Numerical Solution of a Class of Nonlinear Optimal Control Problems Using Linearization and Discretization

ABSTRACT

In this paper, a new approach using linear combination property of intervals and discretization is proposed to solve a class of nonlinear optimal control problems, containing a nonlinear system and linear functional, in three phases. In the first phase, using linear combination property of intervals, changes nonlinear system to an equivalent linear system, in the second phase, using discretization method, the attained problem is converted to a linear programming problem, and in the third phase, the latter problem will be solved by linear programming methods. In addition, efficiency of our approach is confirmed by some numerical examples.

In this paper, a new approach using linear combination property of intervals and discretization is proposed to solve a class of nonlinear optimal control problems, containing a nonlinear system and linear functional, in three phases. In the first phase, using linear combination property of intervals, changes nonlinear system to an equivalent linear system, in the second phase, using discretization method, the attained problem is converted to a linear programming problem, and in the third phase, the latter problem will be solved by linear programming methods. In addition, efficiency of our approach is confirmed by some numerical examples.

KEYWORDS

Linear and Nonlinear Optimal Control, Linear Combination Property of Intervals, Linear Programming, Discretization, Dynamical Control Systems

Linear and Nonlinear Optimal Control, Linear Combination Property of Intervals, Linear Programming, Discretization, Dynamical Control Systems

Cite this paper

nullM. Skandari and E. Tohidi, "Numerical Solution of a Class of Nonlinear Optimal Control Problems Using Linearization and Discretization,"*Applied Mathematics*, Vol. 2 No. 5, 2011, pp. 646-652. doi: 10.4236/am.2011.25085.

nullM. Skandari and E. Tohidi, "Numerical Solution of a Class of Nonlinear Optimal Control Problems Using Linearization and Discretization,"

References

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[13] W. Alt, “Approximation of Optimal Control Problems with Bound Constraints by Control Parameterization,” Control and Cybernetics, Vol. 32, No. 3, 2003, pp. 451-472.

[14] T. M. Gindy, H. M. El-Hawary, M. S. Salim and M. El-Kady, “A Chebyshev Approximation for Solving Optimal Control Problems,” Computers & Mathematics with Applications, Vol 29, No. 6, 1995, pp 35-45. doi:10.1016/0898-1221(95)00005-J

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[17] P. Balasubramaniam, J. A. Samath and N. Kumaresan, “Optimal Control for Nonlinear Singular Systems with Quadratic Performance Using Neural Networks,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp. 1535-1543.

[18] T. Cheng, F. L. Lewis, M. Abu-Khalaf, “A Neural Network Solution for Fixed-Final Time Optimal Control of Nonlinear Systems,” Automatica, Vol. 43, No. 3, 2007, pp. 482-490.

[19] P. V. Medagam and F. Pourboghrat, “Optimal Control of Nonlinear Systems Using RBF Neural Network and Adaptive Extended Kalman Filter,” Proceedings of American Control Conference Hyatt Regency Riverfront, St. Louis, 10-12 June 2009, pp. 355-360.

[20] D. Luenberger, “Linear and Nonlinear Programming,” Kluwer Academic Publishers, Norwell, 1984.

[1] M. Diehl, H. G. Bock and J. P. Schloder, “A Real-Time Iteration Scheme for Nonlinear Optimization in Optimal Feedback Control,” Siam Journal on Control and Optimization, Vol. 43, No. 5, 2005, pp.1714-1736. doi:10.1137/S0363012902400713

[2] M. Diehl, H. G. Bock, J. P. Schloder, R. Findeisen, Z. Nagyc and F. Allgower, “Real-Time Optimization and Nonlinear Model Predictive Control of Processes Governed by Differential-Algebraic Equations,” Journal of Process Control, Vol. 12, No. 4, 2002, pp. 577-585.

[3] M. Gerdts and H. J. Pesch, “Direct Shooting Method for the Numerical Solution of Higher-Index DAE Optimal Control Problems,” Journal of Optimization Theory and Applications, Vol. 117, No. 2, 2003, pp. 267-294. doi:10.1023/A:1023679622905

[4] H. J. Pesch, “A Practical Guide to the Solution of Real- Life Optimal Control Problems,” 1994. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.53.5766&rep=rep1&type=pdf

[5] J. A. Pietz, “Pseudospectral Collocation Methods for the Direct Transcription of Optimal Control Problems,” Master Thesis, Rice University, Houston, April 2003.

[6] O. V. Stryk, “Numerical Solution of Optimal Control Problems by Direct Collocation,” International Series of Numerical Mathematics, Vol. 111, No. 1, 1993, pp. 129-143.

[7] A. H. Borzabadi, A. V. Kamyad, M. H. Farahi and H. H. Mehne, “Solving Some Optimal Path Planning Problems Using an Approach Based on Measure Theory,” Applied Mathematics and Computation, Vol. 170, No. 2, 2005, pp. 1418-1435.

[8] M. Gachpazan, A. H. Borzabadi and A. V. Kamyad, “A Measure-Theoretical Approach for Solving Discrete Optimal Control Problems,” Applied Mathematics and Computation, Vol. 173, No. 2, 2006, pp. 736-752.

[9] A.V. Kamyad, M. Keyanpour and M. H. Farahi, “A New Approach for Solving of Optimal Nonlinear Control Problems,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp. 1461-1471.

[10] A. V. Kamyad, H. H. Mehne and A. H. Borzabadi, “The Best Linear Approximation for Nonlinear Systems,” Applied Mathematics and Computation, Vol. 167, No. 2, 2005, pp. 1041-1061.

[11] K. P. Badakhshan and A. V. Kamyad, “Numerical Solution of Nonlinear Optimal Control Problems Using Nonlinear Programming,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp. 1511-1519.

[12] K. P. Badakhshan, A. V. Kamyad and A. Azemi, “Using AVK Method to Solve Nonlinear Problems with Uncertain Parameters,” Applied Mathematics and Computation, Vol. 189, No. 1, 2007, pp. 27-34.

[13] W. Alt, “Approximation of Optimal Control Problems with Bound Constraints by Control Parameterization,” Control and Cybernetics, Vol. 32, No. 3, 2003, pp. 451-472.

[14] T. M. Gindy, H. M. El-Hawary, M. S. Salim and M. El-Kady, “A Chebyshev Approximation for Solving Optimal Control Problems,” Computers & Mathematics with Applications, Vol 29, No. 6, 1995, pp 35-45. doi:10.1016/0898-1221(95)00005-J

[15] H. Jaddu, “Direct Solution of Nonlinear Optimal Control Using Quasilinearization and Chebyshev Polynomials Problems,” Journal of the Franklin Institute, Vol. 339, No. 4-5, 2002, pp. 479-498.

[16] G. N. Saridis, C. S. G. Lee, “An Approximation Theory of Optimal Control for Trainable Manipulators,” IEEE Transations on Systems, Vol. 9, No. 3, 1979, pp. 152-159.

[17] P. Balasubramaniam, J. A. Samath and N. Kumaresan, “Optimal Control for Nonlinear Singular Systems with Quadratic Performance Using Neural Networks,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp. 1535-1543.

[18] T. Cheng, F. L. Lewis, M. Abu-Khalaf, “A Neural Network Solution for Fixed-Final Time Optimal Control of Nonlinear Systems,” Automatica, Vol. 43, No. 3, 2007, pp. 482-490.

[19] P. V. Medagam and F. Pourboghrat, “Optimal Control of Nonlinear Systems Using RBF Neural Network and Adaptive Extended Kalman Filter,” Proceedings of American Control Conference Hyatt Regency Riverfront, St. Louis, 10-12 June 2009, pp. 355-360.

[20] D. Luenberger, “Linear and Nonlinear Programming,” Kluwer Academic Publishers, Norwell, 1984.