Skip to main content
SpringerLink
Log in

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

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

This paper studies the synchronization of a class of uncertain fractional order (FO) chaotic systems that is applicable in secure communication. A novel hybrid FO controller, based on sliding mode and state feedback techniques combined with fuzzy logic, is developed. The algorithm, derived via the fractional Lyapunov theory, guarantees the stability of the overall system and the convergence of the synchronization errors toward a small residual set. Simulations demonstrate the capability of the proposed control algorithm in secure communications, not only in terms of speed of response, but also by reducing the chattering phenomenon.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • 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 

Download references

Author information

Authors and Affiliations

  1. Faculty of Electrical and Robotic Engineering, Shahrood University of Technology, Shahrood, 36199-95161, Iran

    Seyed Mehdi Abedi Pahnehkolaei & Alireza Alfi

  2. Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Rua Dr. António Bernardino de Almeida 431, 4249-015, Porto, Portugal

    J.A. Tenreiro Machado

Authors
  1. Seyed Mehdi Abedi Pahnehkolaei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alireza Alfi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J.A. Tenreiro Machado
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alireza Alfi.

Additional information

Communicated by Vasily E. Tarasov.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pahnehkolaei, S.M.A., Alfi, A. & Machado, J.T. Fuzzy logic embedding of fractional order sliding mode and state feedback controllers for synchronization of uncertain fractional chaotic systems. Comp. Appl. Math. 39, 182 (2020). https://doi.org/10.1007/s40314-020-01206-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-020-01206-7

Keywords

Mathematics Subject Classification

Search

Navigation