Two numerical methods for nonlinear constrained quadratic optimal control problems using linear B-spline functions

Document Type : Research Article


1 Payame Noor University

2 University of Tabriz


This paper presents two numerical methods for solving the nonlinear constrained optimal control problems including quadratic performance index.
The methods are based upon linear B-spline functions. The properties of B-spline functions are presented. Two operational matrices of integration are introduced for related procedures. These matrices are then utilized to reduce the solution of the nonlinear constrained optimal control to a nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the presented techniques.


1. Avrile, M. Nonlinear programming: Analysis and Methods, Englewood Cli_s, NJ: Prentice-Hall, 1976.
2. Bellman, R. Dynamic Programming, Princeton, NJ: Princeton University Press, 1957.
3. Bellman, R., Kalaba, R. and Kotkin, B. Polynomial Approximation – A New Computational Technique in Dynamic Programming: Allocation Processes, Mathematics of Computation, 17(1963), 155-161.
4. Bellman, R. and Dreyfus, S. E. Applied Dynamic Programming, Princeton University Press, 1971.
5. Betts, J. Issues in the direct transcription of optimal control problem to sparse nonlinear programs, in: Bulirsch, R., Kraft, D. (Eds.), Computational Optimal Control, Germany: Birkhauser, 115(1994), 3-17.
6. Betts, J. Survey of numerical methods for trajectory optimization, J. Guidance Control Dynamics, 21(2) (1998), 193-207.
7. de Boor, C. A practical guide to spline, Springer Verlag, New York, 1978.
8. Chui, C.K. An introduction to wavelets, San Diego, Calif: Academic Press, 1992.
9. Elnegar, G.N. and Kazemi, M.A. Pseudospectral Chebyshev optimal control of constrained nonlinear dynamical systems, Comput. Optim. Applica. 11(1998), 195-217.
10. Foroozandeh, Z. and Shamsi, M. Solution of nonlinear optimal control problems by the interpolating scaling functions, Acta Astronautica, 72(2012), 21-26.
11. Gill, P.E. and Murray, W. Linearly constrained problems including linear and quadratic programming, In: Jacobs D, editor, The State of the Art in Numerical Analysis, London, New York: Academic Press, (1977), 313-363.
12. Gong, Q., Kang, W. and Ross, I.M. A pseudospectral method for the optimal control of constrained feedback linearizable systems, IEEE Trans Automat Control, 51(7) (2006), 1115-1129.
13. Goswami, J.C. and Chan, A.K. Fundamentals of wavelets: theory, algorithms, and applications, John Wiley & Sons, Inc., 1999.
14. Hull, D.G. Optimal Control Theory for Applications, New York: Springer-Verlag, 2003.
15. Jaddu, H. Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials, Journal of the Franklin In stitute, 339 (2002), 479-498.
16. Jaddu, H. and Shimemura, E. Computation of optimal control trajectories using Chebyshev polynomials: parameterization and quadratic programming, Optimal Control Appl Methods, 20 (1999), 21-42.
17. Junkins, J.L. and Turner, J.D. Optimal Spacecraft Rotational Maneuvers, Amsterdam: Elsevier, 1986.
18. Kirk, D.E. Optimal control theory, Englewood Cli_s, NJ: Prentice Hall, 1970.
19. Kleiman, D.L., Fortmann, T. and Athans, M. On the design of linear systems with piecewise-constant feedback gains, IEEE Trans Automat Control, 13 (1968), 354-361.
20. Lancaster, P. Theory of Matrices, New York: Academic Press, 1969.
21. Lakestani, M., Dehghan, M. and Irandoust-pakchin, S. The constructionof operational matrix of fractional derivatives using B-spline functions, Commun Nonlinear Sci Numer Simulat, 17 (2012), 1149-1162.
22. Lakestani, M., Razzaghi, M. and Dehghan, M. Solution of nonlinear fredholm-hammerstein integral equations by using semiorthogonal spline wavelets, Hindawi Publishing Corporation Mathematical Problems in Engineering, 1 (2005), 113-121.
23. Lakestani, M., Razzaghi, M. and Dehghan, M. Semiorthogonal spline wavelets approximation for fredholm integro-di_erential equations, Hindawi Publishing Corporation Mathematical Problems in Engineering, 1
(2006), 1-12.
24. Leitman, G. The Calculus of Variations and Optimal Control, New York: Springer, 1981.
25. Marzban, H.R. and Razzaghi, M. Hybrid functions approach for linearly constrained quadratic optimal control problems, Appl Math Modell, 27(2003), 471-485.
26. Marzban, H.R. and Razzaghi, M. Rationalized Haar approach for nonlinear constrined optimal control problems, Appl Math Modell, 34 (2010), 174-183.
27. Marzban, H.R. and Hoseini, S.M. A composite Chebyshev _nite di_erence method for nonlinear optimal control problems, Commun Nonlinear Sci Numer Simulat, 18 (2013), 1347-1361.
28. Mashayekhi, S., Ordokhani, Y. and Razzaghi M. Hybrid functions approach for nonlinear constrained optimal control problems, Commun Nonlinear Sci Numer Simulat, 17 (2012), 1831-1843.
29. Ordokhani, Y. and Razzaghi, M. Linear quadratic optimal control problems with inequality constraints via rationalized Haar functions, Dyn Contin Discrete Impul Syst Ser B, 12 (2005), 761-773.
30. Razzaghi, M. and Elnagar, G.Linear quadratic optimal control problems via shifted Legendre state parameterization , Internat. J. Systems Sci., 25(1994), 393-399.
31. Teo, K.L. and Wong, K.H. Nonlinearly constrained optimal control problems, J. Austral Math Soc. Ser B, 33 (1992), 507-530.
32. Yen, V. and Nagurka, M. Optimal control of linearly constrained linear systems via state parameterization, Optimal Control Appl Methods, 13(1992), 155-167.