Advertisement

Journal of Optimization Theory and Applications

, Volume 163, Issue 3, pp 957–988 | Cite as

Switching Time and Parameter Optimization in Nonlinear Switched Systems with Multiple Time-Delays

  • Chongyang Liu
  • Ryan Loxton
  • Kok Lay TeoEmail author
Article

Abstract

In this paper, we consider a dynamic optimization problem involving a general switched system that evolves by switching between several subsystems of nonlinear delay-differential equations. The optimization variables in this system consist of: (1) the times at which the subsystem switches occur; and (2) a set of system parameters that influence the subsystem dynamics. We first establish the existence of the partial derivatives of the system state with respect to both the switching times and the system parameters. Then, on the basis of this result, we show that the gradient of the cost function can be computed by solving the state system forward in time followed by a costate system backward in time. This gradient computation procedure can be combined with any gradient-based optimization method to determine the optimal switching times and parameters. We propose an effective optimization algorithm based on this idea. Finally, we consider three numerical examples, one involving the 1,3-propanediol fed-batch production process, to illustrate the effectiveness and applicability of the proposed algorithm.

Keywords

Switched system Time-delay system Switching times  Nonlinear optimization 

Mathematics Subject Classification

49M07 49M37 65K10 

Notes

Acknowledgments

The first author is supported by the Natural Science Foundation for the Youth of China (Grant Nos. 11201267 and 11001153), the Tian Yuan Special Funds of the National Natural Science Foundation of China (Grant No. 11126077), and the Shandong Province Natural Science Foundation of China (Grant Nos. ZR2010AQ016 and ZR2011AL003). The second author is supported by the Natural Science Foundation of China (Grant No. 11350110208) and the Australian Research Council (Discovery Grant DP110100083). The third author is supported by the Australian Research Council (Discovery Grant DP110100083).

References

  1. 1.
    van der Schaft, A., Schumacher, H.: An Introduction to Hybrid Dynamical Systems. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ramadge, P., Wonham, W.: The control of discrete event systems. Proc. IEEE 77, 81–98 (1989)CrossRefGoogle Scholar
  3. 3.
    Astrom, K.J., Puruta, K.: Swing up a pendulum by energy control. Autom. J. IFAC 36, 287–295 (2000)CrossRefGoogle Scholar
  4. 4.
    Hespanha, J., Liberzon, D., Morse, A.S.: Overcoming the limitations of adaptive control by means of logic-based switching. Syst. Control Lett. 49, 49–56 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Howlett, P.: Optimal strategies for the control of a train. Autom. J. IFAC 32, 519–532 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Liu, C.Y., Feng, E.M., Yin, H.C.: Optimal switching control for microbial fed-batch culture. Nonlinear Anal. Hybrid Syst. 2, 1168–1174 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Loxton, R., Teo, K.L., Rehbock, V., Ling, W.K.: Optimal switching instants for a switched-capacitor DC/DC power converter. Autom. J. IFAC 45, 973–980 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Woon, S.F., Rehbock, V., Loxton, R.: Towards global solutions of optimal discrete-valued control problems. Optim. Control Appl. Methods 33, 576–594 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Seidman, T.I.: Optimal control for switching systems. In: Proceedings of the 21st Annual Conference on Information Science and Systems, Baltimore, MD, pp. 485–489 (1987).Google Scholar
  10. 10.
    Branicky, M.S., Borkar, V.S., Mitter, S.K.: A unified framework for hybrid control: model and optimal control theory. IEEE Trans. Autom. Control 43, 31–45 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Sussmann, H.: A maximum principle for hybrid optimal control problems. In: Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, pp. 3972–3977 (1999).Google Scholar
  12. 12.
    Luus, R., Chen, Y.Q.: Optimal switching control via direct search optimization. Asian J. Control 6, 302–306 (2004)CrossRefGoogle Scholar
  13. 13.
    Xu, X.P., Antsaklis, P.J.: Optimal control of switched systems based on parameterization of the switching instants. IEEE Trans. Autom. Control 49, 2–15 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bengea, S.C., Raymond, A.D.: Optimal control of switching systems. Autom. J. IFAC 41, 11–27 (2005)zbMATHGoogle Scholar
  15. 15.
    Giua, A., Seatzu, C., Van Der Mee, C.: Optimal control of autonomous linear systems switched with a preassigned finite sequence. In: Proceedings of the 2001 IEEE International Symposium on Intelligent Control, México City, México, pp. 144–146 (2001).Google Scholar
  16. 16.
    Xu, X.P., Antsaklis, P.J.: Optimal control of switched autonomous systems. In: Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, pp. 4401–4406 (2002).Google Scholar
  17. 17.
    Egerstedt, M., Wardi, Y., Delmotte, F.: Optimal control of switching times in switched dynamical systems. In: Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, HI, pp. 2138–2143 (2003).Google Scholar
  18. 18.
    Lin, Q., Loxton, R., Teo, K.L., Wu, Y.H.: A new computational method for optimizing nonlinear impulsive systems. Dyn. Contin. Discret. Impuls. Syst. Ser. B Appl. Algorithms 18, 59–76 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Loxton, R., Teo, K.L., Rehbock, V.: Computational method for a class of switched system optimal control problems. IEEE Trans. Autom. Control 54, 2455–2460 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Richard, J.P.: Time-delay systems: an overview of some recent advances and open problems. Autom. J. IFAC 29, 1667–1694 (2003)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Meyer, C., Schroder, S., De Doncker, R.W.: Solid-state circuit breakers and current limiters for medium-voltage systems having distributed power systems. IEEE Trans. Power Electr. 19, 1333–1340 (2004)CrossRefGoogle Scholar
  22. 22.
    Kim, D.K., Park, P.G., Ko, J.W.: Output-feedback H\(_\infty \) control of systems over communication networks using a deterministic switching system approach. Autom. J. IFAC 40, 1205–1212 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lin, Q., Loxton, R., Teo, K.L.: The control parameterization method for nonlinear optimal control: A survey. J. Ind. Manag. Optim. 10, 275–309 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Verriest, E.I., Delmotte, F., Egerstedt, M.: Optimal impulsive control of point delay systems with refractory period. In: Proceedings of the 5th IFAC Workshop on Time Delay Systems, Leuven, Belgium (2004).Google Scholar
  25. 25.
    Verriest, E.I.: Optimal control for switched point delay systems with refractory period. In: Proceedings of the 16th IFAC World Congress, Prague, Czech Republic (2005).Google Scholar
  26. 26.
    Delmotte, F., Verriest, E.I., Egerstedt, M.: Optimal impulsive control of delay systems. ESAIM Control Optim. Calc. Var. 14, 767–779 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wu, C.Z., Teo, K.L., Li, R., Zhao, Y.: Optimal control of switched systems with time delay. Appl. Math. Lett. 19, 1062–1067 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ahmed, N.U.: Elements of Finite-dimensional Systems and Control Theory. Longman Scientific and Technical, Essex (1988)zbMATHGoogle Scholar
  29. 29.
    Li, R., Teo, K.L., Wong, K.H., Duan, G.R.: Control parameterization enhancing transform for optimal control of switched systems. Math. Comput. Model. 43, 1393–1403 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Farhadinia, B., Teo, K.L., Loxton, R.: A computational method for a class of non-standard time optimal control problems involving multiple time horizons. Math. Comput. Model. 49, 1682–1691 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Liu, C.Y., Loxton, R., Teo, K.L.: Optimal parameter selection for nonlinear multistage systems with time-delays. Comput. Optim. Appl. doi: 10.1007/s10589-013-9632-x
  32. 32.
    Teo, K.L., Goh, C.J., Wong, K.H.: A Unified Computational Approach to Optimal Control Problems. Longman Scientific and Technical, Essex (1991)zbMATHGoogle Scholar
  33. 33.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  34. 34.
    Schittkowski, K.: A Fortran Implementation of a Sequential Quadratic Programming Algorithm with Distributed and Non-monotone Line Search—User’s Guide. University of Bayreuth, Bayreuth (2007)Google Scholar
  35. 35.
    Hindmarsh, A.C.: Large ordinary differential equation systems and software. IEEE Control Syst. Mag. 2, 24–30 (1982)CrossRefGoogle Scholar
  36. 36.
    Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (1980)CrossRefGoogle Scholar
  37. 37.
    Xiu, Z.L., Song, B.H., Sun, L.H., Zeng, A.P.: Theoretical analysis of effects of metabolic overflow and time delay on the performance and dynamic behavior of a two-stage fermentation process. Biochem. Eng. J. 11, 101–109 (2002)CrossRefGoogle Scholar
  38. 38.
    Mu, Y., Zhang, D.J., Teng, H., Wang, W., Xiu, Z.L.: Microbial production of 1,3-propanediol by Klebsiella pneumoniae using crude glycerol from biodiesel preparation. Biotechnol. Lett. 28, 1755–1759 (2006)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceShandong Institute of Business and TechnologyYantaiChina
  2. 2.Department of Mathematics and StatisticsCurtin UniversityPerthAustralia
  3. 3.Institute of Cyber-Systems and ControlZhejiang UniversityHangzhouChina

Personalised recommendations