Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Neural Processing Letters
Published in: Neural Processing Letters 3/2020

29-09-2020

An Application of Generalized Fuzzy Hyperbolic Model for Solving Fractional Optimal Control Problems with Caputo–Fabrizio Derivative

Authors: Marzieh Mortezaee, Mehdi Ghovatmand, Alireza Nazemi

Published in: Neural Processing Letters | Issue 3/2020

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this paper we present a new approach for solving a class of fractional optimal control problems based on generalized fuzzy hyperbolic model. The fractional derivatives are described in the Caputo–Fabrizio sense. In order to solve this problem, the necessary optimality conditions associated to the fractional optimal control problem is first derived. The solution of these conditions is then approximated by fuzzy solution based on generalized fuzzy hyperbolic model. A learning algorithm is used to achieve the adjustable parameters of the obtained fuzzy solution. In order to confirm the efficiency and accuracy of the proposed approach, some illustrative examples are implemented.
previous article Deep Plot-Aware Generalized Matrix Factorization for Collaborative Filtering
next article Stochastic One-Step Training for Feedforward Artificial Neural Networks

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
Literature
1.
Sun H, Zhang Y, Baleanu D, Chen W (2018) Chen Y A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simulat 64:213–231MATH
2.
Pham VT, Vaidyanathan S, Volos C, Kapitaniak T (2018) Nonlinear dynamical systems with self-excited and hidden attractors. Springer, BerlinMATH
3.
Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New YorkMATH
4.
Riewe F (1996) Nonconservative Lagrangian and Hamiltonian mechanics. Phys Rev E 53(2):1890–1899MathSciNet
5.
6.
Agrawal O (2004) A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn 38:323–337MathSciNetMATH
7.
Bhrawy AH, Doha EH, Baleanu D, Ezz-Eldien SS, Abdelkawy MA (2015) An accurate numerical technique for solving fractional optimal control problems. Proc Roman Acad Ser A 16:47–54MathSciNetMATH
8.
Bhrawy AH, Ezz-Eldien SS (2016) A new Legendre operational technique for delay fractional optimal control problems. Calcolo 53:521–543
9.
Bhrawy AH, Doha EH, Machado JA, Ezz-Eldien SS (2015) An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index. Asian J Control 17(6):2389–2402 MathSciNetMATH
10.
Zaky MA (2018) A Legendre collocation method for distributed-order fractional optimal control problems. Nonlinear Dyn 91(4):2667–2681MATH
11.
Keshavarz E, Ordokhani Y, Razzaghi M (2016) A numerical solution for fractional optimal control problems via Bernoulli polynomials. J Vib Control 22(18):1–15MathSciNetMATH
12.
Behroozifar M, Habibi N (2018) A numerical approach for solving a class of fractional optimal control problems via operational matrix Bernoulli polynomials. J Vib Control 24(12):2494–2511MathSciNetMATH
13.
Zaky MA, Tenreiro Machado JA (2017) On the formulation and numerical simulation of distributed-order fractional optimal control problems. Commun Nonlinear Sci Numer Simul 52:177–189MathSciNetMATH
14.
Heydari MH, Hooshmandasl MR, Maalek Ghaini FM, Cattani C (2016) Wavelets method for solving fractional optimal control problems. Appl Math Comput 286:139–154MathSciNetMATH
15.
Hosseinpour S, Nazemi AR (2016) Solving fractional optimal control problems with fixed or free final states by Haar wavelet collocation method. IMA J Math Control Inf 33(2):543–561MathSciNetMATH
16.
Almeida R, Torres D (2015) A discrete method to solve fractional optimal control problems. Nonlinear Dyn 80:1811–1816MathSciNetMATH
17.
Baleanu D, Jajarmi A, Hajipour M (2017) A new formulation of the fractional optimal control problems involving Mittag–Leffler nonsingular kernel. J Optim Theory Appl 157(3):718–737MathSciNetMATH
18.
Jajarmi A, Hajipour M, Mohammadzadeh E, Baleanu D (2018) A new approach for the nonlinear fractional optimal control problems with external persistent disturbances. J Frankl Inst 355:3938–3967MathSciNetMATH
19.
Agrawal OP, Defterli O, Baleanu D (2010) Fractional optimal control problems with several state and control variable. J Vib Control 16(3):1967–1976MathSciNetMATH
20.
Tohidi E, Saberi Nik H (2015) A Bessel collocation method for solving fractional optimal control problems. Appl Math Model 39(2):455–465MathSciNetMATH
21.
Bhrawy AH, Ezz-Eldien SS, Doha EH, Abdelkawy MA, Baleanu D (2017) Solving fractional optimal control problems within a Chebyshev–Legendre operational technique. Int J Control 90(6):1230–1244MathSciNetMATH
22.
Doha EH, Bhrawy AH, Baleanu D, Ezz-Eldien SS, Hafez RM (2015) An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. Adv Differ Equ 15:1–17MathSciNetMATH
23.
Dehghan M, Hamedi EA, Khosravian-Arab H (2016) A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials. J Vib Control 22(6):1–13MathSciNetMATH
24.
Rabiei K, Ordokhani Y, Babolian E (2017) The Boubaker polynomials and their application to solve fractional optimal control problems. Nonlinear Dyn 88(2):1013–1026MathSciNetMATH
25.
Nemati A, Yousefi S, Soltanian F, Ardabili JS (2016) An efficient numerical solution of fractional optimal control problems by using the Ritz method and Bernstein operational matrix. Asian J Control 18(6):2272–2282MathSciNetMATH
26.
Rahimkhani P, Ordokhani Y, Babolian E (2016) An efficient approximate method for solving delay fractional optimal control problems. Nonlinear Dyn 86(3):1649–1661MathSciNetMATH
27.
Moradi L, Mohammadi F, Baleanu D (2019) A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets. J Vibr Control 25(2):1–15MathSciNet
28.
Bello Salati A, Shamsi F, Torres D (2019) Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems. Commun Nonlinear Sci Numer Simul 67:334–350MathSciNetMATH
29.
Sabermahani S, Ordokhani Y, Youse S (2019) Fractional-order Lagrange polynomials: an application for solving delay fractional optimal control problems.Trans Inst Meas Control 41:2997–3009
30.
Peng L, Zhou Y, Debbouche A (2019) Approximation techniques of optimal control problems for fractional dynamic systems in separable Hilbert spaces. Chaos Solitons Fract 118:234–241MathSciNetMATH
31.
Hassani H, Tenreiro Machado JA, Naraghirad E (2019) Generalized shifted Chebyshev polynomials for fractional optimal control problems. Commun Nonlinear Sci Numer Simul 75:50–61MathSciNetMATH
32.
Lotfi A (2019) Epsilon penalty method combined with an extension of the Ritz method for solving a class of fractional optimal control problems with mixed inequality constraints. Appl Numer Math 135:497–509MathSciNetMATH
33.
Hosseinpour S, Nazemi AR, Tohidi E (2019) Muntz–Legendre spectral collocation method for solving delay fractional optimal control problems. J Comput Appl Math 351:344–363MathSciNetMATH
34.
Rooh UA, Li A, Ali MM (2015) Fuzzy, neural network and expert systems methodologies and applications: a review. J Mob Multimed 11(1):157–176
35.
Yu X, Zhou Z, Gao Q, Li D, Ríha K (2018) Infrared image segmentation using growing immune field and clone threshold. Infrared Phys Technol 88:184–193
36.
Yu X, Ye X, Gao Q (2019) Pipeline image segmentation algorithm and heat loss calculation based ongene-regulated apoptosis mechanism. Int J Press Vessels Pip 172:329–336
37.
Duan L, Fang X, Huang C (2017) Global exponential convergence in a delayed almost periodic nicholsons blowflies model with discontinuous harvesting. Math Methods Appl Sci 41(5):1954–1965MathSciNetMATH
38.
Duan L, Huang L, Guo Z, Fang X (2017) Periodic attractor for reaction–diffusion high-order hopfield neural networks with time-varying delays. Comput Math Appl 73(2):233–245MathSciNetMATH
39.
Yang C, Huang L, Li F (2018) Exponential synchronization control of discontinuous nonautonomous networks and autonomous coupled networks. Complexity 2018:1–10MATH
40.
Huang C, Su R, Cao J, Xiao S (2020) Asymptotically stable of high-order neutral cellular neural networks with proportional delays and D operators. Math Comput Simul 171:127–135MathSciNet
41.
Chen D, Zhang W, Cao J, Huang C (2020) Fixed time synchronization of delayed quaternion-valued memristor-based neural networks. Adv Differ Equ 2020:92MathSciNet
42.
Ghasemi S, Nazemi AR, Hosseinpour S (2017) Nonlinear fractional optimal control problems with neural network and dynamic optimization schemes. Nonlinear Dyn 89(4):2669–2682MathSciNetMATH
43.
Kheyrinataj F, Nazemi AR (2020) Fractional power series neural network for solving delay fractional optimal control problems. Connect Sci 32:53–80
44.
Yavari M, Nazemi AR (2019) An efficient numerical scheme for solving fractional infinite-horizon optimal control problems. ISA Trans 94:108–118
45.
Ghasemi S, Nazemi AR (2018) A neural network method based on Mittag-Leffer function for solving a class of fractional optimal control problems. AUT J Model Simul 50:211–218
46.
Kheyrinataj F, Nazemi AR (2020) Fractional Chebyshev functional link neural network-optimization method for solving delay fractional optimal control problems with Atangana-Baleanu derivative. Optim Control Appl Methods 41:808–832MathSciNet
47.
Yavari M, Nazemi AR (2020) On fractional infinite-horizon optimal control problems with a combination of conformable and Caputo-Fabrizio fractional derivatives. ISA Trans 101:78–90
48.
Boulkaibet I, Belarbi K, Bououden S, Marwala T, Chadli M (2017) A new T–S fuzzy model predictive control for nonlinear processes. Expert Syst Appl 88:132–151
49.
Zhu Q, Azar AT (2015) Complex system modelling and control through intelligent soft computations. Springer, BerlinMATH
50.
Wu ZG, Dong SH, Shi P, Su H, Huang T, Lu R (2017) Fuzzy-model-based nonfragile guaranteed cost control of nonlinear Markov jump systems. IEEE Trans Syst Man Cybern Syst 47(8):1–10
51.
Mirzajani S, PourmahmoodAghababa M, Heydari A (2019) Adaptive T–S fuzzy control design for fractional-order systems withparametric uncertainty and input constraint. Fuzzy Sets Syst 365(15):22–39MATH
52.
Zhang HG, Yongbing Q (2001) Modeling, identification, and control of a class of nonlinear systems. IEEE Trans Fuzzy Syst 9(2):349–354MATH
53.
Zhang HG, Wang Z, Liu D (2003) Chaotifying fuzzy hyperbolic model using adaptive inverse optimal control approach. Int J Bifurc Chaos 12:32–43
54.
Takagi T, Sugeno M (1985) Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Syst Man Cybern 15(1):116–132MATH
55.
Dang QV et al (2017) Robust stabilizing controller design for Takagi–Sugeno fuzzy descriptor systems under state constraints and actuator saturation. Fuzzy Sets Syst 329:77–90MathSciNetMATH
56.
Muthukumar P, Balasubramaniam P, Ratnavelu K (2016) T–S fuzzy predictive control for fractional order dynamical systems and its applications. Nonlinear Dyn 86(2):751–763MATH
57.
Dong J, Fu Y (2017) A design method for T–S fuzzy systems with partly immeasurable premise variables subject to actuator saturation. Neurocomputing 225:164–173
58.
Shen H, Su L, Park JH (2017) Reliable mixed/passive control for T–S fuzzy delayed systems based on a semi-Markov jump model approach. Fuzzy Sets Syst 314:79–98MathSciNetMATH
59.
Chen YY, Chang YT, Chen BS (2009) Fuzzy solutions to partial differential equations: adaptive approach. IEEE Trans Fuzzy Syst 17(1):116–127
60.
Zhou Ya, Wan Xiaoxiao, Huang Chuangxia, Yang Xinsong (2020) Finite-time stochastic synchronization of dynamic networks with nonlinear coupling strength via quantized intermittent control. Appl Math Comput 376:125157MathSciNetMATH
61.
Pakdaman M, Effati S (2016) Approximating the solution of optimal control problems by fuzzy systems. Neural Process Lett 43(3):667–686
62.
Kosko B (1994) Fuzzy systems as universal approximators. IEEE Trans Comput 43(11):1329–1333MATH
63.
64.
Wang LX (1992) Fuzzy systems are universal approximators. In: Proceedings of the IEEE international conference on fuzzy systems. San Diego, pp 1163–1170
65.
Ying H (1994) Sufficient conditions on general fuzzy systems as function approximators. Automatica 30:521–525 MATH
66.
Zeng XJ, Singh MG (1995) Approximation theory of fuzzy systems–MIMO case. IEEE Trans Fuzzy Syst 3(4):219–235
67.
Kreinovich V, Nguyen HT, Yam Y (2000) Fuzzy systems are universal approximators for a smooth function and its derivatives. Int J Intell Syst 15(6):565–574MATH
68.
Zak SH (2003) Systems and control. Oxford University Press, Oxford
69.
Zhang M, Zhang H (2006) Robust adaptive fuzzy control scheme for nonlinear system with uncertainty. J Control Theory Appl 4(2):209–216MathSciNetMATH
70.
Zhang HG, Wang ZL, Li M, Quan B, Zhang MJ (2004) Generalized fuzzy hyperbolic model: a universal approximator. Acta Autom Sin 30(3):416–422MathSciNet
71.
Zhang JL, Zhang HG, Luo YH, Liang HJ (2013) Nearly optimal control scheme using adaptive dynamic programming based on generalized fuzzy hyperbolic model. Acta Autom Sin 39(2):142–148MathSciNetMATH
72.
Sun Q, Wang Q, Yang J, Qiu Y, Zhang H (2014) Chaotic dynamics in smart grid and suppression scheme via generalized fuzzy hyperbolic model. Math Probl Eng 2014:Article ID 761271
73.
Cui Y, Zhang HG, Wang Y, Gao W (2016) Adaptive control for a class of uncertain strict-feedback nonlinear systems based on a generalized fuzzy hyperbolic model. Fuzzy Sets Syst 302:52–64MathSciNetMATH
74.
Zhang H, Liu D (2006) Fuzzy modeling and fuzzy control. Springer, Berlin MATH
75.
ZhangM, Zhang H and Liu D (2004) A generalized fuzzy hyperbolicmodeling and control scheme. IEEE Int Conf Fuzzy Syst 3:1203–1207
76.
Zhang M, Zhang H (2005) Modeling and control based on generalized fuzzy hyperbolic model. In: American control conference (2005)
77.
78.
Yepez-Martinez H, Gomez-Aguilar JF (2019) A new modified definition of Caputo–Fabrizio fractional-order derivative and their applications to the multi step homotopy analysis method. J Comput Appl Math 346:247–260MathSciNetMATH
79.
Yildiz TA, Jajarmi A, Yildiz B, Baleanu D (2019) New aspects of time fractional optimal control problems within operators with nonsingular kernel. Discrete Continu Dyn Syst 13:407–428MathSciNetMATH
80.
Bastos N (2018) Calculus of variations involving Caputo–Fabrizio fractional differentiation. Stat Optim Inf Comput 6:12–21MathSciNet
81.
Caputo M, Fabrizio M (2015) A new definition of fractional derivative without singular kernel Progr. Fract Differ Appl 1(2):73–85
82.
Atanacković TM, Pilipović S, Zorica D (2018) Properties of the Caputo–Fabrizio fractional derivative and its distributional settings. Fract Calc Appl Anal 21(1):29–44MathSciNetMATH
83.
Bazaraa MS, Sherali HD, Shetty CM (2006) Nonlinear programming—theory and algorithms, 3rd edn. Wiley, NJ, p 2006MATH
84.
Nocedal J, Wright S (2006) Numerical optimization, 2nd edn. Springer, NewYorkMATH
85.
Lee KY, El-Sharkawi KY (2008) Modern heuristic optimization techniques: theory and applications to power systems. IEEE Press Series Power Eng
86.
Mei W, Bullo W (2017) LaSalle invariance principle for discrete-time dynamical systems: a concise and self-contained tutorial. arXiv:​1710.​03710
Metadata
Title
An Application of Generalized Fuzzy Hyperbolic Model for Solving Fractional Optimal Control Problems with Caputo–Fabrizio Derivative
Authors
Marzieh Mortezaee
Mehdi Ghovatmand
Alireza Nazemi
Publication date
29-09-2020
Publisher
Springer US
Published in
Neural Processing Letters / Issue 3/2020
Print ISSN: 1370-4621
Electronic ISSN: 1573-773X
DOI
https://doi.org/10.1007/s11063-020-10334-4

Other articles of this Issue 3/2020

Neural Processing Letters 3/2020 Go to the issue

OriginalPaper

A New Robust Fuzzy Clustering Approach: DBKIFCM

OriginalPaper

Region Stability and Stabilization of Recurrent Neural Network with Parameter Disturbances

OriginalPaper

Output Layer Multiplication for Class Imbalance Problem in Convolutional Neural Networks

OriginalPaper

Complementary Boundary Estimation Network for Temporal Action Proposal Generation

OriginalPaper

Parameter-Free Extreme Learning Machine for Imbalanced Classification

OriginalPaper

Convolutional Sparse Coded Dynamic Brain Functional Connectivity