An Improved Shooting Method for a Class of Switching Optimal Control Problems

علوم کاربردی و محاسباتی در مکانیک

Article 9, Volume 31, Issue 1 - Serial Number 21, February 2020, Pages 123-138   

Document Type: Original Article

DOI: 10.22067/fum-mech.v31i1.77967

Author

Fariba Vahidifar

Abstract

Optimal control appears in some of the well-known areas of science in recent decades. In optimal control field, a classical subject is the switching optimal control problems which arise in some well-known application areas, such as aerospace engineering, cranes and industrial robots. In these problems, the input control jumps from one boundary to another and appears linearly in the objective function and the dynamical equations. What’s important in solving these problems, is finding the switching points with high accuracy. In this paper, an improved shooting method for solving these problems is investigated. For this purpose, the well-known indirect shooting method for numerical solution of this class of optimal control problems is first reviewed and it’s deficiency in obtaining the switching points accurately, is expressed. Then, with redesigning the method by applying a control parameterization, we transform the problem to the solution of the shooting equation, in which, the values of the switching points and the initial values of the costate variables are unknown parameters. This leads the switching points are captured accurately and thereby accurate solution of the problem is obtained. Numerical results of five benchmark examples are presented and efficiency of the method is reported.

Keywords

Switching Optimal Control; Pontryagin’s Minimum Principle; Shooting Method; Control Parameterization

Supplementary Files

EA9-mehrpooya-Nahayi.pdf

References

1. Ledzewicz, U. and Schttler, H., "Analysis of a cell-cycle specific model for cancer chemotherapy", Journal of Biological Systems, Vol. 10, pp. 183–206, (2002).

2. Woon, S. Rehbock, V. and Loxton, R. ''Towards global solutions of optimal discrete-valued control problems'', Optimal Control Applications and Methods, 33, pp. 576–594, (2012).

3. Xia, L. and Shihada, B. ''Power and delay optimisation in multi-hop wireless networks'', International Journal of Control, 87, pp. 1252–1265, (2014).

4. Kaya, C. and Noakes, J. ''Computations and time-optimal controls'', Optimal Control Applications and Methods, 17, pp. 171–185, (1996).

5. Kaya, C. and Noakes, J. ''Computational method for time-op timal switching control'', Journal of Optimization Theory and Applications, 117, pp. 69–92, (2003)

6. Maurer, H. Buskens, C.Kim, J.-H and Kaya, C. ''Optimization methods for the verification of second order sufficient conditions for bang-bang controls'', Optimal Control Applications and Methods, 26, pp. 129–156, (2005).

7. Lin, Q. Loxton, R. Teo, K. and Wu, Y. ''A new computational method for a class of free terminal time optimal control problems'', Pacific Journal of Optimization, 7, pp. 63–81, (2011).

8. Lin, Q. Loxton, R. Teo, K. and Wu, Y ''Optimal control computation for nonlinear systems with state-dependent stopping criteria'', Automatica, 48, pp. 2116–2129, (2012).

9. Shamsi, M. ''A modified pseudospectral scheme for accurate solution of bang-bang optimal control problems'', Optimal Control Applications and Methods, 32, pp. 668–680, (2011).

10. Gerdts, M. ''A nonsmooth newton’s method for control-state constrained optimal control problems'', Mathematics and Computers in Simulation, 79, pp. 925–936, (2008).

11. Riedinger, P. and Morarescu, I.-C., ''A numerical framework for optimal control of switched input affine nonlinear systems subject to path constraint'', Mathematics and Computers in Simulation, 95, pp. 63–77, (2014).

12. Yu, C. Li, B. Loxton, R. and Teo, K. ''Optimal discrete-valued control computation'', Journal of Global Optimization, 56, pp. 503–518, (2013).

13. Lee, H. Teo, K. Rehbock, V. and Jennings, L., ''Control parametrization enhancing technique for optimal discrete-valued control problems'', Automatica, 35, pp. 1401–1407, (1999).

14. Li, R. Feng, Z. Teo, K. and Duan, G., ''Tracking control of linear switched systems'', ANZIAM Journal, 49, pp. 187–203, (2008).

15. Loxton, R. Teo, K. Rehbock, V. and Yiu, K., ''Optimal control problems with a continuous inequality constraint on the state and the control'', Automatica, 45, pp. 2250–2257, (2009).

16. Wong, K. and Tang, W., ''Optimal control of switched impulsive systems with time delay'', ANZIAM Journal, 53, 292–307, (2012).

17. Kirk, D., ''Optimal Control Theory'', Prentice-Hall, Englewood Cliffs, NJ, (1970).

18. Bryson Jr., A.E. and Ho, Y.C., ''Applied optimal control: Optimization, estimation, and control'', Hemisphere Publishing Corp. Washington, D. C., (1975).

19. Betts, J., ''Survey of numerical methods for trajectory optimization'', Journal of Guidance, Control, and Dynamics, 21, pp. 193–207, (1998).

20. Betts, J. T. ''Practical Methods for Optimal Control and Estimation Using Nonlinear Programming'', 2nd edition, Cambridge University Press, New York, NY, USA, (2009).

21. Maurer, H. and Osmolovskii, N., ''Second order sufficient conditions for time-optimal bang-bang control'', SIAM Journal on Control and Optimization, 42, pp. 2239–2263, (2004).

22. Oberle. H. and Grimm, W., ''BNDSCO -A program for the numerical solution of optimal control problems'', Technical Report 515-89/22, Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, Germany, (1989).

23. Chernousko, F., Akulenko, L. and Bolotnik, N., ''Time-optimal control for robotic manipulators'', Optimal Control Applications and Methods, 10, pp. 293–311, (1989).

24. Geering, H., Guzzella, L., Hepner, S. and Onder, C., ''Time-optimal motions of robots in assembly tasks'', IEEE Transactions on Automatic Control, 31, pp. 512–518, (1986).

25. Mezzadri, F and Galligani, E., ''A chebyshev technique for the solution of optimal control problems with nonlinear programming methods'', Mathematics and Computers in Simulation, 121, pp. 95–108, (2016).

26. Teo, K., Goh, C.J. and Wong, K.H., ''A Unified Computational Approach to Optimal Control Problems'', Longman Scientific and Technical, (1991).

27. Woon, S., Rehbock, V. and Loxton, R., ''Global optimization method for continuous-time sensor scheduling'', Nonlinear Dynamics and Systems Theory, 10, pp. 175–188, (2010)