  30. 30.
    Leineweber, D., Bauer, I., Bock, H., Schlöder, J.: An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part I: theoretical aspects. Comput. Chem. Eng. 27, 157–166 (2003)CrossRefGoogle Scholar
  31. 31.
    McShane, E.: The calculus of variations from the beginning to optimal control theory. SIAM J. Control Optim. 27(5), 916–939 (1989)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Merakeb, K., Messine, F., Aïdène, M.: A branch and bound algorithm for minimizing the energy consumption of an electrical vehicle. 4OR (2013, available online)Google Scholar
  33. 33.
    Osborne, M.: On shooting methods for boundary value problems. J. Math. Anal. Appl. 27, 417–433 (1969)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Royden, H., Fitzpatrick, P.: Real Analysis. Prentice Hall, Upper Saddle River (2010)MATHGoogle Scholar
  35. 35.
    Sager, S.: Numerical methods for mixed-integer optimal control problems. Der andere Verlag, Tönning, Lübeck, Marburg (2005). ISBN 3-89959-416-9Google Scholar
  36. 36.
    Sager, S.: Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control. J. Process Control 19(8), 1238–1247 (2009)CrossRefGoogle Scholar
  37. 37.
    Sager, S., Bock, H., Diehl, M.: The integer approximation error in mixed-integer optimal control. Math. Program. A 133(1–2), 1–23 (2012)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Sager, S., Jung, M., Kirches, C.: Combinatorial integral approximation. Math. Methods Oper. Res. 73(3), 363–380 (2011). doi:10.1007/s00186-011-0355-4 CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Sager, S., Reinelt, G., Bock, H.: Direct methods with maximal lower bound for mixed-integer optimal control problems. Math. Program. 118(1), 109–149 (2009)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Sargent, R., Sullivan, G.: The development of an efficient optimal control package. In: Stoer, J. (ed.) Proceedings of the 8th IFIP Conference on Optimization Techniques (1977), Part 2. Springer, Heidelberg (1978)Google Scholar
  41. 41.
    Sciarretta, A., Guzzella, L.: Control of hybrid electric vehicles. IEEE Control Syst. 27(2), 60–70 (2007)CrossRefGoogle Scholar
  42. 42.
    Scott, J.K., Chachuat, B., Barton, P.I.: Nonlinear convex and concave relaxations for the solutions of parametric ODEs. Optim. Control Appl. Methods 34, 145–163 (2013). doi:10.1002/oca.2014 CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Sturm, J.F.: Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999)CrossRefMathSciNetGoogle Scholar
  44. 44.
    Tauchnitz, N.: Das Pontrjaginsche Maximumprinzip für eine Klasse hybrider Steuerungsprobleme mit Zustandsbeschränkungen und seine Anwendung. Ph.D. thesis, BTU Cottbus (2010)Google Scholar
  45. 45.
    Terwen, S., Back, M., Krebs, V.: Predictive powertrain control for heavy duty trucks. In: Proceedings of IFAC Symposium in Advances in Automotive Control, pp. 451–457. Salerno, Italy (2004)Google Scholar
  46. 46.
    van Keulen, T., van Mullem, D., de Jager, B., Kessels, J.T.B.A., Steinbuch, M.: Design, implementation, and experimental validation of optimal power split control for hybrid electric trucks. Control Eng. Pract. 20(5), 547–558 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Sebastian Sager
    • 1
  • Mathieu Claeys
    • 2
    • 3
  • Frédéric Messine
    • 4
  1. 1.Faculty of MathematicsOtto-von-Guericke-Universität MagdeburgMagdeburgGermany
  2. 2.Department of EngineeringUniversity of CambridgeCambridgeUK
  3. 3.LAASUniversité de ToulouseToulouseFrance
  4. 4.LAPLACE-ENSEEIHTUniversité de ToulouseToulouseFrance

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