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

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.

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References

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  MATH  Article  Google Scholar 

  3. 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 

  4. 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

  5. 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

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

  7. 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

    MATH  Article  Google Scholar 

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

    MATH  Article  Google Scholar 

  9. 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

    MathSciNet  Article  Google Scholar 

  10. 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

  11. 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 

  12. 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

    MathSciNet  Article  Google Scholar 

  13. 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 

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

  15. 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

    MathSciNet  MATH  Article  Google Scholar 

  16. 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

    MathSciNet  MATH  Article  Google Scholar 

  17. 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 

  18. 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

    MathSciNet  MATH  Article  Google Scholar 

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

    Google Scholar 

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

    MATH  Google Scholar 

  21. 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

    MathSciNet  Article  MATH  Google Scholar 

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

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

    MathSciNet  MATH  Article  Google Scholar 

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

    Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  26. 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 

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

  28. 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 

  29. 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

    MathSciNet  MATH  Article  Google Scholar 

  30. 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

    MathSciNet  Article  Google Scholar 

  31. 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 

  32. 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 

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

  34. 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

    MathSciNet  MATH  Article  Google Scholar 

  35. 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 

  36. 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

    MathSciNet  Article  MATH  Google Scholar 

  37. 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

    MathSciNet  Article  Google Scholar 

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

    Article  Google Scholar 

  39. 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 

  40. 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

    MathSciNet  Article  MATH  Google Scholar 

  41. 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

  42. 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

    MathSciNet  Article  Google Scholar 

  43. 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

  44. 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 

  45. 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,

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

    MathSciNet  MATH  Article  Google Scholar 

  47. 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

  48. 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 

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

    MathSciNet  MATH  Article  Google Scholar 

  50. 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 

  51. 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

    MathSciNet  MATH  Article  Google Scholar 

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

  53. 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 

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

    MathSciNet  MATH  Article  Google Scholar 

  55. 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 

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

    MathSciNet  MATH  Article  Google Scholar 

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

    MATH  Article  Google Scholar 

  58. 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 

  59. 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 

  60. 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

    MathSciNet  Article  MATH  Google Scholar 

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

    MATH  Book  Google Scholar 

  62. 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 

  63. 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 

  64. 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 

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

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Mohammad Reza Soltanpour.

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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

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Keywords

  • Fractional order
  • Chaos
  • Terminal
  • Finite time
  • Fixed time
  • Observer
  • Disturbance observer
  • Liu
  • Financial
  • Chen
  • Arnodo–Coullet
  • Modified Jerk
  • Lu