Top

Soft Computing

Published in:

19-05-2020 | Methodologies and Application

Solving variable-order fractional differential algebraic equations via generalized fuzzy hyperbolic model with application in electric circuit modeling

Authors: Marzieh Mortezaee, Mehdi Ghovatmand, Alireza Nazemi

Published in: Soft Computing | Issue 22/2020

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

search-config

AI-assisted search

Off

Abstract

In this paper, a new approach based on a generalized fuzzy hyperbolic model is used for the numerical solution of variable-order fractional differential algebraic equations. The fractional derivative is described in the Atangana–Baleanu sense that is a new derivative with fractional order based on the generalized Mittag–Leffler function. First, by using fuzzy solutions with adjustable parameters, the variable-order fractional differential algebraic equations are reduced to a problem consisting of solving a system of algebraic equations. For adjusting the parameters of fuzzy solutions, an unconstrained optimization problem is then considered. A learning algorithm is also presented for solving the unconstrained optimization problem. Finally, some numerical examples are given to verify the efficiency and accuracy of the proposed approach.

previous article Bipolar N-soft set theory with applications

next article Applications of probabilistic hesitant fuzzy rough set in decision support system

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 "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

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

Atangana A, Alqahtani RT (2016) Stability analysis of nonlinear thin viscous fluid sheet flow equation with local fractional variable-order derivative. J Comput Theor Nanosci 13:2710–17

Bazaraa MS, Sherali HD, Shetty CM (2006) Nonlinear programming: theory and algorithms, 3rd edn. Wiley, HobokenMATH

Bendtsen C, Thomsen PG (1999) Numerical solution of differential algebraic equations, technical report, Department of Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark

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

Buckley JJ (1992) Universal fuzzy controllers. Automatica 28:1245–1248MathSciNetMATH

Cao J, Qiu Y, Song G (2017) A compact finite difference scheme for variable order subdiffusion equation. Commun Nonlinear Sci Numer Simul 48:140–149MathSciNetMATH

Chen YY, Chang YT, Chen BS (2009) Fuzzy solutions to partial differential equations: adaptive approach. IEEE Trans Fuzzy Syst 17(1):116–127

Chen Y-M, Wei Y-Q, Liu D-Y, Yu H (2015) Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets. Appl Math Lett 46:83–88MathSciNetMATH

Coronel-Escamilla A, Gómez-Aguilar JF, Torres L, Escobar-Jiménez RF (2018) A numerical solution for a variable-order reaction–diffusion model by using fractional derivatives with non-local and non-singular kernel. Phys A 491:406–24MathSciNet

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

Dang QV et al (2017) Robust stabilizing controller design for Takagi–Sugeno fuzzy descriptor systems under state constraints and actuator satu-ration. Fuzzy Sets Syst 329:77–90MATH

Deng W, Zhao H, Zou L, Li G, Yang X, Wu D (2017a) A novel collaborative optimization algorithm in solving complex optimization problems. Soft Comput 21:4387–4398

Deng W, Zhao H, Yang X, Xiong J, Sun M, Li B (2017b) Study on an improved adaptive PSO algorithm for solving multi-objective gate assignment. Appl Soft Comput 59:288–302

Deng W, Xu J, Zhao H (2019) An improved ant colony optimization algorithm based on hybrid strategies for scheduling problem. IEEE Access 7:20281–20292

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

Fu Z, Chen W, Ling L (2015) Method of approximate particular solutions for constant- and variable-order fractional diffusion models. Eng Anal Bound Elem 57:37–46MathSciNetMATH

Ghanbari B, Gómez-Aguilar JF (2018) Modeling the dynamics of nutrient–phytoplankton–zooplankton system with variable-order fractional derivatives. Chaos, Solitons Fractals 116:114–120MathSciNetMATH

Ghanbari F, Ghanbari K, Mokhtary P (2018) Generalized Jacobi–Galerkin method for nonlinear fractional differential algebraic equations. Comput Appl Math 37(4):5456–5475MathSciNetMATH

Ghomanjani F (2017) A new approach for solving fractional differential–algebraic equations. J Taibah Univ Sci 11:1158–1164

Gómez-Aguilar JF (2018) Analytical and numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations. Phys A 494:52–75MathSciNet

Jia Y, Xu M, Lin Y (2017) A numerical solution for variable order fractional functional differential equation. Appl Math Lett 64:125–130MathSciNetMATH

Karabacak M, Celik E (2013) The numerical solution of fractional differential-algebraic equations (FDAEs). New Trends Math Sci 1(2):1–6

Kosko B (1994) Fuzzy systems as universal approximators. IEEE Trans Comput 43(11):1329–1333MATH

Lee KY, El-Sharkawi MA (2008) Modern heuristic optimization techniques: theory and applications to power systems. Wiley, Hoboken

Li X, Wu B (2017) A new reproducing kernel method for variable order fractional boundary value problems for functional differential equations. J Comput Appl Math 311:387–393MathSciNetMATH

Li X, Li H, Wu B (2017) A new numerical method for variable order fractional functional differential equations. Appl Math Lett 68:80–86MathSciNetMATH

Liu H, Fu Y, Li B (2017) Discrete waveform relaxation method for linear fractional delay differential-algebraic equations. Discrete Dyn Nat Soc 6306570:9 pages

Liu S, Guo Z, Zhang HG (2008) Fuzzy hyperbolic neural network model and its application in H1 filter design, In: Sun F et al (eds) Proceedings of the 5th international symposium on neural networks: advances in neural networks, part I, LNCS 5263, Springer, Berlin, 2008, pp 222–230

Mei W, Bullo F (2017) Lasalle invariance principle for discrete-time dynamical systems: a concise and self-contained tutorial. arXiv preprint arXiv:1710.03710

Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New YorkMATH

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

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

Nocedal J, Wright S (2006) Numerical Optimization, second edn. Springer- Verlag, Berlin, NewYorkMATH

Pakdaman M, Effati S (2016) Approximating the solution of optimal control problems by fuzzy systems. Neural Process Lett 43(3):667–686

Pham VT, Vaidyanathan S, Volos C, Kapitaniak T (2018) Nonlinear dynamical systems with self-excited and hidden attractors. Springer, BerlinMATH

Sh DM, Hassani H (2017) An optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation. Nonlinear Dyn 88:1587–1598MathSciNetMATH

Shen S, Liu F, Anh V, Turner I, Chen J (2013) A characteristic difference method for the variable-order fractional advection–diffusion equation. J Appl Math Comput 42:371–386MathSciNetMATH

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

Shiri B, Baleanu D (2019) System of fractional differential algebraic equations with applications. Chaos Solitons Fractals 120:203–212MathSciNet

Solís-Pérez JE, Gómez-Aguilar JF, Atangana A (2018) Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag–Leffler laws. Chaos Solitons Fractals 114:175–185MathSciNetMATH

Sun H, Chen W, Li CH, Chen Y (2012) Finite difference schemes for variable-order time fractional diffusion equation. Int J Bifurc Chaos 22:1250085MathSciNetMATH

Sun Q, Wang Y, 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, 7 pages

Sun H, Zhang Y, Baleanu D, Chen W, Chen Y (2018) A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simul 64:213–231

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

Wang LX (1992) Fuzzy systems are universal approximators. In: Proceedings 1EEE international conference on fuzzy systems (San Diego), pp 1163–1170

Wang L, Chen N (2014) The predictor-corrector solution for fractional order differential algebraic equation. In: CFDA’14 international conference on fractional differentiation and its applications

Wang SH, Jhu WL, Yung CF, Wang PF (2011) Numerical solutions of differential algebraic equations and its applications in solving TPPC problems. J Mar Sci Technol 19:76–88

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

Xu T, Lu S, Chen W, Chen H (2018) Finite difference scheme for multi-term variable-order fractional diffusion equation. Adv Differ Equ 103:1–13MathSciNetMATH

Yaghoobi S, Moghaddam BP, Ivaz K (2017) An efficient cubic spline approximation for variable-order fractional differential equations with time delay. Nonlinear Dyn 87:815–826MathSciNetMATH

Yang XJ (2017) Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems. Therm Sci 21:1161–1171

Ying H (1994) Sufficient conditions on general fuzzy systems as function approximators. Automatica 30:521–525MATH

Zak SH (2003) Systems and control. Oxford University Press, Oxford

Zeng XJ, Singh MG (1995) Approximation theory of fuzzy systems-MIMO case. IEEE Trans Fuzzy Syst 3(2):219–235

Zhang X-S (2000) Neural networks in optimization. Kluwer Academic Publishers, DordrechtMATH

Zhang HG, Yongbing Q (2001) Modeling, identification, and control of a class of nonlinear systems. IEEE Trans Fuzzy Syst 9(2):349–354MATH

Zhang M, Zhang H (2005) Modeling and control based on generalized fuzzy hyperbolic model. In: American control conference

Zhang M, Zhang H (2006) Robust adaptive fuzzy control scheme for nonlinear system with uncertainty. J Control Theory Appl 4(2):209–216MathSciNetMATH

Zhang HG, Wang Z, Liu D (2003) Chaotifying fuzzy hyperbolic model using adaptive inverse optimal control approach. Int J Bifurc Chaos 12:32–43

Zhang HG, Wang ZL, Li M, Quan B, Zhang MG (2004a) Generalized fuzzy hyperbolic model: a universal approximator. Acta Autom Sin 30(3):416–422MathSciNet

Zhang HG, Wang ZL, Li M, Quan B, Zhang MJ (2004b) Generalized fuzzy hyperbolic model: a universal approximator. Acta Autom Sin 30(3):416–422MathSciNet

Zhang M, Zhang H, Liu D (2004c) A generalized fuzzy hyperbolic modeling and control scheme. In: IEEE international conference on fuzzy systems

Zhang HG, Wang Z, Liu D (2005) Chaotifying fuzzy hyperbolic model using impulsive and nonlinear feedback control approaches. Int J Bifurc Chaos 15(8):2603–2610MathSciNetMATH

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

Zhao H, Liu H, Xu J, Deng W (2019a) Performance prediction using high-order differential mathematical morphology gradient spectrum entropy and extreme learning machine. IEEE Trans Instrum Meas. https://doi.org/10.1109/TIM.2019.2948414CrossRef

Zhao H, Zheng J, Xu J, Deng W (2019b) Fault diagnosis method based on principal component analysis and broad learning system. IEEE Access 7:99263–99272

Zhao H, Zheng J, Deng W, Song Y (2020) Semi-supervised broad learning system based on manifold regularization and broad network. IEEE Trans Circuits Syst I Regul Pap 67(3):983–994MathSciNet

Zhu Q, Azar AT (2015) Complex system modelling and control through intelligent soft computations. Springer, BerlinMATH

Zurigat M, Momani SH, Alawneh A (2010) Analytical approximate solutions of systems of fractional algebraic differential equations by homotopy analysis method. Comput Math Appl 59:1227–1235MathSciNetMATH

Title: Solving variable-order fractional differential algebraic equations via generalized fuzzy hyperbolic model with application in electric circuit modeling
Authors: Marzieh Mortezaee
Mehdi Ghovatmand
Alireza Nazemi
Publication date: 19-05-2020
Publisher: Springer Berlin Heidelberg
Published in: Soft Computing / Issue 22/2020
Print ISSN: 1432-7643
Electronic ISSN: 1433-7479
DOI: https://doi.org/10.1007/s00500-020-04969-7

Springer Professional

Abstract

Please log in to get access to your license.

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

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 22/2020

On the weighted Gini–Simpson index: estimating feasible weights using the optimal point and discussing a link with possibility theory

Identifying cancer-associated modules from microRNA co-expression networks: a multiobjective evolutionary approach

Pedestrian overpass utilization modeling based on mobility friction, safety and security, and connectivity using machine learning techniques

Forecasting hybrid neural network with variational learning rate and q-DSCID synchronization evaluation for energy market

Truss-sizing optimization attempts with CSA: a detailed evaluation

Shadowed sets with higher approximation regions

Premium Partner