Fuzzy logic embedding of fractional order sliding mode and state feedback controllers for synchronization of uncertain fractional chaotic systems

• Abedi Pahnehkolaei SM, Alfi A, Machado JT (2017) Chaos suppression in fractional systems using adaptive fractional state feedback control. Chaos, Solitons & Fractals 103:488–503

  MathSciNet  MATH  Google Scholar 

• Aguila-Camacho N, Duarte-Mermoud MA, Gallegos JA (2014) Lyapunov functions for fractional order systems. Commun Nonlinear Sci Numer Simul 19(9):2951–2957

  MathSciNet  MATH  Google Scholar 

• Alaviyan Shahri ES, Alfi A, Machado JT (2017) Stabilization of fractional-order systems subject to saturation element using fractional dynamic output feedback sliding mode control. J Comput Nonlinear Dyn 12(3):031014–019

  Google Scholar 

• Alaviyan Shahri ES, Alfi A, Tenreiro Machado J (2018) Stability analysis of a class of nonlinear fractional-order systems under control input saturation. Int J Robust Nonlinear Control 28(7):2887–2905

  MathSciNet  MATH  Google Scholar 

• Alfi A (2012) Chaos suppression on a class of uncertain nonlinear chaotic systems using an optimal H\(\infty \) adaptive PID controller. Chaos Solitons Fractals 45(3):351–357

  MathSciNet  MATH  Google Scholar 

• Alligood KT, Sauer TD, Yorke JA, Crawford J (1997) Chaos: an introduction to dynamical systems. Phys Today 50:67

  Google Scholar 

• Antão R, Mota A, Martins RE, Machado JT (2017) Type-2 fuzzy logic: uncertain systems’ modeling and control. Springer, New York

  Google Scholar 

• Badri P, Sojoodi M (2019) Stability and stabilization of fractional-order systems with different derivative orders: an LMI approach. Asian J Control 21:2270–2279

  MathSciNet  MATH  Google Scholar 

• Baleanu D, Golmankhaneh AK, Golmankhaneh AK, Baleanu MC (2009) Fractional electromagnetic equations using fractional forms. Int J Theor Phys 48(11):3114–3123

  MathSciNet  MATH  Google Scholar 

• Bettayeb M, Al-Saggaf UM, Djennoune S (2017) Single channel secure communication scheme based on synchronization of fractional-order chaotic Chua’s systems. Trans Inst Meas Control 40(13):3651–3664

  Google Scholar 

• Bohannan GW (2000) Application of fractional calculus to polarization dynamics in solid dielectric materials. Ph.D. thesis, Montana State University-Bozeman, College of Letters and Science

• Boulkroune A, M’saad M (2012) Fuzzy adaptive observer-based projective synchronization for nonlinear systems with input nonlinearity. J Vib Control 18(3):437–450

  MathSciNet  MATH  Google Scholar 

• Candido R, Silva MT, Eisencraft M (2018) A new encoding and switching scheme for chaos-based communication. Comput Appl Math 37(1):253–266

  MathSciNet  MATH  Google Scholar 

• Chen CS, Chen HH (2009) Robust adaptive neural-fuzzy-network control for the synchronization of uncertain chaotic systems. Nonlinear Anal Real World Appl 10(3):1466–1479

  MathSciNet  MATH  Google Scholar 

• Chen D, Liu W (2016) Chaotic behavior and its control in a fractional-order energy demand-supply system. J Comput Nonlinear Dyn 11(6):061010–016

  Google Scholar 

• Chen HK (2002) Chaos and chaos synchronization of a symmetric gyro with linear-plus-cubic damping. J Sound Vib 255(4):719–740

  MathSciNet  MATH  Google Scholar 

• Chen L, Pan W, Wang K, Wu R, Machado JT, Lopes AM (2017) Generation of a family of fractional order hyper-chaotic multi-scroll attractors. Chaos Solitons Fractals 105:244–255

  MathSciNet  MATH  Google Scholar 

• Chen L, Pan W, Wu R, Tenreiro Machado J, Lopes AM (2016) Design and implementation of grid multi-scroll fractional-order chaotic attractors. Chaos Interdiscip J Nonlinear Sci 26(8):084303–084311

  MathSciNet  MATH  Google Scholar 

• Dabiri A, Moghaddam BP, Machado JT (2018) Optimal variable-order fractional PID controllers for dynamical systems. J Comput Appl Math 339:40–48

  MathSciNet  MATH  Google Scholar 

• Dami L, Benhayoun M, Benzaouia A (2019) Stabilization of positive 2D fractional-order continuous-time systems with delays. Circuits Syst Signal Process 38(5):1962–1981

  MathSciNet  Google Scholar 

• David SA, Machado JAT, Quintino DD, Balthazar J (2016) Partial chaos suppression in a fractional order macroeconomic model. Math Comput Simul 122:55–68

  MathSciNet  Google Scholar 

• David SA, Quintino DD, Inacio C Jr, Machado JT (2018) Fractional dynamic behavior in ethanol prices series. J Comput Appl Math 339:85–93

  MathSciNet  MATH  Google Scholar 

• Diethelm K, Ford NJ, Freed AD (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29(1):3–22

  MathSciNet  MATH  Google Scholar 

• Ding XL, Nieto JJ (2018) Controllability of nonlinear fractional delay dynamical systems with prescribed controls. Nonlinear Anal Model Control 23(1):1–18

  MathSciNet  MATH  Google Scholar 

• Eshaghi S, Ghaziani RK, Ansari A (2020) Hopf bifurcation, chaos control and synchronization of a chaotic fractional-order system with chaos entanglement function. Math Comput Simul 172:321–340

  MathSciNet  Google Scholar 

• Faieghi MR, Delavari H (2012) Chaos in fractional-order Genesio-Tesi system and its synchronization. Commun Nonlinear Sci Numer Simul 17(2):731–741

  MathSciNet  MATH  Google Scholar 

• Golmankhaneh AK, Arefi R, Baleanu D (2015) Synchronization in a nonidentical fractional order of a proposed modified system. J Vib Control 21(6):1154–1161

  MathSciNet  MATH  Google Scholar 

• Hajipour M, Jajarmi A, Baleanu D (2018) An efficient non-standard finite difference scheme for a class of fractional chaotic systems. J Comput Nonlinear Dyn 13:0210131–021013

  Google Scholar 

• Hamri Ne, Ouahabi R (2017) Modified projective synchronization of different chaotic systems using adaptive control. Comput Appl Math 36(3):1315–1332

  MathSciNet  MATH  Google Scholar 

• Huang S, Wang B (2017) Stability and stabilization of a class of fractional-order nonlinear systems for \( 0<\alpha <2\). Nonlinear Dyn 88(2):973–984

  MathSciNet  MATH  Google Scholar 

• Jiang CM, Liu ST, Zhang FF (2018) Complex modified projective synchronization for fractional-order chaotic complex systems. Int J Autom Comput 15(5):603–615

  Google Scholar 

• Jovic B, Berber S, Unsworth C (2006) A novel mathematical analysis for predicting master-slave synchronization for the simplest quadratic chaotic flow and ueda chaotic system with application to communications. Phys D Nonlinear Phenom 213(1):31–50

  MathSciNet  MATH  Google Scholar 

• Khanzadeh A, Mohammadzaman I (2018) Comment on “fractional-order fixed-time nonsingular terminal sliding mode synchronization and control of fractional-order chaotic systems”. Nonlinear Dyn 94(4):3145–3153

  MATH  Google Scholar 

• Li C, Liao X, Wong KW (2004) Chaotic lag synchronization of coupled time-delayed systems and its applications in secure communication. Phys D Nonlinear Phenom 194(3–4):187–202

  MathSciNet  MATH  Google Scholar 

• Li C, Peng G (2004) Chaos in Chen’s system with a fractional order. Chaos Solitons Fractals 22(2):443–450

  MathSciNet  MATH  Google Scholar 

• Li H, Zhang L, Hu C, Jiang Y, Teng Z (2017) Dynamic analysis of a fractional-order single-species model with diffusion. Nonlinear Anal Model Control 22(3):303–316

  MathSciNet  MATH  Google Scholar 

• Li R, Cao J, Alsaedi A, Alsaadi F (2017) Stability analysis of fractional-order delayed neural networks. Nonlinear Anal Model Control 22(4):505–520

  MathSciNet  MATH  Google Scholar 

• Li X, Ma SQ (2017) Adaptive synchronization control of the fractional order hyper chaotic systems with unknown parameters. International conference on fuzzy information and engineering. Springer, New York, pp 340–348

  Google Scholar 

• Liu H, Pan Y, Li S, Chen Y (2018) Synchronization for fractional-order neural networks with full/under-actuation using fractional-order sliding mode control. Int J Mach Learn Cybern 9(7):1219–1232

  Google Scholar 

• Lu JG (2005) Chaotic dynamics and synchronization of fractional-order Arneodós systems. Chaos Solitons Fractals 26(4):1125–1133

  MATH  Google Scholar 

• Machado J (1996) Variable structure control of manipulators with joints having flexibility and backlash. SAMS 23:93–101

  MATH  Google Scholar 

• Machado JT (2012) The effect of fractional order in variable structure control. Comput Math Appl 64(10):3340–3350

  MathSciNet  MATH  Google Scholar 

• Mobayen S, Baleanu D (2017) Linear matrix inequalities design approach for robust stabilization of uncertain nonlinear systems with perturbation based on optimally-tuned global sliding mode control. J Vib Control 23(8):1285–1295

  MathSciNet  MATH  Google Scholar 

• Moghaddam B, Dabiri A, Lopes AM, Machado JT (2019) Numerical solution of mixed-type fractional functional differential equations using modified lucas polynomials. Comput Appl Math 38(2):46

  MathSciNet  MATH  Google Scholar 

• Moghaddam BP, Machado JT, Babaei A (2018) A computationally efficient method for tempered fractional differential equations with application. Comput Appl Math 37(3):3657–3671

  MathSciNet  MATH  Google Scholar 

• Mohammadzadeh A, Ghaemi S, Kaynak O et al (2019) Robust predictive synchronization of uncertain fractional-order time-delayed chaotic systems. Soft Comput 23(16):6883–6898

  MATH  Google Scholar 

• Pahnehkolaei SMA, Alfi A, Machado JT (2019) Delay independent robust stability analysis of delayed fractional quaternion-valued leaky integrator echo state neural networks with quad condition. Appl Math Comput 359:278–293

  MathSciNet  MATH  Google Scholar 

• Rajendran ML, Balachandran K, Trujillo JJ (2017) Controllability of nonlinear stochastic neutral fractional dynamical systems. Nonlinear Anal Model Control 22(5):702–718

  MathSciNet  MATH  Google Scholar 

• Sheikhan M, Shahnazi R, Garoucy S (2013) Synchronization of general chaotic systems using neural controllers with application to secure communication. Neural Comput Appl 22(2):361–373

  Google Scholar 

• Singh AK, Yadav VK, Das S (2017) Synchronization between fractional order complex chaotic systems. Int J Dyn Control 5(3):756–770

  MathSciNet  Google Scholar 

• Song X, Song S, Li B, Tejado Balsera I (2018) Adaptive projective synchronization for time-delayed fractional-order neural networks with uncertain parameters and its application in secure communications. Trans Inst Meas Control 40(10):3078–3087

  Google Scholar 

• Sundarapandian V, Suresh R (2011) Global chaos synchronization of hyperchaotic Qi and Jia systems by nonlinear control. Int J Distrib Parallel Syst 2(2):83–94

  Google Scholar 

• Tabasi M, Balochian S (2018) Synchronization of the chaotic fractional-order Genesio-Tesi systems using the adaptive sliding mode fractional-order controller. J Control Autom Electr Syst 29(1):15–21

  Google Scholar 

• Tanaka K, Sugeno M (1992) Stability analysis and design of fuzzy control systems. Fuzzy Sets Syst 45(2):135–156

  MathSciNet  MATH  Google Scholar 

• Teodoro GS, Machado JT, De Oliveira EC (2019) A review of definitions of fractional derivatives and other operators. J Comput Phys 388:195–208

  MathSciNet  Google Scholar 

• Valério D, Costa JSD (2014) Fractional sliding mode control of MIMO nonlinear noncommensurable plants. J Vib Control 20(7):1052–1065

  MathSciNet  Google Scholar 

• Wang HO, Tanaka K, Griffin MF (1996) An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Trans Fuzzy Syst 4(1):14–23

  Google Scholar 

• Xin B, Zhang J (2015) Finite-time stabilizing a fractional-order chaotic financial system with market confidence. Nonlinear Dyn 79(2):1399–1409

  MathSciNet  MATH  Google Scholar 

• Xiong L, Li P, Ma M, Wang Z, Wang J (2020) Output power quality enhancement of PMSG with fractional order sliding mode control. Int J Electr Power Energy Syst 115:105402

  Google Scholar 

• Xu B, Chen D, Zhang H, Wang F (2015) Modeling and stability analysis of a fractional-order Francis hydro-turbine governing system. Chaos Solitons Fractals 75:50–61

  MathSciNet  MATH  Google Scholar 

• Yang HY, Yang Y, Han F, Zhao M, Guo L (2019) Containment control of heterogeneous fractional-order multi-agent systems. J Franklin Inst 356:752–765

  MathSciNet  MATH  Google Scholar 

• Zaky M, Machado JT (2017) On the formulation and numerical simulation of distributed-order fractional optimal control problems. Commun Nonlinear Sci Numer Simul 52:177–189

  MathSciNet  Google Scholar 

• Zhang W, Wu R, Cao J, Alsaedi A, Hayat T (2017) Synchronization of a class of fractional-order neural networks with multiple time delays by comparison principles. Nonlinear Anal Model Control 22:636–645

  MathSciNet  MATH  Google Scholar 

• Zhou P, Cai H, Yang C (2016) Stabilization of the unstable equilibrium points of the fractional-order BLDCM chaotic system in the sense of Lyapunov by a single-state variable. Nonlinear Dyn 84(4):2357–2361

  MathSciNet  MATH  Google Scholar 

• Zhou T, Li C (2005) Synchronization in fractional-order differential systems. Phys D Nonlinear Phenom 212(1–2):111–125

  MathSciNet  MATH  Google Scholar 

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