Advertisement

Müntz–Legendre neural network construction for solving delay optimal control problems of fractional order with equality and inequality constraints

  • Farzaneh KheyrinatajEmail author
  • Alireza Nazemi
Methodologies and Application
  • 14 Downloads

Abstract

In this paper, an artificial intelligence approach using neural networks is described to solve a class of delay optimal control problems of fractional order with equality and inequality constraints. In the proposed method, a functional link neural network based on the Müntz–Legendre polynomial is developed. The problem is first transformed into an equivalent problem with a fractional dynamical system without delay, using a Padé approximation. According to the Pontryagin’s minimum principle for optimal control problems of fractional order and by constructing an error function, the authors then define an unconstrained minimization problem. The authors use trial solutions for the states, Lagrange multipliers and control functions where these trial solutions are constructed by a single-layer Müntz–Legendre neural network model. The authors then exploit an unconstrained optimization scheme for adjusting the network parameters (weights and bias) and to minimize the computed error function. Some numerical examples are given to illustrate the effectiveness of the proposed method.

Keywords

Müntz–Legendre polynomial Functional link neural network Delay optimal control problems Fractional order Padé approximation Pontryagin minimum principle Error function Optimization scheme 

Notes

Funding

This study was not funded by any grant.

Compliance with ethical standard

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

References

  1. Agrawal OP (2004) A general formulation and solution scheme for fractional and optimal control problems. Nonlinear Dyn 38:323–337MathSciNetzbMATHCrossRefGoogle Scholar
  2. Agrawal OP (2008) A formulation and a numerical scheme for fractional optimal control problems. J Vib Control 14:1291–1299MathSciNetzbMATHCrossRefGoogle Scholar
  3. Alipour M, Rostamy D, Baleanu D (2012) Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices. J Vib Control 19(16):2523–2540MathSciNetzbMATHCrossRefGoogle Scholar
  4. Badalyan GV (1956) Generalization of Legendre polynomials and some of their applications. Rus Armen Summ 9:3–22Google Scholar
  5. Balakrishnan D, Puthusserypady S (2005) Multilayer perceptrons for the classification of brain computer interface data. In: Bioengineering, proceedings of the northeast conference, pp 118–119Google Scholar
  6. Baleanu D, Defterli O, Agrawal OP (2009) A central difference numerical scheme for fractional optimal control problems. J Vib Control 15(4):583–597MathSciNetzbMATHCrossRefGoogle Scholar
  7. Banks HT (1979) Approximation of nonlinear functional differential equation control systems. J Optim Theory Appl 29(3):383–408MathSciNetzbMATHCrossRefGoogle Scholar
  8. Banks HT, Burns JA (1978) Hereditary control problems: numerical method based on averaging approximations. SIAM J Control Optim 16:169–208MathSciNetzbMATHCrossRefGoogle Scholar
  9. Bazaraa MS, Sherali HD, Shetty CM (2006) Nonlinear programming-theory and algorithms, 3rd edn. Wiley, Hoboken, NJzbMATHCrossRefGoogle Scholar
  10. Berg J, Nyström K (2018) A unified deep artificial neural network approach to partial differential equations in complex geometries. Neurocomputing 317:28–41CrossRefGoogle Scholar
  11. Bhrawy AH, Ezz-Eldien SS (2016) A new Legendre operational technique for delay fractional optimal control problems. Calcolo 53:521–543MathSciNetzbMATHCrossRefGoogle Scholar
  12. Bohannan GW (2008) Analog fractional order controller in temperature and motor control applications. J Vib Control 14(9–10):1487–1498MathSciNetCrossRefGoogle Scholar
  13. Borwein P, Erdélyi T, Zhang J (1994) Müntz systems and orthogonal Müntz–Legendre polynomials. Trans Am Math Soc 342:523–542zbMATHGoogle Scholar
  14. Bozzo DG, Kristjanpoller W (2019) An adaptive forecasting approach for copper price volatility through hybrid and non-hybrid models. Appl Soft Comput 74:466–478CrossRefGoogle Scholar
  15. Carini A, Sicuranza GL (2014) Fourier nonlinear filters. Signal Process 94:183–194zbMATHCrossRefGoogle Scholar
  16. Chakravarty S, Dash PKA (2012) PSO based integrated functional link net and interval type-2 fuzzy logic system for predicting stock market indices. Appl Soft Comput 12(2):931–941 noise processes, Signal Process, 90(3) (2010) 834–847CrossRefGoogle Scholar
  17. Cheney EW (1982) Introduction to approximation theory. AMS Chelsea Publishing, ProvidencezbMATHGoogle Scholar
  18. Comminiello D, Scarpiniti M, Scardapane S, Parisi R, Uncini A (2015) Improving nonlinear modeling capabilities of functional link adaptive filters. Neural Netw 69:51–59zbMATHCrossRefGoogle Scholar
  19. Cuyt A (1999) How well can the concept of Padé approximant be generalized to the multivariate case? J Comput Appl Math 105:25–50MathSciNetzbMATHCrossRefGoogle Scholar
  20. Dadkhah M, Farahi MH (2016) Optimal control of time delay systems via hybrid of block-pulse functions and orthonormal Taylor series. Int J Appl Comput Math 2:137–152MathSciNetzbMATHCrossRefGoogle Scholar
  21. Dankovic B, Jovanovic Z, Milojkovic M (2005) Dynamic systems identification using Müntz function neural networks with distributed dynamics. In: TELSIKS 2005–2005 uth international conference on telecommunication in modernsatellite, Cable and Broadcasting Services, 28–30 SeptemberGoogle Scholar
  22. Dehuri S, Cho SB (2010) A comprehensive survey on functional link neural networks and an adaptive PSO-BP learning for CFLNN. Neural Comput Appl 19(2):187–205CrossRefGoogle Scholar
  23. Effati S, Pakdaman M (2010) Artificial neural network approach for solving fuzzy differential equations. Inf Sci 180:1434–1457MathSciNetzbMATHCrossRefGoogle Scholar
  24. Effati S, Pakdaman M (2010) Optimal control problem via neural networks. Neural Comput Appl 23:2093–2100CrossRefGoogle Scholar
  25. Effati S, Rakhshan SA, Saqi S (2018) Formulation of Euler-Lagrange equations for multidelay fractional optimal control problems. J Comput Nonlinear Dyn 13:061007CrossRefGoogle Scholar
  26. Ejlali N, Hosseini SM (2017) A pseudospectral method for fractional optimal control problems. J Optim Theory Appl 174:83–107MathSciNetzbMATHCrossRefGoogle Scholar
  27. Elnagar GN, Kazemi MA (2001) Numerical solution of time-delayed functional differential equation control systems. J Comput Appl Math 130:75–90MathSciNetzbMATHCrossRefGoogle Scholar
  28. Esmaeili S, Shamsi M, Luchko Y (2011) Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials. Comput Math Appl 62:918–929MathSciNetzbMATHCrossRefGoogle Scholar
  29. Facchinei F, Jiang H, Qi L (1999) A smoothing method for mathematical programs with equilibrium constraints. Math Program 35:107–134MathSciNetzbMATHCrossRefGoogle Scholar
  30. Galaviz-Aguilar JA, Roblin P, Cardenas-Valdez JR, Z-Flores E, Trujillo L, Nunez-Perez JC, Schutze O (2019) Comparison of a genetic programming approach with ANFIS for power amplifier behavioral modeling and FPGA implementation. Soft Comput 23:2463–2481CrossRefGoogle Scholar
  31. Ghasemi S, Nazemi AR, Hosseinpour S (2017) Nonlinear fractional optimal control problems with neural network and dynamic optimization schemes. Nonlinear Dyn 89:2669–2682MathSciNetzbMATHCrossRefGoogle Scholar
  32. Ghomanjani F, Farahi MH, Gachpazan M (2014) Optimal control of time-varying linear delay systems based on the Bezier curves. Comput Appl Math 33:687–715MathSciNetzbMATHCrossRefGoogle Scholar
  33. Haddadi N, Ordokhani Y, Razzaghi M (2012) Optimal control of delay systems by using a hybrid functions approximation. J Optim Theory Appl 153:338–356MathSciNetzbMATHCrossRefGoogle Scholar
  34. He S, Reif K, Unbehauen R (2000) Multi-layer neural networks for solving a class of partial differential equations. Neural Netw 13:385–396CrossRefGoogle Scholar
  35. Hosseinpour S, Nazemi A (2017) A collocation method via block-pulse functions for solving delay fractional optimal control problems. IMA J Math Control Inf 34:1215–1237MathSciNetzbMATHGoogle Scholar
  36. Hosseinpour S, Nazemi A, Tohidi E (2019) Müntz-Legendre spectral collocation method for solving delay fractional optimal control problems. J Comput Appl Math 351:344–363MathSciNetzbMATHCrossRefGoogle Scholar
  37. Ibrahim D (2016) An overview of soft computing. Proced Comput Sci 102:34–38CrossRefGoogle Scholar
  38. Jafarian A, Rostami F, Golmankhaneh AK, Baleanu D (2017) Using ANNs approach for solving fractional order Volterra integro-differential equations. Int J Comput Intell Syst 10:470–480CrossRefGoogle Scholar
  39. Jajarmi A, Baleanu D (2017) Suboptimal control of fractional-order dynamic systems with delay argument. J Vib Control 24:2430–2446MathSciNetzbMATHCrossRefGoogle Scholar
  40. Jarad F, Maraaba T, Baleanu D (2010) Fractional variational principles with delay within Caputo derivatives. Rep Math Phys 65(1):17–28MathSciNetzbMATHCrossRefGoogle Scholar
  41. Jarad F, Maraaba T, Baleanu D (2010) Fractional variational optimal control problems with delayed arguments. Nonlinear Dyn 62(3):609–614MathSciNetzbMATHCrossRefGoogle Scholar
  42. Kilbas AAA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, AmsterdamzbMATHGoogle Scholar
  43. Kim JW, Park BJ, Yoo H, Lee JH, Lee JM (2018) Deep reinforcement learning based finite-horizon optimal tracking control for nonlinear system. IFAC PapersOnLine 51:257–262CrossRefGoogle Scholar
  44. Kumar M, Yadav N (2011) Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: a survey. Comput Math Appl 62:3796–3811MathSciNetzbMATHCrossRefGoogle Scholar
  45. Lazoa MJ, Krumreich CE (2014) The action principle for dissipative systems. J Math Phys 55:122902MathSciNetzbMATHCrossRefGoogle Scholar
  46. Lee KY, El-Sharkawi MA (2008) Modern heuristic optimization techniques: theory and applications to power systems. IEEE Press Series on Power EngineeringGoogle Scholar
  47. Lee S, Ha J, Zokhirova M, Moon H, Lee J (2018) Background information of deep learning for structural engineering. Arch Comput Methods Eng 25(1):121–129MathSciNetzbMATHCrossRefGoogle Scholar
  48. Li M, Liu J, Jiang Y, Feng W (2012) Complex-Chebyshev functional link neural network behavioral model for broadband wireless power amplifiers. IEEE Trans Microw Theory Techn 60(6):1979–1989CrossRefGoogle Scholar
  49. Maleki M, Hashim I (2014) Adaptive pseudospectral methods for solving constrained linear and nonlinear time-delay optimal control problems. J Franklin Inst 351:811–839MathSciNetzbMATHCrossRefGoogle Scholar
  50. Mall S, Chakraverty S (2015) Numerical solution of nonlinear singular initial value problems of emden-fowler type using Chebyshev neural network method. Neurocomputing 149:975–982CrossRefGoogle Scholar
  51. Mall S, Chakraverty S (2016) Hermite functional link neural network for solving the Van der Pol-Duffing oscillator equation. Neural Comput 28:1574–1598MathSciNetzbMATHCrossRefGoogle Scholar
  52. Mall S, Chakraverty S (2016) Application of Legendre neural network for solving ordinary differential equations. Appl Soft Comput 43:347–356CrossRefGoogle Scholar
  53. Mall S, Chakraverty S (2017) Single layer Chebyshev neural network model for solving elliptic partial differential equations. Neural Process Lett 45(3):825–840CrossRefGoogle Scholar
  54. Mall S, Chakraverty S (2018) Artificial neural network approach for solving fractional order initial value problems. arXiv:1810.04992v2
  55. Marzban HR, Pirmoradian H (2018) A direct approach for the solution of nonlinear optimal control problems with multiple delays subject to mixed state-control constraints. Appl Math Model 53:189–213MathSciNetCrossRefGoogle Scholar
  56. Mokhtary P, Ghoreishi F, Srivastava HM (2016) The Müntz–Legendre tau method for fractional differential equations. Appl Math Model 40:671–684MathSciNetCrossRefGoogle Scholar
  57. Moradi L, Mohammadi F, Baleanu D (2018) A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets. J Vib Control 25:310–324MathSciNetCrossRefGoogle Scholar
  58. Naderpour H, Mirrashid M (2019) Shear failure capacity prediction of concrete beam-column joints in terms of ANFIS and GMDH. In: Practice periodical on structural design and construction 24:04019006.  https://doi.org/10.1061/(asce)sc.1943-5576.0000417 CrossRefGoogle Scholar
  59. Nazemi A, Effati S (2013) An application of a merit function for solving convex programming problems. Comput Ind Eng 66:212–221CrossRefGoogle Scholar
  60. Negarchi N, Nouri K (2018) Numerical solution of Volterra-Fredholm integral equations using the collocation method based on a special form of the Müntz–Legendre polynomials. J Comput Appl Math 344:15–24MathSciNetzbMATHCrossRefGoogle Scholar
  61. Nocedal J, Wright S (2006) Numerical optimization, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  62. Palanisamy KR, Prasada RG (1983) Optimal control of linear systems with delays in state and control via Walsh functions. IEE Proc D Control Theory Appl 130:300–312zbMATHCrossRefGoogle Scholar
  63. Pan SH, Chen JS (2010) A semi smooth Newton method for the SOCCP based on a one-parametric class of SOC complementarity functions. Comput Optim Appl 45:59–88MathSciNetzbMATHCrossRefGoogle Scholar
  64. Pao Y (1989) Adaptive pattern recognition and neural networks. Addison-Wesley Publishing Company, Reading, MAzbMATHGoogle Scholar
  65. Patel V, Gandhi V, Heda S, George NV (2016) Design of adaptive exponential functional link network-based nonlinear filters. IEEE Trans Circuits Syst 63(9):1434–1442MathSciNetCrossRefGoogle Scholar
  66. Patra JC, Chin WC, Meher PK, Chakraborty G (2008) Legendre-FLANN-based nonlinear channel equalization in wireless communication system. In: IEEE international conference on systems, man and cybernetics, pp 1826–1831Google Scholar
  67. Peng H, Wang X, Zhang S, Chen B (2017) An iterative symplectic pseudospectral method to solve nonlinear state-delayed optimal control problems. Commun Nonlinear Sci Numer Simul 48:95–114MathSciNetCrossRefGoogle Scholar
  68. Peterson LE, Larine KV (2008) Hermite/Laguerre neural networks for classification of artificial fingerprints from optical coherence tomography. In: IEEE seventh international conference on machine learning and applications, pp 637–643Google Scholar
  69. Rahimkhani P, Ordokhani Y, Babolian E (2016) An efficient approximate method for solving delay fractional optimal control problems. Nonlinear Dyn 86:1649–1661MathSciNetzbMATHCrossRefGoogle Scholar
  70. Ricky TQ, Chen Y, Rubanova J, Bettencourt D (2018) Duvenaud, Neural Ordinary Differential Equations, 32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montral, CanadaGoogle Scholar
  71. Safaie E, Farahi MH, Ardehaie MF (2015) An approximate method for numerically solving multi-dimensional delay fractional optimal control problems by Bernstein polynomials. Comput Appl Math 34:831–846MathSciNetzbMATHCrossRefGoogle Scholar
  72. Spille C, Ewert SD, Kollmeier B, Meyer BT (2018) Predicting speech intelligibility with deep neural networks. Comput Speech Lang 48:51–66CrossRefGoogle Scholar
  73. Sweilam NH, Al-Ajami TM (2014) Legendre spectral-collocation method for solving some types of fractional optimal control problems. J Adv Res 6(3):393–403CrossRefGoogle Scholar
  74. Sweilam NH, Al-Ajami TM, Hoppe RHW (2013) Numerical solution of some types of fractional optimal control problems. Hindawi Publishing Corporation, LondonCrossRefGoogle Scholar
  75. Taslakyan AK (1984) Some properties of Legendre quasi polynomials with respect to a Müntz system. Rus Armen Summ Érevan Univ 2:179–189MathSciNetGoogle Scholar
  76. Trajkovic D, Nikolic V, Antic D, Nikolic S, Peric S (2013) Application of the hybrid bond graphs and orthogonal rational filters for sag voltage effect reduction. Elektronika ir Elektrotechnika 19Google Scholar
  77. Turut V, Güzel N (2013a) On solving partial differential equations of fractional order by using the variational iteration method and multivariate Padé approximations. Eur J Pure Appl Math 6:147–171MathSciNetzbMATHGoogle Scholar
  78. Turut V, Güzel N (2013b) Multivariate Padé approximation for solving nonlinear partial differential equations of fractional order. In: Abstracts and applied analysisGoogle Scholar
  79. Van Der Pol B, Bremmer H (1955) Operational calculus based on the two-sided Laplace integral. Cambridge University Press, LondonzbMATHGoogle Scholar
  80. Wang Q, Lu DC, Fang YY (2015) Stability analysis of impulsive fractional differential systems with delay. Appl Math Lett 40:1–6MathSciNetzbMATHCrossRefGoogle Scholar
  81. Wong KH, Jennings LS, Benyah F (2002) The control parametrization enhancing transform for constrained time-delayed optimal control problems. ANZIAM J 43(E):E154–E185MathSciNetzbMATHCrossRefGoogle Scholar
  82. Yin K, Zhao H, Lu L (2018) Functional link artificial neural network filter based on the q-gradient for nonlinear active noise control. J Sound Vib 435:205–217CrossRefGoogle Scholar
  83. Zhang L, Suganthan PN (2015) A comprehensive evaluation of random vector functional link networks. Inf Sci 367:1097–1105Google Scholar
  84. Ziaei E, Farahi MH (2018) The approximate solution of non-linear time-delay fractional optimal control problems by embedding process. IMA J Math Control InfGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesShahrood University of TechnologyShahroodIran

Personalised recommendations