Direct solution of nonlinear constrained quadratic optimal control problems using B-spline functions

Yousef Edrisi Tabriz; Mehrdad Lakestani

Kybernetika (2015)

  • Volume: 51, Issue: 1, page 81-98
  • ISSN: 0023-5954

Abstract

top
In this paper, a new numerical method for solving the nonlinear constrained optimal control with quadratic performance index is presented. The method is based upon B-spline functions. The properties of B-spline functions are presented. The operational matrix of derivative ( 𝐃 φ ) and integration matrix ( 𝐏 ) are introduced. These matrices are utilized to reduce the solution of nonlinear constrained quadratic optimal control to the solution of nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.

How to cite

top

Edrisi Tabriz, Yousef, and Lakestani, Mehrdad. "Direct solution of nonlinear constrained quadratic optimal control problems using B-spline functions." Kybernetika 51.1 (2015): 81-98. <http://eudml.org/doc/270062>.

@article{EdrisiTabriz2015,
abstract = {In this paper, a new numerical method for solving the nonlinear constrained optimal control with quadratic performance index is presented. The method is based upon B-spline functions. The properties of B-spline functions are presented. The operational matrix of derivative ($\mathbf \{D\}_\phi $) and integration matrix ($\mathbf \{P\}$) are introduced. These matrices are utilized to reduce the solution of nonlinear constrained quadratic optimal control to the solution of nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.},
author = {Edrisi Tabriz, Yousef, Lakestani, Mehrdad},
journal = {Kybernetika},
keywords = {optimal control problem; B-spline functions; derivative matrix; collocation method; optimal control problem; B-spline functions; derivative matrix; collocation method},
language = {eng},
number = {1},
pages = {81-98},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Direct solution of nonlinear constrained quadratic optimal control problems using B-spline functions},
url = {http://eudml.org/doc/270062},
volume = {51},
year = {2015},
}

TY - JOUR
AU - Edrisi Tabriz, Yousef
AU - Lakestani, Mehrdad
TI - Direct solution of nonlinear constrained quadratic optimal control problems using B-spline functions
JO - Kybernetika
PY - 2015
PB - Institute of Information Theory and Automation AS CR
VL - 51
IS - 1
SP - 81
EP - 98
AB - In this paper, a new numerical method for solving the nonlinear constrained optimal control with quadratic performance index is presented. The method is based upon B-spline functions. The properties of B-spline functions are presented. The operational matrix of derivative ($\mathbf {D}_\phi $) and integration matrix ($\mathbf {P}$) are introduced. These matrices are utilized to reduce the solution of nonlinear constrained quadratic optimal control to the solution of nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.
LA - eng
KW - optimal control problem; B-spline functions; derivative matrix; collocation method; optimal control problem; B-spline functions; derivative matrix; collocation method
UR - http://eudml.org/doc/270062
ER -

References

top
  1. Betts, J., 10.1007/978-3-0348-8497-6_1, In: Computational Optimal Control (R. Bulirsch and D. Kraft, eds.), Birkhauser, 1994, pp. 3-17. MR1287613DOI10.1007/978-3-0348-8497-6_1
  2. Betts, J., 10.2514/2.4231, J. Guidance, Control, and Dynamics 21 (1998), 193-207. Zbl1158.49303DOI10.2514/2.4231
  3. Boor, C. De., A Practical Guide to Spline., Springer-Verlag, New York 1978. MR0507062
  4. Elnegar, G. N., Kazemi, M. A., 10.1023/A:1018694111831, Comput. Optim. Appl. 11 (1998), 195-217. MR1652069DOI10.1023/A:1018694111831
  5. Foroozandeh, Z., Shamsi, M., 10.1016/j.actaastro.2011.10.004, Acta Astronautica 72 (2012), 21-26. DOI10.1016/j.actaastro.2011.10.004
  6. Gong, Q., Kang, W., Ross, I. M., 10.1109/tac.2006.878570, IEEE Trans. Automat. Control 51 (2006), 1115-1129. MR2238794DOI10.1109/tac.2006.878570
  7. Goswami, J. C., Chan, A. K., 10.1002/9780470926994, John Wiley and Sons Inc. 1999. Zbl1214.65071MR2799281DOI10.1002/9780470926994
  8. Jaddu, H., 10.1016/s0016-0032(02)00028-5, J. Franklin Inst. 339 (2002), 479-498. Zbl1010.93507MR1931507DOI10.1016/s0016-0032(02)00028-5
  9. Jaddu, H., Shimemura, E., 10.1002/(sici)1099-1514(199901/02)20:1<21::aid-oca644>3.3.co;2-4, Optimal Control Appl. Methods 20 (1999), 21-42. MR1690446DOI10.1002/(sici)1099-1514(199901/02)20:1<21::aid-oca644>3.3.co;2-4
  10. Lancaster, P., Theory of Matrices., Academic Press, New York 1969. Zbl0558.15001MR0245579
  11. Lakestani, M., Dehghan, M., Irandoust-Pakchin, S., 10.1016/j.cnsns.2011.07.018, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 3, 1149-1162. Zbl1276.65015MR2843781DOI10.1016/j.cnsns.2011.07.018
  12. Lakestani, M., Razzaghi, M., Dehghan, M., 10.1155/mpe.2005.113, Hindawi Publishing Corporation Mathematical Problems in Engineering 1 (2005), 113-121. Zbl1073.65568MR2144111DOI10.1155/mpe.2005.113
  13. Lakestani, M., Razzaghi, M., Dehghan, M., 10.1155/mpe/2006/96184, Hindawi Publishing Corporation Mathematical Problems in Engineering 1 (2006), 1-12. Zbl1200.65112DOI10.1155/mpe/2006/96184
  14. Marzban, H. R., Razzaghi, M., 10.1016/s0307-904x(03)00050-7, Appl. Math. Modell. 27 (2003), 471-485. Zbl1020.49025DOI10.1016/s0307-904x(03)00050-7
  15. Marzban, H. R., Razzaghi, M., 10.1016/j.apm.2009.03.036, Appl. Math. Modell. 34 (2010), 174-183. MR2566686DOI10.1016/j.apm.2009.03.036
  16. Marzban, H. R., Hoseini, S. M., 10.1016/j.cnsns.2012.10.012, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 1347-1361. Zbl1282.65075MR3016889DOI10.1016/j.cnsns.2012.10.012
  17. Mashayekhi, S., Ordokhani, Y., Razzaghi, M., 10.1016/j.cnsns.2011.09.008, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 1831-1843. Zbl1239.49043MR2855473DOI10.1016/j.cnsns.2011.09.008
  18. Mehra, R. K., Davis, R. E., 10.1109/tac.1972.1099881, IEEE Trans. Automat. Control 17 (1972), 69-72. Zbl0268.49038DOI10.1109/tac.1972.1099881
  19. Ordokhani, Y., Razzaghi, M., Linear quadratic optimal control problems with inequality constraints via rationalized Haar functions., Dynam. Contin. Discrete Impuls. Syst. Ser. B 12 (2005), 761-773. Zbl1081.49026MR2179602
  20. Powell, M. J. D., 10.1093/comjnl/7.2.155, Comput. J. 7 (1964), 155-162. MR0187376DOI10.1093/comjnl/7.2.155
  21. Razzaghi, M., Elnagar, G., 10.1080/00207729408928967, Int. J. Systems Sci. 25 (1994), 393-399. MR1262503DOI10.1080/00207729408928967
  22. Schittkowskki, K., 10.1007/bf02022087, Ann. Oper. Res. 5 (1986), 2, 485-500. MR0948031DOI10.1007/bf02022087
  23. Schumaker, L., Spline Functions: Basic Theory., Cambridge University Press, 2007. Zbl1123.41008MR2348176
  24. Teo, K. L., Wong, K. H., 10.1017/s0334270000007207, J. Austral. Math. Soc. Ser. B 33 (1992), 507-530. Zbl0764.49017MR1154823DOI10.1017/s0334270000007207
  25. Vlassenbroeck, J., 10.1016/0005-1098(88)90094-5, Automatica 24 (1988), 499-506. MR0956571DOI10.1016/0005-1098(88)90094-5
  26. Yen, V., Nagurka, M., 10.1115/1.2896367, J. Dynam. Syst. Measure Control 11 (1991), 206-215. DOI10.1115/1.2896367
  27. Yen, V., Nagurka, M., 10.1002/oca.4660130206, Optimal Control Appl. Methods 13 (1992), 155-167. MR1197736DOI10.1002/oca.4660130206

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.