Skip to main content
SpringerLink
Log in

Terminal observer and disturbance observer for the class of fractional-order chaotic systems

  • Methodologies and Application
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

In this paper, a terminal fractional-order observer and a terminal disturbance observer is proposed to estimate internal states and external disturbances of the class of fractional-order chaotic systems. The estimation of states within fixed time is achieved by employing a nonlinear feedback in terms of the observer error. The fixed convergence time is not relevant to the initial conditions and can be adjusted to any desired values by tuning the designable parameters. Finally, the numerical simulations are performed on fractional-order chaotic Liu, Chen, and Financial systems to validate the theoretical results. Moreover, some numerical simulations are provided to compare the obtained theoretical results with the other methods in the literature.

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
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

References

  • Aghababa MP, Hashtarkhani B (2015) A new adaptive observer design for a class of nonautonomous complex chaotic systems. Complexity 21(2):145–153

    Article  MathSciNet  Google Scholar 

  • Bhat SP, Bernstein DS (2000) Finite-time stability of continuous autonomous systems. SIAM J Control Optim 38(3):751–766

    Article  MathSciNet  MATH  Google Scholar 

  • Boroujeni EA, Momeni HR (2012) Observer based control of a class of nonlinear fractional-order systems using LMI. Int J Sci Eng Investig 1(1):48–52

    Google Scholar 

  • Boulkroune A, Bouzeriba A, Bouden T, Azar AT (2016a) Fuzzy adaptive synchronization of uncertain fractional-order chaotic systems. In: Advances in chaos theory and intelligent control. Springer, pp 681–697

  • Boulkroune A, Hamel S, Azar AT, Vaidyanathan S (2016b) Fuzzy control-based function synchronization of unknown chaotic systems with dead-zone input. In: Advances in chaos theory and intelligent control. Springer, pp 699-718

  • Butzer PL, Westphal U (2000) An introduction to fractional calculus. In: Applications of Fractional Calculus in Physics. pp 1-85

  • Cafagna D, Grassi G (2012) Observer-based projective synchronization of fractional systems via a scalar signal: application to hyperchaotic Rössler systems. Nonlinear Dyn 68(1):117–128

    Article  MATH  Google Scholar 

  • Chen M, Zhou D, Shang Y (2005) A new observer-based synchronization scheme for private communication. Chaos, Solitons Fractals 24(4):1025–1030

    Article  MATH  Google Scholar 

  • Chen M, Wu Q, Jiang C (2012) Disturbance-observer-based robust synchronization control of uncertain chaotic systems. Nonlinear Dyn 70(4):2421–2432. https://doi.org/10.1007/s11071-012-0630-9

    Article  MathSciNet  Google Scholar 

  • Cheng Z-F, Shi D-P (2010) Chaos in a fractional-order nonlinear financial system. In: Intelligent Systems and Applications (ISA), 2010 2nd International Workshop on, 2010. IEEE, pp 1-3

  • Cruz-Victoria JC, Martínez-Guerra R, Pérez-Pinacho CA, Gómez-Cortés GC (2015) Synchronization of nonlinear fractional order systems by means of PI rα reduced order observer. Appl Math Comput 262:224–231

    MathSciNet  MATH  Google Scholar 

  • Defoort M, Polyakov A, Demesure G, Djemai M, Veluvolu K (2015) Leader-follower fixed-time consensus for multi-agent systems with unknown non-linear inherent dynamics. IET Control Theory Appl 9(14):2165–2170

    Article  MathSciNet  Google Scholar 

  • Delavari H, Senejohnny D, Baleanu D (2012) Sliding observer for synchronization of fractional order chaotic systems with mismatched parameter. Open Physics 10(5):1095–1101

    Article  Google Scholar 

  • Diethelm K (2010) The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Springer,

  • Diethelm K, Ford NJ, Freed AD (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29(1–4):3–22

    Article  MathSciNet  MATH  Google Scholar 

  • Diethelm K, Ford NJ, Freed AD, Luchko Y (2005) Algorithms for the fractional calculus: a selection of numerical methods. Comput Methods Appl Mech Eng 194(6–8):743–773

    Article  MathSciNet  MATH  Google Scholar 

  • Djeghali N, Djennoune S, Bettayeb M, Ghanes M, Barbot J-P (2016) Observation and sliding mode observer for nonlinear fractional-order system with unknown input. ISA Trans 63:1–10

    Article  Google Scholar 

  • Filali RL, Benrejeb M, Borne P (2014) On observer-based secure communication design using discrete-time hyperchaotic systems. Commun Nonlinear Sci Numer Simul 19(5):1424–1432

    Article  MathSciNet  MATH  Google Scholar 

  • Guce IK (2013) On fractional derivatives: the non-integer order of the derivative. International Journal of Scientific & Engineering, Research 4(3):1

    Google Scholar 

  • Hardy G, Littlewood J, Polya G (1952) Inequalities. Press, Cambridge U

    MATH  Google Scholar 

  • Hassan MF (2016) Synchronization of uncertain constrained hyperchaotic systems and chaos-based secure communications via a novel decomposed nonlinear stochastic estimator. Nonlinear Dyn 83(4):2183–2211. https://doi.org/10.1007/s11071-015-2474-6

    Article  MathSciNet  MATH  Google Scholar 

  • Hilfer R (2000) Applications of fractional calculus in physics. World Scientific,

  • Ibrir S (2006) On observer design for nonlinear systems. Int J Syst Sci 37(15):1097–1109

    Article  MathSciNet  MATH  Google Scholar 

  • Jun-Jie L, Chong-Xin L (2007) Realization of fractional-order Liu chaotic system by circuit. Chin Phys 16(6):1586

    Article  Google Scholar 

  • Kalman RE (1960) A new approach to linear filtering and prediction problems. J Basic Eng 82(1):35–45

    Article  MathSciNet  Google Scholar 

  • Khan H, Abou SC, Sepehri N (2005) Nonlinear observer-based fault detection technique for electro-hydraulic servo-positioning systems. Mechatronics 15(9):1037–1059

    Article  Google Scholar 

  • Kilbas A, Srivastava H, Trujillo J (2006) Theory and applications of fractional differential equations Elsevier. Amsterdam,

  • Laghrouche S, Liu J, Ahmed FS, Harmouche M, Wack M (2015) Adaptive second-order sliding mode observer-based fault reconstruction for PEM fuel cell air-feed system. IEEE Trans Control Syst Technol 23(3):1098–1109

    Article  Google Scholar 

  • Lan Y-H, Zhou Y (2013) Non-fragile observer-based robust control for a class of fractional-order nonlinear systems. Syst Control Lett 62(12):1143–1150

    Article  MathSciNet  MATH  Google Scholar 

  • Lan Y-H, Huang H-X, Zhou Y (2012) Observer-based robust control of a (1 ≤ a < 2) fractional-order uncertain systems: a linear matrix inequality approach. IET Control Theory Appl 6(2):229–234

    Article  MathSciNet  Google Scholar 

  • Lan Y-H, Gu H-B, Chen C-X, Zhou Y, Luo Y-P (2014) An indirect Lyapunov approach to the observer-based robust control for fractional-order complex dynamic networks. Neurocomputing 136:235–242

    Article  Google Scholar 

  • Li C, Yan J (2007) The synchronization of three fractional differential systems. Chaos, Solitons Fractals 32(2):751–757. https://doi.org/10.1016/j.chaos.2005.11.020

    Article  Google Scholar 

  • Li C, Deng WJAM, Computation (2007) Remarks on fractional derivatives. 187 (2):777-784

  • Li H, Gao Y, Shi P, Lam H-K (2016a) Observer-based fault detection for nonlinear systems with sensor fault and limited communication capacity. IEEE Trans Autom Control 61(9):2745–2751

    Article  MathSciNet  MATH  Google Scholar 

  • Li L, Ding SX, Qiu J, Yang Y, Zhang Y (2016b) Weighted fuzzy observer-based fault detection approach for discrete-time nonlinear systems via piecewise-fuzzy Lyapunov functions. IEEE Trans Fuzzy Syst 24(6):1320–1333

    Article  Google Scholar 

  • Liu X, Yuan S (2012) Reduced-order fault detection filter design for switched nonlinear systems with time delay. Nonlinear Dyn 67(1):601–617. https://doi.org/10.1007/s11071-011-0013-7

    Article  MathSciNet  MATH  Google Scholar 

  • Lü L, Li Y, Sun A (2013) Parameter identification and chaos synchronization for uncertain coupled map lattices. Nonlinear Dyn 73(4):2111–2117. https://doi.org/10.1007/s11071-013-0927-3

    Article  MathSciNet  Google Scholar 

  • Luenberger DG (1964) Observing the state of a linear system. IEEE transactions on military electronics 8(2):74–80

    Article  Google Scholar 

  • Luo S, Li S, Tajaddodianfar F, Hu J (2018) Observer-based adaptive stabilization of the fractional-order chaotic MEMS resonator. Nonlinear Dyn 92(3):1079–1089. https://doi.org/10.1007/s11071-018-4109-1

    Article  MATH  Google Scholar 

  • Mahmoud GM, Aly SA, AL-Kashif MA (2008) Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system. Nonlinear Dyn 51(1):171–181. https://doi.org/10.1007/s11071-007-9200-y

    Article  MathSciNet  MATH  Google Scholar 

  • Matignon D, d’Andréa-Novel B Some results on controllability and observability of finite-dimensional fractional differential systems. In: Computational engineering in systems applications, 1996. Citeseer, pp 952-956

  • Mengue AD, Essimbi BZ (2012) Secure communication using chaotic synchronization in mutually coupled semiconductor lasers. Nonlinear Dyn 70(2):1241–1253. https://doi.org/10.1007/s11071-012-0528-6

    Article  MathSciNet  Google Scholar 

  • N’Doye I, Darouach M, Voos H Observer-based approach for fractional-order chaotic synchronization and communication. In: 2013 European Control Conference (ECC), 2013a. IEEE, pp 4281-4286

  • N’Doye I, Voos H, Darouach M (2013) Observer-based approach for fractional-order chaotic synchronization and secure communication. IEEE Journal on Emerging and Selected Topics in Circuits and Systems 3(3):442–450

    Article  Google Scholar 

  • Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Academic press,

  • Polyakov A (2012) Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans Autom Control 57(8):2106–2110

    Article  MathSciNet  MATH  Google Scholar 

  • Polyakov A Fixed-time stabilization of linear systems via sliding mode control. In: Variable Structure Systems (VSS), 2012 12th International Workshop on, 2012a. IEEE, pp 1-6

  • Rahme S, Meskin N (2015) Adaptive sliding mode observer for sensor fault diagnosis of an industrial gas turbine. Control Engineering Practice 38:57–74

    Article  Google Scholar 

  • Senejohnny DM, Delavari H (2012) Active sliding observer scheme based fractional chaos synchronization. Commun Nonlinear Sci Numer Simul 17(11):4373–4383

    Article  MathSciNet  MATH  Google Scholar 

  • Shao S, Wheeler PW, Clare JC, Watson AJ (2013) Fault detection for modular multilevel converters based on sliding mode observer. IEEE Trans Power Electron 28(11):4867–4872

    Article  Google Scholar 

  • Shao S, Chen M, Yan X (2016) Adaptive sliding mode synchronization for a class of fractional-order chaotic systems with disturbance. Nonlinear Dyn 83(4):1855–1866

    Article  MathSciNet  MATH  Google Scholar 

  • Slotine J, Li W (1998) Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ, 1991. Google Scholar

  • Smith AH, Monti A, Ponci F (2007) Indirect measurements via a polynomial chaos observer. IEEE Trans Instrum Meas 56(3):743–752

    Article  Google Scholar 

  • Spurgeon SK (2008) Sliding mode observers: a survey. Int J Syst Sci 39(8):751–764

    Article  MathSciNet  MATH  Google Scholar 

  • Su H, Luo R, Zeng Y (2017) The observer-based synchronization and parameter estimation of a class of chaotic system via a single output. Pramana 89(5):78. https://doi.org/10.1007/s12043-017-1476-y

    Article  Google Scholar 

  • Tan CP, Yu X, Man Z (2010) Terminal sliding mode observers for a class of nonlinear systems. Automatica 46(8):1401–1404

    Article  MathSciNet  MATH  Google Scholar 

  • Tornambè A (1992) Asymptotic observers for non-linear systems. Int J Syst Sci 23(3):435–442

    Article  MATH  Google Scholar 

  • Wang H, Han Z, Zhang W, Xie Q (2008) Chaotic synchronization and secure communication based on descriptor observer. Nonlinear Dyn 57(1):69. https://doi.org/10.1007/s11071-008-9420-9

    Article  MATH  Google Scholar 

  • Wang H, Zhu X-J, Gao S-W, Chen Z-Y (2011) Singular observer approach for chaotic synchronization and private communication. Commun Nonlinear Sci Numer Simul 16(3):1517–1523

    Article  Google Scholar 

  • Wang J, Ma Q, Zeng L (2013) Observer-based synchronization in fractional-order leader–follower complex networks. Nonlinear Dyn 73(1):921–929. https://doi.org/10.1007/s11071-013-0843-6

    Article  MathSciNet  MATH  Google Scholar 

  • Wiener N (1949) Extrapolation, interpolation, and smoothing of stationary time series, vol 7. MIT press Cambridge, MA

    Book  MATH  Google Scholar 

  • Yao J, Jiao Z, Ma D (2014) Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with backstepping. IEEE Trans Industr Electron 61(11):6285–6293

    Article  Google Scholar 

  • Yu J, Ma Y, Yu H, Lin C (2016) Reduced-order observer-based adaptive fuzzy tracking control for chaotic permanent magnet synchronous motors. Neurocomputing 214:201–209

    Article  Google Scholar 

  • Zhang R, Gong J (2014) Synchronization of the fractional-order chaotic system via adaptive observer. Systems Science & Control Engineering: An Open Access Journal 2(1):751–754

    Article  Google Scholar 

  • Zuo Z, Tie L (2016) Distributed robust finite-time nonlinear consensus protocols for multi-agent systems. Int J Syst Sci 47(6):1366–1375

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Shahid Sattari Aeronautical University of Science and Technology, P.O. Box 1599964511, Shamshiri Street, Tehran, Iran

    Mohammad Reza Soltanpour & Mehrdad Shirkavand

Authors
  1. Mohammad Reza Soltanpour
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mehrdad Shirkavand
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammad Reza Soltanpour.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interests.

Human and animals rights

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

Informed consent

Informed consent was obtained from all individual participants included in the study.

Additional information

Communicated by V. Loia.

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

Soltanpour, M.R., Shirkavand, M. Terminal observer and disturbance observer for the class of fractional-order chaotic systems. Soft Comput 24, 8881–8898 (2020). https://doi.org/10.1007/s00500-019-04418-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-019-04418-0

Keywords

Search

Navigation