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Hybrid Systems, Optimal Control and Hybrid Vehicles pp 233–273Cite as

Direct Methods for Optimal Control

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Part of the book series: Advances in Industrial Control ((AIC))

Abstract

In this chapter another type of methods to solve optimal control problems is discussed. Direct methods transform the original problem via a discretization of the control and the state functions on a time grid to a nonlinear constrained optimization problem. This procedure is known as direct transcription of an optimal control problem and refers to the method of approximating the infinite-dimensional problem by a finite-dimensional one and to solve it with nonlinear programming algorithms.

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References

  1. Alamir M, Attia S (2004) On solving optimal control problems for switched hybrid nonlinear systems by strong variations algorithms. In: 6th IFAC symposium on nonlinear control systems (NOLCOS), Stuttgart, Germany, pp 558–563

    Google Scholar 

  2. Axelsson H, Wardi Y, Egerstedt M (2005) Transition-time optimization for switched systems. In: Proceedings of IFAC world congress

    Google Scholar 

  3. Axelsson H, Wardi Y, Egerstedt M, Verriest E (2008) Gradient descent approach to optimal mode scheduling in hybrid dynamical systems. J Optim Theory Appl 136. doi:10.1007/s10957-007-9305-y

  4. Balinski ML, Wolfe P, Bertsekas DP, Camerini P, Cullum J (1975) Nondifferentiable optimization. North-Holland, Amsterdam

    Google Scholar 

  5. Bengea S, DeCarlo R (2003) Optimal and suboptimal control of switching systems. In: Proceedings of the 42nd IEEE conference on decision and control, pp 5295–5300

    Google Scholar 

  6. Bengea S, Uthaichana K, Žefran M, DeCarlo RA (2011) The control system handbook optimal control of switching systems via embedding into continuous optimal control problems, 2nd edn. CRC Press

    Google Scholar 

  7. Bengea SC, DeCarlo RA (2005) Optimal control of switching systems. Automatica 41(1):11–27

    Google Scholar 

  8. Bertsekas DP (1975) Nondifferentiable optimization via approximation. In: Nondifferentiable optimization. Springer, pp 1–25

    Google Scholar 

  9. Betts J, Huffman W (1999) Exploiting sparsity in the direct transcription method for optimal control. Comput Optim Appl 14(2):179–201

    Article  MathSciNet  MATH  Google Scholar 

  10. Betts JT (1998) Survey of numerical methods for trajectory optimization. J Guid Control Dyn 21(2):193–207

    Article  MATH  Google Scholar 

  11. Betts JT (2010) Practical methods for optimal control and estimation using nonlinear programming. Society for industrial and applied mathematics, 2nd edn. doi:10.1137/1.9780898718577

  12. Bock H, Plitt K (1984) A multiple shooting algorithm for direct solution of optimal control problems. In: Proceedings of the 9th world congress of the international federation of automatic control, vol 9

    Google Scholar 

  13. Boehme TJ, Frank B, Schultalbers M, Schori M, Lampe B (2013) Solutions of hybrid energy-optimal control for model-based calibrations of HEV powertrains. In: SAE world congress, technical Paper 2013-01-1747. doi:10.4271/2013-01-1747

  14. Bonami P, Biegler LT, Conn AR, Cornuéjols G, Grossmann IE, Laird CD, Lee J, Lodi A, Margot F, Sawaya N et al (2008) An algorithmic framework for convex mixed integer nonlinear programs. Discret Optim 5(2):186–204

    Article  MathSciNet  MATH  Google Scholar 

  15. Bonnans JF, Gilbert JC, Lemaréchal C, Sagastizábal CA (2006) Numerical optimization: theoretical and practical aspects. Springer Science & Business Media

    Google Scholar 

  16. Bulirsch R, Nerz E, Pesch H, von Stryk O (1993) Combining direct and indirect methods in optimal control: Range maximization of a hang glider. In: Bulirsch R, Miele A, Stoer J, Well K (eds) Optimal control, ISNM international series of numerical mathematics, vol 111. Birkhäuser, Basel, pp 273–288. doi:10.1007/978-3-0348-7539-4_20

  17. Burer S, Letchford AN (2012) Non-convex mixed-integer nonlinear programming: a survey. Surv Oper Res Manag Sci 17(2):97–106

    MathSciNet  Google Scholar 

  18. Büskens C (1998) Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustandsbeschränkungen. PhD thesis, Universität Münster

    Google Scholar 

  19. Butikofer S (2008) Generalized Newton-type methods for nonsmooth equations in optimization and complementarity problems. PhD thesis, ETH Zürich

    Google Scholar 

  20. Clarke FH (1978) Nonsmooth analysis and optimization. In: Proceedings of the international congress of mathematicians (Helsinki), pp 847–853

    Google Scholar 

  21. Cook W (2012) Markowitz and Manne + Eastman + Land and Doig = branch and bound. Optim Storit 227–238

    Google Scholar 

  22. Duran MA, Grossmann IE (1986) An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math Program 36(3):307–339

    Article  MathSciNet  MATH  Google Scholar 

  23. Egerstedt M, Wardi Y, Axelsson H (2006) Transition-time optimization for switched-mode dynamical systems. IEEE Trans Autom Control 51. doi:10.1109/TAC.2005.861711

  24. Enright P, Conway B (1992) Discrete approximations to optimal trajectories using direct transcription and nonlinear programming. J Guid Control Dyn 15(4):994–1002

    Article  MATH  Google Scholar 

  25. Fletcher R, Leyffer S (1994) Solving mixed integer nonlinear programs by outer approximation. Math Program 66(1–3):327–349

    Article  MathSciNet  MATH  Google Scholar 

  26. Geoffrion AM (1972) Generalized Benders decomposition. J Optim Theory Appl 10(4):237–260

    Article  MathSciNet  MATH  Google Scholar 

  27. Gerdts S (2012) Mixed-integer DAE optimal control problems: necessary conditions and bounds. In: Campbell S, Mehrmann V, Biegler L (eds) Control and optimization with differential-algebraic constraints. SIAM, pp 189–212

    Google Scholar 

  28. Gerdts M (2006) A variable time transformation method for mixed-integer optimal control problems. Optim Control Appl Methods 27(3):169–182

    Article  MathSciNet  Google Scholar 

  29. Gerdts M (2012) Optimal control of ordinary differential equations and differential-algebraic equations. de Gruyter, Berlin

    MATH  Google Scholar 

  30. Grossmann IE (2002) Review of nonlinear mixed-integer and disjunctive programming techniques. Optim Eng 3(3):227–252

    Article  MathSciNet  MATH  Google Scholar 

  31. Grossmann IE, Kravanja Z (1995) Mixed-integer nonlinear programming techniques for process systems engineering. Comput Chem Eng 19:189–204

    Article  Google Scholar 

  32. Hedlund S, Rantzer A (2002) Convex dynamic programming for hybrid systems. IEEE Trans Autom Control 47:1536–1540

    Article  MathSciNet  Google Scholar 

  33. Hwang I, Li J, Du D (2008) A numerical algorithm for optimal control of a class of hybrid systems: differential transformation based approach. Int J Control 81:277–293

    Article  MathSciNet  MATH  Google Scholar 

  34. Grossmann IE, ZK, (1993) Mixed-integer nonlinear programming: a survey of algorithms and applications. IMA Vol Math Appl 93:73–100

    Google Scholar 

  35. Kamgarpour M, Tomlin C (2012) On optimal control of non-autonomous switched systems with a fixed mode sequence. Automatica 48(6):1177–1181

    Article  MathSciNet  MATH  Google Scholar 

  36. Kirches C (2011) Fast numerical methods for mixed-integer nonlinear model-predictive control. Springer

    Google Scholar 

  37. Land AH, Doig AG (1960) An automatic method of solving discrete programming problems. Econ: J Econ Soc 497–520

    Google Scholar 

  38. Lemaréchal C (1989) Nondifferentiable optimization. In: Handbooks in operations research and management science. Elsevier Science Publishers B.V., North-Holland

    Google Scholar 

  39. Lewis A, Overton M (2012) Nonsmooth optimization via Quasi-Newton methods. Math Program 141:135–163

    Article  MathSciNet  MATH  Google Scholar 

  40. Leyffer S (1993) Deterministic methods for mixed integer nonlinear programming. PhD thesis, University of Dundee

    Google Scholar 

  41. Leyffer S (2001) Integrating SQP and branch-and-bound for mixed integer nonlinear programming. Comput Optim Appl 18(3):295–309

    Article  MathSciNet  MATH  Google Scholar 

  42. Leyffer S (2006) Complementarity constraints as nonlinear equations: theory and numerical experience. In: Optimization with multivalued mappings. Springer, pp 169–208

    Google Scholar 

  43. Linderoth JT, Savelsbergh MW (1999) A computational study of search strategies for mixed integer programming. INFORMS J Comput 11(2):173–187

    Article  MathSciNet  MATH  Google Scholar 

  44. Little JD, Murty KG, Sweeney DW, Karel C (1963) An algorithm for the traveling salesman problem. Oper Res 11(6):972–989

    Article  MATH  Google Scholar 

  45. Luks̆an L, Vlc̆ek J (1999) Globally convergent variable metric method for convex and nonsmooth unconstrained minimization. J Optim Theory Appl 102:593–613

    Google Scholar 

  46. Maurer H, Büskens C, Kim JH, Kaya C (2005) Optimization methods for the verification of second order sufficient conditions for bang-bang controls. Optim Control Appl Methods 26(3):129–156

    Article  MathSciNet  Google Scholar 

  47. Mordukhovich B (1978) On difference approximations of optimal control systems. J Appl Math Mech 42(3):452–461

    Article  MathSciNet  MATH  Google Scholar 

  48. Mordukhovich B (2006) Variational analysis and generalized differentiation II, applications. Grundlehren der mathematischen Wissenschaften. Springer, Berlin

    Book  Google Scholar 

  49. Nüesch T, Elbert P, Flankl M, Onder C, Guzzella L (2014) Convex optimization for the energy management of hybrid electric vehicles considering engine start and gearshift costs. Energies 7:834–856

    Article  Google Scholar 

  50. Quesada I, Grossmann IE (1992) An LP/NLP based branched and bound algorithm for convex MINLP optimization problems. Comput Chem Eng 16:937–947

    Article  Google Scholar 

  51. Rall L (1981) Automatic differentiation: techniques and applications, vol 120. Lecture notes in computer science. Springer, Berlin

    Google Scholar 

  52. Riedinger P, Daafouz J, Iung C (2003) Suboptimal switched controls in context of singular arcs. In: Proceedings of the 42nd IEEE conference on decision and control, vol 6. IEEE, pp 6254–6259

    Google Scholar 

  53. SA Attia JR V Azhmyakov (2007) State jump optimization for a class of hybrid autonomous systems. In: Proceedings of the 16th IEEE international conference on control applications, pp 1408–1413

    Google Scholar 

  54. Sager S (2005) Numerical methods for mixed-integer optimal control problems. PhD thesis, Universität Heidelberg

    Google Scholar 

  55. Sager S (2009) Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control. J Process Control 19(8):1238–1247

    Article  Google Scholar 

  56. Sager S, Bock HG, Diehl M (2007) Solving mixed–integer control problems by sum up rounding with guaranteed integer gap. Preprint, IWR, University of Heidelberg. http://www.ub.uni-heidelberg.de/archiv/8384

  57. Sager S, Jung M, Kirches C (2011) Combinatorial integral approximation. Math Methods Oper Res 73(3):363–380

    Article  MathSciNet  MATH  Google Scholar 

  58. Sager S, Bock H, Diehl M (2012) The integer approximation error in mixed-integer optimal control. Math Program 133(1–2):1–23

    Article  MathSciNet  MATH  Google Scholar 

  59. Schäfer R (2014) Gemischt-ganzzahlige Optimalsteuerung, Sensitivitätsanalyse und Echtzeitoptimierung von Parallel-Hybridfahrzeugen. Master’s thesis, Universität Bremen

    Google Scholar 

  60. Schori M (2015) Solution of optimal control problems for switched systems. Algorithms and applications for hybrid vehicles. PhD thesis, Universität Rostock

    Google Scholar 

  61. Schori M, Boehme TJ, Frank B, Schultalbers M (2014) Control optimization of discontinuous hybrid systems using embedding. Discret Event Syst 12:326–331

    Google Scholar 

  62. Schori M, Boehme TJ, Jeinsch T, Lampe B (2015) Switching time optimization for discontinuous switched systems. Proceedings of the 2015 European control conference (ECC), July 15–17. Linz. IEEE, pp 1742–1747

    Google Scholar 

  63. Shaikh MS (2004) Optimal control of hybrid systems: theory and algorithms. PhD thesis, Department of Electrical and Computer Engineering, McGill University, Montreal

    Google Scholar 

  64. Shor NZ (1985) Minimization methods for non-differentiable functions. Springer Ser Comput Math 3. doi:10.1007/978-3-642-82118-9

  65. von Stryk O (1995) Numerische Lösung optimaler Steuerungsprobleme: Diskretisierung, Parameteroptimierung und Berechnung der adjungierten Variablen. Fortschritt-Berichte VDI-Verlag 8

    Google Scholar 

  66. von Stryk O, Bulirsch R (1992) Direct and indirect methods for trajectory optimization. Ann Oper Res 37:357–373

    Article  MathSciNet  MATH  Google Scholar 

  67. von Stryk O, Glocker M (2000) Decomposition of mixed-integer optimal control problems using branch and bound and sparse direct collocation. In: The 4th international conference on automation of mixed processes: hybrid dynamic systems, pp 99–104

    Google Scholar 

  68. Tavernini L (1987) Differential automata and their discrete simulators. Nonlinear Anal Theory Methods Appl 11:583–665

    Article  MathSciNet  MATH  Google Scholar 

  69. Tavernini L (2009) Generic asymptotic error estimates for the numerical simulation of hybrid systems. Nonlinear Anal: Hybrid Syst 3(2):108–123

    MathSciNet  MATH  Google Scholar 

  70. Till J, Engell S, Panek S, Stursberg O (2004) Applied hybrid system optimization: an empirical investigation of complexity. Control Eng Pract 12(10):1291–1303

    Article  Google Scholar 

  71. Tsang T, Himmelblau D, Edgar T (1975) Optimal control via collocation and non-linear programming. Int J Control 21(5):763–768

    Article  MATH  Google Scholar 

  72. Turau V (2009) Algorithmische Graphentheorie. Oldenbourg Verlag

    Google Scholar 

  73. Xu X, Antsaklis PJ (2003) Optimal control of hybrid autonomous systems with state jumps. In: Proceedings of the American control conference. IEEE, pp 5191–5196

    Google Scholar 

  74. Xu X, Antsaklis PJ (2000) A dynamic programming approach for optimal control of switched systems. In: Proceedings of the 39th IEEE conference on decision and control, vol 2. IEEE, pp 1822–1827

    Google Scholar 

  75. Xu X, Antsaklis PJ (2000) Optimal control of switched systems: new results and open problems. In: Proceedings of the American control conference, vol 4. IEEE, pp 2683–2687

    Google Scholar 

  76. Xu X, Antsaklis PJ (2001) Switched systems optimal control formulation and a two stage optimization methodology. In: Proceedings of the 9th Mediterranean conference on control and automation

    Google Scholar 

  77. Xu X, Antsaklis PJ (2004) Optimal control of switched systems based on parameterization of the switching instants. IEEE Trans Autom Control 49. doi:10.1109/TAC.2003.821417

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    Thomas J. Böhme & Benjamin Frank

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Correspondence to Thomas J. Böhme .

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Böhme, T.J., Frank, B. (2017). Direct Methods for Optimal Control. In: Hybrid Systems, Optimal Control and Hybrid Vehicles. Advances in Industrial Control. Springer, Cham. https://doi.org/10.1007/978-3-319-51317-1_8

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  • DOI: https://doi.org/10.1007/978-3-319-51317-1_8

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