The Numerical Solution of Nonlinear Optimal Control Problems by Using Operational Matrix of Bernstein Polynomials

Document Type : Original Article

Authors
Najmeh Ghaderi 1 ΒΆ
Mohammad Hadi Farahi 2
1 Department of Applied Mathematics, Factually of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran
2 Department of Applied Mathematics, Factually of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran, and The Center of Excellence on Modelling and Control Systems (CEMCS), Mashhad, Iran
10.22034/maco.2.1.2
Abstract
A numerical approach based on Bernstein polynomials is presented to unravel optimal control of nonlinear systems. The operational matrices of differentiation, integration and product are introduced. Then, these matrices are implemented to decrease the solution of nonlinear optimal control problem to the solution of the quadratic programming problem which can be solved with many algorithms and softwares. This method is easy to implement it with an accurate solution. Some examples are included to demonstrate the validity and applicability of the technique.

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Keywords

[1] O. P. Agrawal, General formulation for the numarical solution of optimal control problems, Int. J. Control. 20(2) (1989) 627–638.
[2] K. P. Badakhshan and A. V. Kamyad, Numerical solution of nonlinear optimal control problems using nolinear programming, AMC. 187 (2007) 1511–1519.
[3] M. V. Basin, P. Shi and D. C. Alvarez, Central suboptimal h∞ control design for nonlinear polynomial systems, Int. J. Systems Science. 42 (2011) 801–808.
[4] G. Bashein and M. Enns, Computation of optimal control by a method combining quasi-linearization and quadratic programming, Int. J. Control. 16(1) (1972) 177–187.
[5] M. Behroozifar, S. A. Yousefi and A. N. Ranjbar, Numerical solution of optimal control of time-varying Singular Systems via operational Matrices, Int. J. Engineering. 27(4) (2014) 523–532.
[6] M. I. Bhatti, Solution of fractional harmonic oscillator in a fractional B-poly basis, PTS. 2(2) (2014), 8–13.
[7] M. I. Bhatti and P. Bracken, Solutions of differential equations in a Bernstein polynomial basis, J. Comput. Appl. Math. 205 (2007) 272–280.
[8] T. Bullock and G. Franklin, A second order feedback method for optimal control computations, IEEE Trans. Autom. Control. 12(6) (1967) 666–673.
[9] M. Dehghan, S. A. Yousefi and K. Rashedi, Ritz-Galerkin method for solving an inverse heat conduction problem with a nonlinear source term via Bernstein multiscaling functions and cubic B spline function. Inverse Probl. Sci. En. 21(3) (2013) 500–523.
[10] M. R. Eslahchi and M. Dehghan, Application of Taylor series in obtaining the orthogonal operational matrix, Comput. Math. with Appl. 61(9) (2011) 2596–2604.
[11] H. Jaddu, Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev ploynomial, J. Franklin Inst. 339 (2002) 479–498.
[12] N. Ghaderi and M. H. Farahi, The numerical solution of some optimal control systems with constant and pantograph delays via Bernstein polynomials. IJMSI, 15(2) (2020) 163–181.
[13] M. H. Gonzaleza and M. V. Basina, Discrete time optimal control for stochastic nonlinear polynomial systems. Int. J. Gen. Syst., 43(3–4) (2014) 359–371.
[14] A. Kayedi-Bardeh, M. R. Eslahchi and M. Dehghan, A method for obtaining the operational matrix of the fractional Jacobi functions and applications. J. Vib. Control, 20(5) (2014) 736–748.
[15] T. T. Lee and Y. F. Chang, Parameter and optimal control of nonlinear systems via general orothogonal ploynomials. Analysis, Int. J. Control, 4(44) (1986) 1089–1102.
[16] S. Mashayekhi, Y. Ordokhani and Y. Razzaghi, Hybrid function approach for nonlinear constrained optimal control problem. Commun Nonlinear Sci Numer Simulat., 17 (2012) 1831–1843.
[17] B. M. Mohan and S. K. Kar, Optimal control of nonlinear systems via orthogonal function, Automation and Signals(ICEAS), Bhubaneswar, India., 28(4) (1994) 28–30.
[18] J. E. Rubio Control and optimization; the linear treatment of nonlinear problems, Manchester University Press, (1985).
[19] E. R. Pinch Optimal Control and the calculus of variation, Oxford University Press, (1993).
[20] A. Saadatmandi and M. Dehghan, A new operational matrix for solving fractionalorder differential equations. Comput. Math. with Appl. 59(3), (2010), 1326–1336.
[21] W. Shienyu, Convergence of Block pluse series approximation for optimal control problem, Int. J. Systems Science. 21(7) (1990) 1335–1368.
[22] J. Vlassenbroeck and R. V. Dooren, A Chebyshev technique for solving nonlinear optimal control problems, IEEE Trans. Automatic Control. 33(4) (1988) 333–340.
[23] G. A. Watson, Approximation theory and numerical methods. John Wiley & Sons, (1980)
[24] S. A. Yousefi and M. Behroozifar, Operational matrices of Bernstein polynomials and their applications, Int. J. Syst. Sci. 41, (2010), 709–716.
[25] S. A. Yousefi, M. Behroozifar and M. Dehghan, The operational matrices of Bernstein polynomial for solving the parabolic equation subject to specification of the mass, J. Comput. Appl. Math 235, (2011), 5272–5283.
[26] S. A. Yousefi, M. Behroozifar and M. Dehghan, Numerical solution of the nonlinear age-structured population models by using the operational matrices of Bernstein polynomials, Appl. Math Model. 36(3), (2012), 945–963.
 
Mathematical Analysis and Convex Optimization
Volume 2, Issue 1
June 2021
Pages 11-27