# Optimal Control of Nonlinear Systems Using the Shifted Legendre Polynomials

Keywords:
Non-linear systems, Legendre Polynomials, Optimal Control, Numerical Methods

### Abstract

A numerical technique based on Legendre Polynomials for finding optimal control of nonlinear systems with quadratic performance index is presented. An operational matrix of integration and product matrix are introduced and are used to reduce the nonlinear differential equations to the solution of nonlinear algebraic equations. The optimal solution of two classes of first and second order nonlinear systems is considered. In the case of second-order nonlinear systems, a new approach is introduced to find the optimal solution. In both cases, numerical examples are given and compared with the Taylor polynomial to confirm the accuracy of the proposed method.### References

[1] S. P. Banks, Mathematical theories of nonlinear systems, Prentice Hall, 1988.

[2] J. Slotine, W. Li, Applied nonlinear control, Prentice Hall, 1991.

[3] M. Samavat, A. K. Sedeegh , S. P. Banks, On the approximation of pseudo linear systems by linear time varying systems, Int. J. Eng. , Volume 17, No 1, pp. 29-32, 2004.

[4] W. L. Chen, Y. P. Shih, Parameter estimation of bilinear systems via Walsh functions, Journal of the Franklin Institute, Volume 305, Issue 5, pp. 249-257, 1978.

[5] K. Maleknejad, M. Shahrezaee, H. Khatami, Numerical solution of integral equations system of the second kind by block–pulse functions, Applied Mathematics and Computation, 166, pp. 15–24, 2005.

[6] X. T. Wang, Yuan Min Li, Numerical solutions of integro-differential systems by hybrid of general block-pulse functions and the second Chebyshev polynomials, Applied Mathematics and Computation, 209, pp. 266–272, 2009.

[7] H. Jaddu and E. Shimemura, Solution of nonlinear optimal control problem using Chebyshev polynomials. In Proceeding of the 2nd Asian ControlConference, Seod, Korea, pages 1-417-420, 1997.

[8] S. G. Mouroutsos, P. D. Sparis , Taylor series approach to system identification, analysis and optimal control, Journal of the Franklin Institute, Volume 319, Issue 3, pp. 359-371, 1985.

[9] M. Gülsu, M. Sezer, A Taylor polynomial approach for solving differential-difference equations, Journal of Computational and Applied Mathematics 186, pp. 349–364, 2006

[10] S.Yalcinbas, Taylor polynomial solution of nonlinear Volterra–Fredholm integral equations, Appl. Math. Comput. 127(2002) 195–206.

[11] M.L. Nagurka, V. Yen, Fourier-based optimal control of nonlinear dynamic systems, Trans. ASME J. Dyn. Syst. Meas. Control 112 (1) (1990) 17–26.

[12] B. A. Ardekani, A. Keyhani, Identification of non-linear systems using the exponential Fourier series, Int. J. Control, VOL. 50, No. 4, pp. 1553-1558, 1989.

[13] B. A. Ardekani, M. Samavat, H. Rahmani, Parameter identification of time-delay systems via exponential Fourier series, Int. J. Sys. Sci, Vol. 22, No. 7, pp. 1301-1306, 1991.

[14] M. Samavat, A. J. Rashidi, A new algorithm for Analysis and Parameter Identification of time varying systems, ACC proceedings, 1995.

[15] R. Ebrahimi, M. Samavat, M. A. Vali, A. A. Gharavisi, Application of Fourier series direct method to the optimal control of singular systems, ICGST –ACSE Journal, Volume 7, Issue 2, 2007.

[16] S.Dong-Her, K.Fan-Chu, Analysis and parameter estimation of a scaled system via shifted Legendre polynomials, International Journal of Systems Science, Volume17, No 3, pp.401- 408,1986

[17] M. Razzaghi, S. Yousefi, Legendre Wavelets method for the Solution of Nonlinear Problems in the calculus of Variations, Mathematical and Computer Modelling 34, pp. 45-54, 2001.

[18] J.Wang, S.Wang, Approximation of nonlinear functional via general ortoghonal polynominals and application to control problems, Int. J. Sys. Sci. Vol. 23, No. 8, pp. 1261-1276, 1992.

[2] J. Slotine, W. Li, Applied nonlinear control, Prentice Hall, 1991.

[3] M. Samavat, A. K. Sedeegh , S. P. Banks, On the approximation of pseudo linear systems by linear time varying systems, Int. J. Eng. , Volume 17, No 1, pp. 29-32, 2004.

[4] W. L. Chen, Y. P. Shih, Parameter estimation of bilinear systems via Walsh functions, Journal of the Franklin Institute, Volume 305, Issue 5, pp. 249-257, 1978.

[5] K. Maleknejad, M. Shahrezaee, H. Khatami, Numerical solution of integral equations system of the second kind by block–pulse functions, Applied Mathematics and Computation, 166, pp. 15–24, 2005.

[6] X. T. Wang, Yuan Min Li, Numerical solutions of integro-differential systems by hybrid of general block-pulse functions and the second Chebyshev polynomials, Applied Mathematics and Computation, 209, pp. 266–272, 2009.

[7] H. Jaddu and E. Shimemura, Solution of nonlinear optimal control problem using Chebyshev polynomials. In Proceeding of the 2nd Asian ControlConference, Seod, Korea, pages 1-417-420, 1997.

[8] S. G. Mouroutsos, P. D. Sparis , Taylor series approach to system identification, analysis and optimal control, Journal of the Franklin Institute, Volume 319, Issue 3, pp. 359-371, 1985.

[9] M. Gülsu, M. Sezer, A Taylor polynomial approach for solving differential-difference equations, Journal of Computational and Applied Mathematics 186, pp. 349–364, 2006

[10] S.Yalcinbas, Taylor polynomial solution of nonlinear Volterra–Fredholm integral equations, Appl. Math. Comput. 127(2002) 195–206.

[11] M.L. Nagurka, V. Yen, Fourier-based optimal control of nonlinear dynamic systems, Trans. ASME J. Dyn. Syst. Meas. Control 112 (1) (1990) 17–26.

[12] B. A. Ardekani, A. Keyhani, Identification of non-linear systems using the exponential Fourier series, Int. J. Control, VOL. 50, No. 4, pp. 1553-1558, 1989.

[13] B. A. Ardekani, M. Samavat, H. Rahmani, Parameter identification of time-delay systems via exponential Fourier series, Int. J. Sys. Sci, Vol. 22, No. 7, pp. 1301-1306, 1991.

[14] M. Samavat, A. J. Rashidi, A new algorithm for Analysis and Parameter Identification of time varying systems, ACC proceedings, 1995.

[15] R. Ebrahimi, M. Samavat, M. A. Vali, A. A. Gharavisi, Application of Fourier series direct method to the optimal control of singular systems, ICGST –ACSE Journal, Volume 7, Issue 2, 2007.

[16] S.Dong-Her, K.Fan-Chu, Analysis and parameter estimation of a scaled system via shifted Legendre polynomials, International Journal of Systems Science, Volume17, No 3, pp.401- 408,1986

[17] M. Razzaghi, S. Yousefi, Legendre Wavelets method for the Solution of Nonlinear Problems in the calculus of Variations, Mathematical and Computer Modelling 34, pp. 45-54, 2001.

[18] J.Wang, S.Wang, Approximation of nonlinear functional via general ortoghonal polynominals and application to control problems, Int. J. Sys. Sci. Vol. 23, No. 8, pp. 1261-1276, 1992.

Published

2014-03-10

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*Majlesi Journal of Electrical Engineering*,

*8*(2). Retrieved from http://mjee.iaumajlesi.ac.ir/index/index.php/ee/article/view/1167

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