International Journal of Intelligent Systems and Applications(IJISA)

ISSN: 2074-904X (Print), ISSN: 2074-9058 (Online)

Published By: MECS Press

IJISA Vol.6, No.6, May. 2014

An Iterative Technique for Solving a Class of Nonlinear Quadratic Optimal Control Problems Using Chebyshev Polynomials

Full Text (PDF, 400KB), PP.53-57

Views:253   Downloads:9


Hussein Jaddu, Amjad Majdalawi

Index Terms

Nonlinear quadratic optimal control problem, Banks Iterative Technique, Chebyshev polynomials, State parameterization


In this paper, a method for solving a class of nonlinear optimal control problems is presented. The method is based on replacing the dynamic nonlinear optimal control problem by a sequence of quadratic programming problems. To this end, the iterative technique developed by Banks is used to replace the original nonlinear dynamic system by a sequence of linear time-varying dynamic systems, then each of the new problems is converted to quadratic programming problem by parameterizing the state variables by a finite length Chebyshev series with unknown parameters. To show the effectiveness of the proposed method, simulation results of a nonlinear optimal control problem are presented.

Cite This Paper

Hussein Jaddu, Amjad Majdalawi,"An Iterative Technique for Solving a Class of Nonlinear Quadratic Optimal Control Problems Using Chebyshev Polynomials", International Journal of Intelligent Systems and Applications(IJISA), vol.6, no.6, pp.53-57, 2014. DOI: 10.5815/ijisa.2014.06.06


[1]Kraft D. On converting optimal control problems into nonlinear programming problems. Computational Mathematical Programming. Vol 15, Ed. Schittkowski, Springer, Berlin, 1985, pp. 261-280.

[2]Betts J. Issues in the direct transcription of optimal control problem to sparse nonlinear programs. Computational Optimal Control, Ed. R. Bulirsch and D. Kraft, Birkhauser, Germany, 1994, pp. 3-17.

[3]Jaddu, H., Numerical methods for solving optimal control problems using Chebyshev polynomials, PHD Thesis 1998. 

[4]Jaddu, H., Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials, Journal of the Franklin Institute, Vol. 339, 2002, pp 479-498.

[5]Jaddu, H and Vlach M. Successive approximation method for non-linear optimal control problems with applications to a container crane problem, Optimal Control Applications and Methods, Vol. 23, 2002, pp 275-288.

[6]Spangelo I. Trajectory optimization for vehicles using control parameterization and nonlinear programming. Phd thesis, 1994.

[7]Goh C J. and Teo K L. Control parameterization: A unified approach to optimal control problems with general constraints, Automatica, 24-1, 1988, pp 3-18.

[8]Vlassenbroeck J and Van Doreen R. A Chebyshev technique for solving nonlinear optimal control problems, IEEE Trans. Automat. Cont., 33,1988, pp 333-340.

[9]Frick P A and Stech D J. Epsilon-Ritz method for solving optimal control problems: Useful parallel solution method, Journal of Optimization Theory and Applications, Vol. 79, 1993, pp 31-58.

[10]Tomas-Rodriguez M and Banks S P. Linear approximations to nonlinear dynamical systems with applications to stability and spectral theory, IMA Journal of Control and Information, 20, 2003, pp 1-15.

[11]Banks S P and Dinesh, K. Approximate optimal control and stability of nonlinear finite and infinite-dimensional systems. Ann. Op. Res., 98, 2000, pp 19-44.

[12]Tomas-Rodriguez M and Banks S P. An iterative approach to eigenvalue assignment for nonlinear systems, Proceedings of the 45th IEEE Conference on Decision & Control, 2006, pp 977-982.

[13]Navarro Hernandez C, Banks S P and Aldeen M. Observer design for nonlinear systems using linear approximations. IMA J. Math. Cont Inf(20), 2003, pp359-370

[14]Tomas-Rodriguez M, Banks S P and M.U. Salamci M U. Sliding mode control for nonlinear systems: An iterative approach, Proceedings of the 45th IEEE Conference on Decision & Control, 2006, pp 4963-4968

[15]Jaddu H and Shimemura E. Computation of optimal control trajectories using Chebyshev polynomials: parameterization and quadratic programming, Optimal Control Applications and methods, 20, 1999, pp.21-42

[16]Fox L and Parker I B. Chebyshev Polynomials in Numerical Analysis, Oxford University Press, England, 1968. 

[17]Bullock T and Franklin G. A second order feedback method for optimal control computations, IEEE Trans. Automat. Cont., Vol. 12, 1967, pp 666-673.

[18]Bashein G and Enns M. Computation of optimal control by a method combining quasi-linearization and quadratic programming, Int. J. Control, Vol. 16, 1972, pp 177-187.