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Method for solving bang-bang and singular optimal control problems using adaptive Radau collocation

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Abstract

A method is developed for solving bang-bang and singular optimal control problems using adaptive Legendre–Gauss–Radau collocation. The method is divided into several parts. First, a structure detection method is developed that identifies switch times in the control and analyzes the corresponding switching function for segments where the solution is either bang-bang or singular. Second, after the structure has been detected, the domain is decomposed into multiple domains such that the multiple-domain formulation includes additional decision variables that represent the switch times in the optimal control. In domains classified as bang-bang, the control is set to either its upper or lower limit. In domains identified as singular, the objective function is augmented with a regularization term to avoid the singular arc. An iterative procedure is then developed for singular domains to obtain a control that lies in close proximity to the singular control. The method is demonstrated on four examples, three of which have either a bang-bang and/or singular optimal control while the fourth has a smooth and nonsingular optimal control. The results demonstrate that the method of this paper provides accurate solutions to problems whose solutions are either bang-bang or singular when compared against previously developed mesh refinement methods that are not tailored for solving nonsmooth and/or singular optimal control problems, and produces results that are equivalent to those obtained using previously developed mesh refinement methods for optimal control problems whose solutions are smooth.

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Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon request.

References

  1. Betts, J.T.: Practical Methods for Optimal Control and Estimation Using Nonlinear Programming. SIAM, New York (2010)

    Book  Google Scholar 

  2. Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 47(1), 99–131 (2002). https://doi.org/10.1137/S0036144504446096

    Article  MathSciNet  MATH  Google Scholar 

  3. Biegler, L.T., Zavala, V.M.: Large-scale nonlinear programming using IPOPT: an integrating framework for enterprise-wide optimization. Comput. Chem. Eng. 33(3), 575–582 (2008). https://doi.org/10.1016/j.compchemeng.2008.08.006

    Article  Google Scholar 

  4. Benson, D.A., Huntington, G.T., Thorvaldsen, T.P., Rao, A.V.: Direct trajectory optimization and costate estimation via an orthogonal collocation method. J. Guid. Control. Dyn. 29(6), 1435–1440 (2006). https://doi.org/10.2514/1.20478

    Article  Google Scholar 

  5. Garg, D., Patterson, M.A., Hager, W.W., Rao, A.V., Benson, D.A., Huntington, G.T.: A unified framework for the numerical solution of optimal control problems using pseudospectral methods. Automatica 46(11), 1843–1851 (2010). https://doi.org/10.1016/j.automatica.2010.06.048

    Article  MathSciNet  MATH  Google Scholar 

  6. Garg, D., Hager, W.W., Rao, A.V.: Pseudospectral methods for solving infinite-horizon optimal control problems. Automatica 47(4), 829–837 (2011). https://doi.org/10.1016/j.automatica.2011.01.085

    Article  MathSciNet  MATH  Google Scholar 

  7. Garg, D., Patterson, M.A., Darby, C.L., Francolin, C., Huntington, G.T., Hager, W.W., Rao, A.V.: Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems via a radau pseudospectral method. Comput. Optim. Appl. 49(2), 335–358 (2011). https://doi.org/10.1007/s10589-009-9291-0

    Article  MathSciNet  MATH  Google Scholar 

  8. Rao, A.V., Benson, D.A., Darby, C.L., Francolin, C., Patterson, M.A., Sanders, I., Huntington, G.T.: Algorithm 902: GPOPS. A MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method. ACM Trans. Math. Software 37(2) (2010). https://doi.org/10.1145/1731022.1731032

  9. Kameswaran, S., Biegler, L.T.: Convergence rates for direct transcription of optimal control problems using collocation at Radau points. Comput. Optim. Appl. 41(1), 81–126 (2008). https://doi.org/10.1007/s10589-007-9098-9

    Article  MathSciNet  MATH  Google Scholar 

  10. Patterson, M.A., Hager, W.W., Rao, A.V.: A \(ph\) mesh refinement method for optimal control. Opt. Control Appl. Methods 36(4), 398–421 (2015). https://doi.org/10.1002/oca.2114

    Article  MathSciNet  MATH  Google Scholar 

  11. Elnagar, G., Kazemi, M.A., Razzaghi, M.: The pseudospectral legendre method for discretizing optimal control problems. IEEE Trans. Autom. Control 40(10), 1793–1796 (1995). https://doi.org/10.1109/9.467672

    Article  MathSciNet  MATH  Google Scholar 

  12. Hager, W.W., Hou, H., Rao, A.V.: Convergence rate for a gauss collocation method applied to unconstrained optimal control. J. Optim. Theory Appl. 169(3), 801–824 (2016). https://doi.org/10.1007/s10957-016-0929-7

    Article  MathSciNet  MATH  Google Scholar 

  13. Hager, W.W., Hou, H., Rao, A.V.: Lebesgue constants arising in a class of collocation methods. IMA J. Numer. Anal. 37(4), 1884–1901 (2017). https://doi.org/10.1093/imanum/drw060

    Article  MathSciNet  MATH  Google Scholar 

  14. Hager, W.W., Liu, J., Mohapatra, S., Rao, A.V., Wang, X.-S.: Convergence rate for a Gauss collocation method applied to constrained optimal control. SIAM J. Control. Optim. 56, 1386–1411 (2018). https://doi.org/10.1137/16M1096761

    Article  MathSciNet  MATH  Google Scholar 

  15. Hager, W.W., Hou, H., Mohapatra, S., Rao, A.V., Wang, X.-S.: Convergence rate for a Radau hp-collocation method applied to constrained optimal control. Comput. Optim. Appl. 74, 274–314 (2019). https://doi.org/10.1007/s10589-019-00100-1

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, F., Hager, W.W., Rao, A.V.: Adaptive mesh refinement method for optimal control using nonsmoothness detection and mesh size reduction. J. Franklin Inst. 352(10), 4081–4106 (2015). https://doi.org/10.1016/j.jfranklin.2015.05.028

    Article  MathSciNet  MATH  Google Scholar 

  17. Gong, Q., Fahroo, F., Ross, I.M.: Spectral algorithm for pseudospectral methods in optimal control. J. Guid. Control. Dyn. 31(3), 460–471 (2008). https://doi.org/10.2514/1.32908

    Article  Google Scholar 

  18. Miller, A.T., Hager, W.W., Rao, A.V.: Mesh refinement method for solving optimal control problems with nonsmooth solutions using jump function approximations. Opt. Control Appl. Methods (2021). https://doi.org/10.1002/oca.2719

    Article  MathSciNet  MATH  Google Scholar 

  19. Schlegel, M., Marquardt, W.: Direct sequential dynamic optimization with automatic switching structure detection. IFAC Proc. Vol. 37(9), 419–424 (2004). https://doi.org/10.1016/s1474-6670(17)31845-1

    Article  Google Scholar 

  20. Schlegel, M., Marquardt, W.: Detection and exploitation of the control switching structure in the solution of dynamic optimization problems. J. Process Control 16(3), 275–290 (2006). https://doi.org/10.1016/j.jprocont.2005.06.008

    Article  Google Scholar 

  21. Wang, P., Yang, C., Yuan, Z.: The combination of adaptive pseudospectral method and structure detection procedure for solving dynamic optimization problems with discontinuous control profiles. Ind. Eng. Chem. Res. 53(17), 7066–7078 (2014). https://doi.org/10.1021/ie404148j

    Article  Google Scholar 

  22. Chen, W., Biegler, L.T.: Nested direct transcription optimization for singular optimal control problems. AIChE J. 62(10), 3611–3627 (2016). https://doi.org/10.1002/aic.15272

    Article  Google Scholar 

  23. Chen, W., Ren, Y., Zhang, G., Biegler, L.T.: A simultaneous approach for singular optimal control based on partial moving grid. AIChE J. 65(6), e16584 (2019). https://doi.org/10.1002/aic.16584

    Article  Google Scholar 

  24. Darby, C.L., Hager, W.W., Rao, A.V.: An hp-adaptive pseudospectral method for solving optimal control problems. Opt. Control Appl. Methods 32(4), 476–502 (2010). https://doi.org/10.1002/oca.957

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu, F., Hager, W.W., Rao, A.V.: Adaptive mesh refinement method for optimal control using decay rates of legendre polynomial coefficients. IEEE Trans. Control Syst. Technol. 26(4), 1475–1483 (2018). https://doi.org/10.1109/tcst.2017.2702122

    Article  Google Scholar 

  26. Agamawi, Y.M., Hager, W.W., Rao, A.V.: Mesh refinement method for solving bang-bang optimal control problems using direct collocation. AIAA Scitech 2020 Forum 0378 (2020). https://doi.org/10.2514/6.2017-1506

  27. Aghaee, M., Hager, W.W.: The switch point algorithm. SIAM J. Control. Optim. 59(4), 2570–2593 (2021)

    Article  MathSciNet  Google Scholar 

  28. Kaya, C., Noakes, J.: Computational method for time-optimal switching control. J. Optim. Theory Appl. 117(1), 69–92 (2003). https://doi.org/10.1023/a:1023600422807

    Article  MathSciNet  MATH  Google Scholar 

  29. Mehrpouya, M.A., Khaksar-e Oshagh, M.: An efficient numerical solution for time switching optimal control problems. Comput.l Methods Differ. Equ. 9(1) (2021). https://doi.org/10.22034/cmde.2020.33529.1542

  30. Aronna, M.S., Bonnans, J.F., Martinon, P.: A shooting algorithm for optimal control problems with singular Arcs. J. Optim. Theory Appl. 158(2), 419–459 (2013). https://doi.org/10.1007/s10957-012-0254-8

    Article  MathSciNet  MATH  Google Scholar 

  31. Mehra, R., Davis, R.: A generalized gradient method for optimal control problems with inequality constraints and singular arcs. IEEE Trans. Autom. Control 17(1), 69–79 (1972). https://doi.org/10.1109/tac.1972.1099881

    Article  MATH  Google Scholar 

  32. Jacobson, D., Gershwin, S., Lele, M.: Computation of optimal singular controls. IEEE Trans. Autom. Control 15(1), 67–73 (1970). https://doi.org/10.1109/tac.1970.1099360

    Article  MathSciNet  Google Scholar 

  33. Maurer, H.: Numerical solution of singular control problems using multiple shooting techniques. J. Optim. Theory Appl. 18(2), 235–257 (1976). https://doi.org/10.1007/bf00935706

    Article  MathSciNet  MATH  Google Scholar 

  34. Andrés-Martínez, O., Flores-Tlacuahuac, A., Kameswaran, S., Biegler, L.T.: An efficient direct/indirect transcription approach for singular optimal control. AIChE J. 65(3), 937–946 (2018). https://doi.org/10.1002/aic.16487

    Article  Google Scholar 

  35. Caponigro, M., Ghezzi, R., Piccoli, B., Trélat, E.: Regularization of chattering phenomena via bounded variation controls. IEEE Trans. Autom. Control 63(7), 2046–2060 (2018). https://doi.org/10.1109/TAC.2018.2810540

    Article  MathSciNet  MATH  Google Scholar 

  36. Mall, K., Grant, M.J., Taheri, E.: Uniform trigonometrization method for optimal control problems with control and state constraints. J. Spacecr. Rocket. 57(5), 995–1007 (2020). https://doi.org/10.2514/1.a34624

    Article  Google Scholar 

  37. Fabien, B.C.: Indirect solution of inequality constrained and singular optimal control problems via a simple continuation method. J. Dyn. Syst. Meas. Contr. 136(2), 021003 (2013). https://doi.org/10.1115/1.4025596

    Article  Google Scholar 

  38. Andrés-Martínez, O., Biegler, L.T., Flores-Tlacuahuac, A.: An indirect approach for singular optimal control problems. Comput. Chem. Eng. 139, (2020). https://doi.org/10.1016/j.compchemeng.2020.106923

  39. Maga, L., Reverberi, A., et al.: A pattern recognition approach to the solution of optimal singular control problems. Chem. Eng. J. 68(1), 35–40 (1997). https://doi.org/10.1016/S1385-8947(97)00068-5

    Article  Google Scholar 

  40. Athans, M., Falb, P.L.: Optimal Control: An Introduction to the Theory and Its Applications. Courier Corporation, North Chelmsford (2013)

    Google Scholar 

  41. Kirk, D.E.: Optimal Control Theory: An Introduction. Courier Corporation, North Chelmsford (2004)

    Google Scholar 

  42. Bryson, A.E., Ho, Y.: Applied Optimal Control: Optimization, Estimation, and Control. Hemisphere Publishing Corporation, London (1975)

    Google Scholar 

  43. Schättler, H., Ledzewicz, U.: Geometric Optimal Control: Theory, Methods and Examples, vol. 38. Springer, Berlin (2012)

    Book  Google Scholar 

  44. Kelley, H., Kopp, R.E., Moyer, H.G.: Topics in Optimization, edited by Leitman (1967)

  45. Kopp, R.E., Moyer, H.G.: Necessary conditions for singular extremals. AIAA J. 3(8), 1439–1444 (1965). https://doi.org/10.2514/3.3165

    Article  MATH  Google Scholar 

  46. Archibald, R., Gelb, A., Yoon, J.: Polynomial fitting for edge detection in irregularly sampled signals and images. SIAM J. Numer. Anal. 43(1), 259–279 (2005). https://doi.org/10.1137/s0036142903435259

    Article  MathSciNet  MATH  Google Scholar 

  47. Fritsch, F.N., Carlson, R.E.: Monotone piecewise cubic interpolation. SIAM J. Numer. Anal. 17(2), 238–246 (1980). https://doi.org/10.1137/0717021

    Article  MathSciNet  MATH  Google Scholar 

  48. Patterson, M.A., Rao, A.V.: GPOPS-II. ACM Trans. Math. Software 41(1), 1–37 (2014). https://doi.org/10.1145/2558904

    Article  Google Scholar 

  49. Weinstein, M.J., Rao, A.V.: Algorithm 984: ADiGator, a toolbox for the algorithmic differentiation of mathematical functions in MATLAB using source transformation via operator overloading. ACM Trans. Math. Software 44(2), 1–25 (2017). https://doi.org/10.1145/3104990

    Article  MathSciNet  MATH  Google Scholar 

  50. Dolan, E.D., More, J.J., Munson, T.S.: Benchmarking optimization software with COPS 3.0. Tech. rep., Argonne National Laboratory, Argonne, Illinois (2004). https://doi.org/10.2172/834714

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Acknowledgements

The authors gratefully acknowledge support for this research from the U.S. National Science Foundation under grants DMS-1819002 and CMMI-2031213, the U.S. Office of Naval Research under grant N00014-19-1-2543, and from Lockheed-Martin Corporation under contract 4104177872.

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  1. Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL, 32611-6250, USA

    Elisha R. Pager & Anil V. Rao

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Correspondence to Anil V. Rao.

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Pager, E.R., Rao, A.V. Method for solving bang-bang and singular optimal control problems using adaptive Radau collocation. Comput Optim Appl 81, 857–887 (2022). https://doi.org/10.1007/s10589-022-00350-6

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