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Optimal Control Formulation for Complementarity Dynamical Systems

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Abstract

In this paper, we show that the complementarity dynamical systems can be reformulated as optimal control problems. By using this reformulation, we present a pseudospectral scheme to discretize the complementarity dynamical systems. Applying this discretization, the complementarity dynamical system is reduced to a sequence of nonlinear programming problems. Numerical examples and comparison with two other methods are included to demonstrate the capability of the proposed method.

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References

  1. Moreau, J.J.: Les liaisons unilatérales et le principe de Gauss. Comptes Rendus de l’ des Sci 256, 871–874 (1963)

    Google Scholar 

  2. Moreau, J.J.: Quadratic programming in mechanics: dynamics of one-sided constraints. J. SIAM Control 4(1), 153–158 (1966)

    Article  MathSciNet  Google Scholar 

  3. Moreau, J.J.: Unilateral contact and dry friction in finite freedom dynamics. In: Nonsmooth Mechanics and Applications, pp. 1–82. Springer (1988)

  4. Schatzman, M.: A class of nonlinear differential equations of second order in time. Nonlinear Anal. 2(3), 355–373 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  5. Monteiro-Marques, M.D.P.: Differential inclusions in nonsmooth mechanical problems. In: Progress in Nonlinear Differential Equations and Their Applications, vol. 9. Birkhäuser (1993)

  6. Vasca, F., Iannelli, L., Çamlibel, M.K., Frasca, R.: A new perspective for modeling power electronics converters: complementarity framework. IEEE Trans. Power Electron. 24(2), 456–468 (2009)

    Article  Google Scholar 

  7. Zhong, R., Yuan, F., Pan, T.: A linear complementarity system approach to macroscopic freeway traffic modeling. In: Proceedings of the 2013 IEEE International Conference on Cyber Technology in Automation, Control and Intelligent Systems, pp. 18–23 (2013)

  8. Schumacher, J.M.: Complementarity systems in optimization. Math. Program. Ser. B 101(1), 263–295 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brogliato, B., Daniilidis, A., Lemaréchal, C., Acary, V.: On the equivalence between complementarity systems, projected systems and differential inclusions. Syst. Control Lett. 55(1), 45–51 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Heemels, W.P.M.H., Schumacher, J.M., Weiland, S.: Projected dynamical systems in a complementarity formalism. Oper. Res. Lett. 27(2), 83–91 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Çamlibel, M.K., Pang, J.S., Shen, J.: Lyapunov stability of complementarity and extended systems. SIAM J. Optim. 17(4), 1056–1101 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Çamlibel, M.K.: Popov–Belevitch–Hautus type tests for the controllability of linear complementarity systems. Syst. Control Lett. 56(3), 381–387 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Çamlibel, M.K., Pang, J.S., Shen, J.: Conewise linear systems: nonzenoness and observability. SIAM J. Control Optim. 45(5), 1769–1800 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. van der Schaft, A.J., Schumacher, J.M.: Complementarity modeling of hybrid systems. IEEE Trans. Autom. Control 43(4), 483–490 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Pang, J.S., Stewart, D.E.: Differential variational inequalities. Math. Program. 113(2), 345–424 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Brogliato, B., Thibault, L.: Existence and uniqueness of solutions for non-autonomous complementarity dynamical systems. J. Convex Anal. 17(3–4), 961–990 (2010)

    MATH  MathSciNet  Google Scholar 

  17. Brogliato, B., Goeleven, D.: Existence, uniqueness of solutions and stability of nonsmooth multivalued Lur’e dynamical systems. J. Convex Anal. 20(3), 881–900 (2013)

    MATH  MathSciNet  Google Scholar 

  18. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Cambridge (1992)

    MATH  Google Scholar 

  19. Heemels, W.P.M.H., Schumacher, J.M., Weiland, S.: Applications of complementarity systems. In: Proceedings of the European Control Conference, Karlsruhe, Germany (1999)

  20. Çamlıbel, M.K., Heemels, W.P.M.H., Schumacher, J.M.: On linear passive complementarity systems. Eur. J. Control 8(3), 220–237 (2002)

    Article  MATH  Google Scholar 

  21. Heemels, W.P.M.H., Çamlibel, M.K., Schumacher, J.M., Brogliato, B.: Observer-based control of linear complementarity systems. Int. J. Robust Nonlinear Control 21(10), 1193–1218 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. Brogliato, B., Goeleven, D.: Well-posedness, stability and invariance results for a class of multivalued Lur’e dynamical systems. Nonlinear Anal. 74(1), 195–212 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Çamlıbel, M.K., Iannelli, L., Vasca, F.: Passivity and complementarity. Math. Program. 145(1–2), 531–563 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mangasarian, O.L.: Equivalence of the complementarity problem to a system of nonlinear equations. SIAM J. Appl. Math. 31, 89–92 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kanzow, C.: Some noninterior continuation methods for linear complementarity problems. SIAM J. Matrix Anal. Appl. 17(4), 851–868 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. Galántai, A.: Properties and construction of NCP functions. Comput. Optim. Appl. 52, 805–824 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Miri, S.M., Effati, S.: On generalized convexity of nonlinear complementarity functions. J. Optim. Theory Appl. 164, 723–730 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kang, W.: Rate of convergence for the Legendre pseudospectral optimal control of feedback linearizable systems. J. Control Theory Appl. 8(4), 391–405 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  29. Ross, I.M., Karpenko, M.: A review of pseudospectral optimal control: from theory to flight. Annu. Rev. Control 36(2), 182–197 (2012)

    Article  Google Scholar 

  30. Tang, X., Liu, Z., Hu, Y.: New results on pseudospectral methods for optimal control. Automatica 65, 160–163 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  31. Kang, W., Ross, I.M., Gong, Q.: Pseudospectral optimal control and its convergence theorems. In: Analysis and Design of Nonlinear Control Systems, pp. 109–124. Springer (2008)

  32. 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)

    Article  MATH  MathSciNet  Google Scholar 

  33. Nocedal, J., Wright, S.: Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)

    Google Scholar 

  34. Acary, V., Bonnefon, O., Brogliato, B.: Nonsmooth modeling and simulation for switched circuits. Lecture Notes in Electrical Engineering, vol. 69. Springer, Berlin (2011)

  35. Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems. Lecture Notes in Applied and Computational Mechanics, vol. 35. Springer, Berlin (2008)

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Acknowledgements

The authors are grateful to the associate editor and the anonymous referees for their valuable comments and suggestions in improving the quality of this paper.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

    S. Mohsen Miri & Sohrab Effati

  2. Center of Excellence on Soft Computing and Intelligent Information Processing, Ferdowsi University of Mashhad, Mashhad, Iran

    Sohrab Effati

Authors
  1. S. Mohsen Miri
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  2. Sohrab Effati
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Correspondence to Sohrab Effati.

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Communicated by Aris Daniilidis.

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Miri, S.M., Effati, S. Optimal Control Formulation for Complementarity Dynamical Systems. J Optim Theory Appl 175, 356–372 (2017). https://doi.org/10.1007/s10957-017-1178-0

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  • DOI: https://doi.org/10.1007/s10957-017-1178-0

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