International Journal of Nonlinear Analysis and Applications

A novel scheme for solving multi-delay fractional optimal control problems

Document Type : Research Paper

Author

Department of Mathematics, Payame Noor University (PNU), Tehran, Iran

Abstract

In this paper, we consider the problems of suboptimal control for a class of fractional-order optimal control problems with multi-delay argument. The fractional derivative in these problems is in the Caputo sense. To solve the problem, first by a suitable approximation, we replace the Caputo derivative to integer order derivative. The optimal control law consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence, is obtained by a sensitivity approach. The feed back term is determined by solving Riccati matrix differential equation. By using a finite sum of the series, we can obtain a suboptimal control law. Finally, numerical results are included to demonstrate the validity and applicability of the present technique.

Keywords

[1] A. Alizadeh and S. Effati, Numerical schemes for fractional optimal control problems, J. Dyn. Sys. Meas. Control , 139(8) (2017) 081002.
[2] O. P. Agrawal and D. A. Baleanu, Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems, J. Vib. Control , 13(9-10) (2007) 1269-1281.
[3] M. Basin and J. Rodriguez-Gonzalez, Optimal control for linear systems with multiple time delay in control input, IEEE Trans. Autom. control , 51(1) (2006) 91-97.
[4] T. Binazadeh and M. Yousefi, Designing a cascade-control structure using fractional-order controllers: timedelay fractional-order proportional-derivative controller and fractional-order sliding-mode controller, J. Eng. Mech., 143(7) (2017) 04017037.
[5] A. H. Bhrawy and S. S. Ezz-Eldien, A new Legendre operational technique for delay fractional optimal control problems, Calcolo, 53 (2016) 521-543.
[6] M. Dadkhah and M. H. Farahi, Optimal control of time delay systems via hybrid of block pulse functions andorthogonal Taylor series, Int. J. Appl. Comput. Math. , 2(1) (2016) 137-152.
[7] S. Effati, S. A. Rakhshan and S. Saqi, Formulation of Euler-Lagrange Equations for Multidelay Fractional Optimal Control Problems, J. Comput Nonlinear Dyn., 13(1) (2018) 061007.
[8] M. Ghanes, J. Deleon and J. P. Barbot, Simultaneous observation and identification for nonlinear systems under unknown time-varying delays, J. Franklin. Inst., 353(10) (2016) 2305-2318.
[9] L. Gollmann and H. Maurer, Theory and applications of optimal control problems with mul-tiple time delays, J. Ind. Manag Optim., 10(2) (2014) 413-441.
[10] K. Gu, V. L. Kharitonov and J. Chen, Stability of time-delay systems, Birkhauser, Boston. 2003.
[11] S. Hosseinpour, A. Nazemi and E. Tohidi, M¨untz-Legendre spectral collocation method for solving delay fractional optimal control problems, J Comput. Appl. Math., 351(1) (2019) 344-363.
[12] G. Hwang and M. Y. Chen, Suboptimal control of linear time-varying multi-delay systems via shifted Legendre polynomials, Int. J. Syst. Sci., 16(12) (1985) 1517-1537.
[13] P. Ignaciuk, DARE solutions for LQ optimal and suboptimal control of systems with multiple input–output delays, J. Franklin Inst., 353(5) (2016) 974-991.
[14] G. L. Kharatishvili, The maximum principle in the theory of optimal process with time-lags, Doki Akad Nauk SSSR., 136 (1961) 39-42.
[15] F. Kheyrinataj and A. Nazemi, M¨untz–Legendre neural network construction for solving delay optimal control problems of fractional order with equality and inequality constraints, Soft Comput., 24 (2020) 9575-9594.
[16] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, in: North-Holland Mathematics Studies, Elsevier Science B.V: Amsterdam, 2006.
[17] H. W. J. Lee and K. H. Wong, Semi-infinite programming approach to nonlinear time-delayed optimal control problems with linear continuous, Optimization Methods and Software., 21(5) (2006) 679-691.
[18] Y. Liang, H. Zhang, K. Zhang and R. Wang, A novel neural network discrete-time optimal control design for nonlinear time-delay systems using adaptive critic designs, Optim. Control Appl. Meth., 41(3) (2020) 748-764.
[19] M. Malek-Zavarei, Near-Optimum design of nonstationary linear systems with state and control delays, J. Optim. Theor. Appl., 30 (1980) 73-88.
[20] M. Malek-Zavarei and M. Jamshidi, Time delay systems analysis, optimization and applications, North-Holland systems and control series, 1987.
[21] M. Maleki and I. Hashim, Adaptive pseudospectral methods for solving constrained linear and nonlinear time-delay optimal control problems, J. Franklin Inst., 351 (2014) 811-839.
[22] H. R. Marzban and S. M. Hoseini, An efficient discretization scheme for solving nonlinear optimal control problems with multiple time delays, Optim. Control Appl. Meth., 37(4) (2016) 682-707.
[23] H. R. Marzban and M. Razzaghi, Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials, J. Franklin Inst., 341 (2004) 279-293.
[24] H. R. Marzban and H. Pirmoradian, A novel approach for the numerical investigation of optimal control problems containing multiple delays, Optim. Control Appl. Meth., 53 (2018) 189-213.
[25] S. M. Mirhosseini-Alizamini, S. Effati and A. Heydari, An iterative method for suboptimal control of linear timedelayed systems, Syst. Control Lett., 82 (2015) 40-50.
[26] S. M. Mirhosseini-Alizamini, S. Effati and A. Heydari, Solution of linear time-varying multi-delay systems via variational iteration method, J. Math. Comput. Sci., 16 (2016) 282-297.
[27] S. M. Mirhosseini-Alizamini, Numerical Solution of the Controlled Harmonic Oscillator by Homotopy Perturbation Method, Control Optim. Appl. Math., 2(1) (2017) 77-91.
[28] S. M. Mirhosseini-Alizamini and S. Effati, An iterative method for suboptimal control of a class of nonlinear timedelayed systems, Int. J. Control., 92(12) (2019) 2869-2885. https://doi.org/10.1080/00207179.2018.1463456
[29] S. M. Mirhosseini-Alizamini, Solving linear optimal control problems of the time-delayed systems by Adomian decomposition method, Iranian J. Num. Anal. Optim., 9(2) (2019) 165-183. https://doi.org/10.22067/ijnao. v9i2.74000
[30] R. Mohammadzadeh and M. Lakestani, Optimal control of linear time-delay systems by a hybrid of block-pulse functions and biothogonal cubic Hermit spline multiwavelets, Optim. Control Appl. Meth., 39(1) (2018) 357-376.
[31] L. Moradi, F. Mohammadi and D. A. Baleanu, Direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets, J. Vib. Control., 25 (2018) 310-324.
[32] S. Nemati, P. M. Lima and S. Sedaghat, Legendre wavelet collocation method combined with the Gauss–Jacobi quadrature for solving fractional delay-type integro-differential equations, Appl. Numerical Math., 149 (2020) 99- 112.
[33] S. Pooseh, R. Almeida and D. F. M Torres, Numerical approximations of fractional derivatives with applications,Asian. J. Control., 15(5) (2013) 698-712.
[34] S. Pooseh, R. Almeida and D. F. M. Torres, Fractional optimal control problems with free terminal time, J. Ind. Manag Optim., 10(2) (2014) 363-381.
[35] S. Sabermahani, Y. Ordokhani and S. A. Yousefi, Fibonacci wavelets and their applications for solving two class of time-varying delay problems, Optim. Control Appl. Meth., 41(2) (2020) 395-416.
[36] S. Soradi-Zeid and M. Yousefi, A New Reconstraction Approach of Riccati Differential Equation for Solving a Class of Fractional Optimal Control Problems, Asian. Res. J. Math., 1(2) (2016) 1-12.
[37] G. Y. Tang, R. Dong and H. W. Gao, Optimal sliding mode control for nonlinear systems with time-delay, Nonlinear Anal. Hyb. sys., 2 (2008) 891-899.
[38] G. Y. Tang, Suboptimal control for nonlinear systems: a successive approximation approach, Sys. Control Lett., 54 (2005) 429-434.
[39] V. Tarasov, Fractional dynamics: application of fractional calculus to dynamics of particles, fields and media, Springer-Verlag Berlin Heidelberg, New York, 2010.
[40] R. J. Vanderbei and D. F. Shanno, An interior-point algorithm for nonconvex nonlinear programming, Comput. Optim. Appl., 13 (1999) 231-252.
[41] A. Wachter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for largescale nonlinear programming, Math. Program., 106 (2006) 25-57.
[42] Q. Wang, F. Chen F. Huang, Maximum principle for optimal control problem of stochastic delay differential equations driven by fractional Brownian motions, Optim. Control Appl. Meth., 37(1) (2016) 90-107.
[43] W. Witayakiattilerd, Optimal regulation of impulsive fractional differential equation with delay and application to nonlinear fractional Heat equation, J. Math. Res., 5(2) (2013) 94.
[44] M. Yousefi, T. Binazadeh, Delay-independent sliding mode control of time-delay linear fractional order systems, Trans. Institute Meas. Control., 10 (2016) 1-16.
[45] M. Yavari and A. Nazemi, An efficient numerical scheme for solving fractional infinite-horizon optimal control problems, ISA Transactions., 94 (2019) 108-118.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 1
March 2022
Pages 2095-2113
  • Receive Date: 11 March 2021
  • Revise Date: 25 June 2021
  • Accept Date: 28 June 2021