Restarted State Parameterization Method for Optimal Control Problems
- Imam Javad University College, Yazd, Iran
- Department of Mathematics, Yazd University, Yazd, Iran
In this paper we introduce an efficient algorithm based on state parameterization method to solve optimal control problems and Van Der Pol oscillator. In fact, state variable can be considered as linear combination of polynomials with unknown coefficients. Using this method, an optimal control problem breaks down into an optimization and will be solved via mathematical programming techniques. By this algorithm, the control and state variables can be approximated as a function of time. Finally, some numerical examples are presented to show the validity and efficiency of the proposed method.
- Optimal control problems
- State parameterization method
- Mathematical programming techniques
- Van Der Pol oscillator.
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, The mathematical theory of optimal processes, , (1962)
R. Bellman, Dynamic Programming, University Press, Princeton, N J (1957)
W. H. Fleming, C. J. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, (1975)
W. H. Fleming, H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, (2006)
D. E. Kirk, Optimal Control Theory An Introduction, Prentice-Hall, Englewood Cliffs (1970)
A. B. Pantelev, A. C. Bortakovski, T. A. Letova, Some Issues and Examples in Optimal Control, MAI Press, Moscow , (in Russian) (1996)
H. J. Kushner, P. G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, 2nd Edition, Springer-Verlag, New York, NY (2001)
H. J. Kushner, A partial history of the early development of continuous-time nonlinear stochastic systems theory, Automatica , 50.2 (2014), 303-334.
A. Jajarmi, N. Pariz, S. Effati, A. V. Kamyad, Infinite horizon optimal control for nonlinear interconnected Large-Scale dynamical systems with an application to optimal attitude control, Asian. J. Control., 15 (2013), 1-12.
B. Kafash, A. Delavarkhalafi, S. M. Karbassi, Application of variational iteration method for Hamilton–Jacobi–Bellman equations, Appl. Math. Modell. , 37 (2013), 3917–3928.
H. M. Jaddu, Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials, J. Franklin Inst., 339 (2002), 479-498.
K. Maleknejad, H. Almasieh, Optimal control of Volterra integral equations via triangular functions, Math. Comput. Modell., 53 (2011), 1902–1909.
E. Hesameddini, A. Fakharzadeh Jahromi, M. Soleimanivareki, H. Alimorad, Differential transformation method for solving a class of nonlinear optimal control problems, The Journal of Mathematics and Computer Science , 5(1) (2012), 67-74.
H. M. Jaddu, Numerical Methods for solving optimal control problems using chebyshev polynomials, PhD thesis, School of Information Science , Japan Advanced Institute of Science and Technology (1998)
B. Kafash, A. Delavarkhalafi, S. M. Karbassi, Application of Chebyshev polynomials to derive efficient algorithms for the solution of optimal control problems, Sci. Iran. , 19 (2012), 795-805.
T. Akman, K. Bülent, Variational time discretization methods for optimal control problems governed by diffusion–convection–reaction equations, Journal of Computational and Applied Mathematics, 272 (2014), 41-56.
Chongyang Liu, Ryan Loxton, Kok Lay Teo, A computational method for solving time-delay optimal control problems with free terminal time, Systems & Control Letters , 72 (2014), 53-60.
P. Chris, et al., Consistent approximation of a nonlinear optimal control problem with uncertain parameters, Automatica , (2014)
H. Sadeghian, et al., On the linear-quadratic regulator problem in one-dimensional linear fractional stochastic systems, Automatica , 50.1 (2014), 282-286.
E. Tohidi, H. Saberi Nik, A Bessel collocation method for solving fractional optimal control problems, Applied Mathematical Modelling , (2014)
B. D. Craven, Control and Optimization, Chapman & Hall, London (1995)
K. L. Teo, L. S. Jennings, H. W. J. Lee, V. Rehbock, The control parametrization enhancing transform for constrained optimal control problems, J. Austral. Math. Soc. Ser. B., 40 (1999), 314–335.
H. W. J. Lee, K. L. Teo, L. S. Jennings, H. W. J. Lee, V. Rehbock, Control parametrization enhancing technique for time optimal control problem, Dynamical Syst. Appl., 6 (1997), 243–261.
J. Vlassenbroeck, A chebyshev polynomial method for optimal control with state constraints, Automatica, 24(4) (1988), 499-506.
R. Van Dooren, J. Vlassenbroeck, Chebyshev series solution of the controlled Duffing oscillator, J. Comput. Phys., 47 (1982), 321–329
S. E. El-Gendi, Chebyshev solution of differential, integral, and integro-differential equations, Comput. J., 12 (1969), 282–287.
T. M. El-Gindy, H. M. El-Hawary, M. S. Salim, M. El-Kady, A Chebyshev approximation for solving opimal control problems, Comput. Math. Appl., 29 (1995), 35–45.
M. El-Kady, Elsayed M. E. Elbarbary., A Chebyshev expansion method for solving nonlinear optimal control problems, Applied Mathematics and Computation, 129 (2002), 171–182.
H. H. Mehne, A. Hashemi Borzabadi, A numerical method for solving optimal control problems using state parametrization, Numer Algor., 42 (2006), 165–169.
P. A. Frick, D. J. Stech, Solution of optimal control problems on a parallel machine using the Epsilon method, Optim. Control Appl. Methods., 16 (1995), 1–17.
B. Kafash, A. Delavarkhalafi, S. M. Karbassi, Numerical Solution of Nonlinear Optimal Control Problems Based on State Parametrization, Iranian Journal of Science and Technology, 35 (A3) (2012), 331-340.
B. Kafash, A. Delavarkhalafi, S. M. Karbassi, K. Boubaker, A Numerical Approach for Solving Optimal Control Problems Using the Boubaker Polynomials Expansion Scheme, Journal of Interpolation and Approximation in Scientific Computing , 2014 (2014), 1-18.