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Multi-Objective Optimal Control: A Direct Approach

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Satellite Dynamics and Space Missions

Part of the book series: Springer INdAM Series ((SINDAMS,volume 34))

Abstract

The chapter introduces an approach to solve optimal control problems with multiple conflicting objectives. The approach proposed in this chapter generates sets of Pareto optimal control laws that satisfy a set of boundary conditions and path constraints. The chapter starts by introducing basic concepts of multi-objective optimisation and optimal control theory and then presents a general formulation of multi-objective optimal control problems in scalar form using the Pascoletti-Serafini scalarisation method. From this scalar form the chapter derives the first order necessary conditions for local optimality and develops a direct transcription method by Finite Elements in Time (DFET) that turns the infinite dimensional multi-objective optimal control problem into a finite dimensional multi-objective nonlinear programming problem (MONLP). The transcription method is proven to be locally convergent under some assumptions on the nature of the optimal control problem. A memetic agent-based optimisation approach is then proposed to solve the MONLP problem and return a partial reconstruction of the globally optimal Pareto set. An illustrative example concludes the chapter.

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References

  1. Coverstone-Carroll, V., Hartmann, J.W., Mason, W.J.: Optimal multi-objective low-thrust spacecraft trajectories. Comput. Methods Appl. Mech. Eng. 186(2), 387–402 (2000)

    Article  Google Scholar 

  2. Yalçın Kaya, C., Maurer, H.: A numerical method for nonconvex multi-objective optimal control problems. Comput. Optim. Appl. 57(3), 685–702 (2014)

    Article  MathSciNet  Google Scholar 

  3. Englander, J.A., Vavrina, M.A., Ghosh, A.R.: Multi-objective hybrid optimal control for multiple-flyby low-thrust mission design. In: 25th AAS/AIAA Space Flight Mechanics Meeting, 11–15 January (2015)

    Google Scholar 

  4. Ober-Blöbaum, S., Ringkamp, M., zum Felde, G.: Solving multiobjective optimal control problems in space mission design using discrete mechanics and reference point techniques. In: IEEE 51st Annual Conference on Decision and Control (CDC), 2012, pp. 5711–5716. IEEE, Piscataway (2012)

    Google Scholar 

  5. Chankong, Y.Y., Haimes, V.: Multiobjective Decision Making. Dover Publications, Inc., Mineola (2008)

    MATH  Google Scholar 

  6. Hillermeier, C.: Nonlinear Multiobjective Optimization. International Series of Numerical Mathematics. Birkhäuser, Basel (2001). https://doi.org/10.1007/978-3-0348-8280-4

    Book  Google Scholar 

  7. Pascoletti, A., Serafini, P.: Scalarizing vector optimization problems. J. Optim. Theory Appl. 42, 499–524 (1984)

    Article  MathSciNet  Google Scholar 

  8. Eichfelder, G.: Adaptive Scalarization Methods in Multiobjective Optimization. Springer, Berlin (2008). https://doi.org/10.1007/978-3-540-79159-1

    Book  Google Scholar 

  9. Zuiani, F., Vasile, M.: Multi agent collaborative search based on tchebycheff decomposition. Comput. Optim. Appl. 56(1), 189–208 (2013)

    Article  MathSciNet  Google Scholar 

  10. Zhang, Q., Li, H.: Moea/d: a multi-objective evolutionary algorithm based on decomposition. IEEE Trans. Evol. Comput. 11(6), 712–731 (2007)

    Article  Google Scholar 

  11. Shapiro, S.: Lagrange and mayer problems in optimal control. Automatica 3(3), 219–230 (1966)

    Article  Google Scholar 

  12. Vasile, M.: Finite elements in time: a direct transcription method for optimal control problems. In: AIAA/AAS Astrodynamics Specialist Conference, Guidance, Navigation, and Control and Co-located Conferences, Toronto, 2–5 August 2010

    Google Scholar 

  13. Vasile, M., Finzi, A.E.: Direct lunar descent optimisation by finite elements in time approach. Int. J. Mech. Control 1(1) (2000)

    Google Scholar 

  14. Hodges, D.H., Bless, R.R.: Weak hamiltonian finite element method for optimal control problems. J. Guid. Control Dyn. 14(1), 148–156 (1991)

    Article  Google Scholar 

  15. Bottasso, C.L., Ragazzi, A.: Finite element and runge-kutta methods for boundary-value and optimal control problems. J. Guid. Control Dyn. 23(4), 749–751 (2000)

    Article  Google Scholar 

  16. Vasile, M., Bernelli-Zazzera, F.: Optimizing low-thrust and gravity assist maneuvers to design interplanetary trajectories. J. Astronaut. Sci. 51(1), 13–35 (2003)

    MathSciNet  MATH  Google Scholar 

  17. Vasile, M., Bernelli-Zazzera, F.: Targeting a heliocentric orbit combining low-thrust propulsion and gravity assist manoeuvres. Oper. Res. Space Air 79, 203–229 (2003)

    Article  Google Scholar 

  18. Ricciardi, L.A., Vasile, M.: Direct transcription of optimal control problems with finite elements on bernstein basis. AIAA J. Guid. Control Dyn. 42(2), 229–243 (2019)

    Article  Google Scholar 

  19. Zuiani, F., Kawakatsu, Y., Vasile, M.: Multi-objective optimisation of many-revolution, low-thrust orbit raising for destiny mission. Adv. Astronaut. Sci. 148, 783–802. In: Proceedings of the 23rd AAS/AIAA Space Flight Mechanics Conference, January (2013)

    Google Scholar 

  20. Ricciardi, L.A., Vasile, M.: Improved archiving and search strategies for multi agent collaborative search. In: Advances in Evolutionary and Deterministic Methods for Design, Optimization and Control in Engineering and Sciences, pp. 435–455. Springer, Cham (2018)

    Google Scholar 

  21. Ricciardi, L.A., Vasile, M., Maddock, C.: Global solution of multi-objective optimal control problems with multi agent collaborative search and direct finite elements transcription. In: IEEE Congress on Evolutionary Computation (CEC), 2016, pp. 869–876. IEEE, Piscataway (2016)

    Google Scholar 

  22. Ricciardi, L.A., Vasile, M., Toso, F., Maddock, C.A.: Multi-objective optimal control of the ascent trajectories of launch vehicles. In: AIAA/AAS Astrodynamics Specialist Conference, pp. 5669 (2016)

    Google Scholar 

  23. Vasile, M., Ricciardi, L.: A direct memetic approach to the solution of multi-objective optimal control problems. In: IEEE Symposium Series on Computational Intelligence (SSCI), 2016, pp. 1–8. IEEE, Piscataway (2016)

    Google Scholar 

  24. McKay, M.D., Beckman, R.J., Conover, W.J.: A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2), 239–245 (1979)

    MathSciNet  MATH  Google Scholar 

  25. Bryson, A.E.: Applied Optimal Control: Optimization, Estimation and Control. CRC Press, Boca Raton (1975)

    Google Scholar 

  26. Van Veldhuizen, D.A.: Multiobjective evolutionary algorithms: classifications, analyses, and new innovations. Ph.D. dissertation, Air Force Institute of Technology, Wright-Patterson AFB, Ohio (1999)

    Google Scholar 

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Acknowledgements

The research was partially funded by an ESA NPI grant (ref TEC-ECN-SoW-20140806) and Airbus Defence & Space. The author would like to thank Mr Lorenzo Ricciardi for the contribution to the development and implementation of the algorithms used for the generation of the numerical results in this chapter.

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Authors and Affiliations

  1. University of Strathclyde, Glasgow, UK

    Massimiliano Vasile

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  1. Massimiliano Vasile
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Correspondence to Massimiliano Vasile .

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Editors and Affiliations

  1. Department of Mathematics, University of Pisa, Pisa, Italy

    Giulio Baù

  2. Department of Mathematics, University of Rome Tor Vergata, Roma, Italy

    Alessandra Celletti

  3. Faculty of Mathematics, Alexandru Ioan Cuza University of Iasi, Iasi, Romania

    Cătălin Bogdan Galeș

  4. Department of Mathematics, University of Pisa, Pisa, Italy

    Giovanni Federico Gronchi

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Vasile, M. (2019). Multi-Objective Optimal Control: A Direct Approach. In: Baù, G., Celletti, A., Galeș, C., Gronchi, G. (eds) Satellite Dynamics and Space Missions. Springer INdAM Series, vol 34. Springer, Cham. https://doi.org/10.1007/978-3-030-20633-8_6

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