Nonlinear Dynamics

, Volume 61, Issue 3, pp 383–397 | Cite as

Fractional order control of a coupled tank

  • H. Delavari
  • A. N. Ranjbar
  • R. Ghaderi
  • S. Momani
Original paper


In this paper, a hybrid system that combines the advantages in terms of robustness of the fractional control and the Sliding Mode Control (SMC) will be proposed. The proposed fractional order SMC is applied to a level control in a nonlinear coupled tank, as a case study. To investigate the capability of the method, a Sliding Mode Controller is alternatively designed. Primarily a sliding surface based on linear compensation networks PD or PID is designed. The work is followed by designation of a fractional form of these networks, PD μ or PI λ D μ . Finally, the performance of the proposed technique is also investigated under disturbance and variation in parameters of system. The simulation results indicate the significance of the fractional order sliding mode controllers.


Fractional controllers Nonlinear control Sliding mode control Coupled tanks 


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  1. 1.
    Ahn, H.S., Chen, Y., Podlubny, I.: Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality. Appl. Math. Comput. 187, 27–34 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Valério, D., Costa, J.: Tuning of fractional PID controllers with Ziegler–Nichols-type rules. Signal Process. 86, 2771–2784 (2006) zbMATHCrossRefGoogle Scholar
  3. 3.
    Feliu-Batlle, V., Rivas Pérez, R., Sánchez Rodríguez, L.: Fractional robust control of main irrigation canals with variable dynamic parameters. Control Eng. Pract. 15, 673–686 (2007) CrossRefGoogle Scholar
  4. 4.
    Calderón, A.J., Vinagre, B.M., Feliu, V.: Fractional order control strategies for power electronic buck converters. Signal Process. 86, 2803–2819 (2006) zbMATHCrossRefGoogle Scholar
  5. 5.
    Kusnezov, D., Bulgac, A., Dang, G.D.: Quantum Levy processes and fractional kinetics. Phys. Rev. Lett. 82, 1136 (1999) CrossRefGoogle Scholar
  6. 6.
    Sun, H.H., Abdelwahad, A.A., Onaral, B.: Linear approximation of transfer function with a pole of fractional power. IEEE Trans. Automat. Control 29(5), 441–444 (1984) zbMATHCrossRefGoogle Scholar
  7. 7.
    Laskin, N.: Fractional market dynamics. Physica A 287, 482 (2000) CrossRefMathSciNetGoogle Scholar
  8. 8.
    El-Sayed, A.M.A.: Fractional-order diffusion-wave equation. Int. J. Theor. Phys. 35(2), 311 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Heaviside, O.: Electromagnetic Theory. Chelsea, New York (1971) Google Scholar
  10. 10.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) zbMATHGoogle Scholar
  11. 11.
    Yu, C., Gao, G.: Existence of fractional differential equations. J. Math. Anal. Appl. 310, 26–29 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ladaci, S., Loiseau, J.J., Charef, A.: Fractional order adaptive high-gain controllers for a class of linear systems. Commun. Nonlinear Sci. Numer. Simul. 13, 707–714 (2008) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Oustaloup, A., Bluteau, B., Nouillant, M.: First generation scalar CRONE control: application to a two DOF manipulator and comparison with nonlinear decoupling control. Int. Conf. Syst., Man Cybern. 4, 453–458 (1993) Google Scholar
  14. 14.
    Oustaloup, A.: Fractional order sinusoidal oscillators: optimization and their use in highly linear FM modulation. IEEE Trans. Circ. Syst. 28(10), 1007–1009 (1981) CrossRefGoogle Scholar
  15. 15.
    Serrier, P., Moreau, X., Sabatier, J., Oustaloup, A.: Taking into account the non-linearities in the CRONE approach: application to vibration isolation. In: 32nd Annual Conference on Industrial Electronics, IECON 2006, pp. 5360–5365 (2006) Google Scholar
  16. 16.
    Lanusse, P., Benlaoukli, H., Nelson-Gruel, D., Oustaloup, A.: Fractional-order control and interval analysis of SISO systems with time-delayed state. Control Theory Appl., IET 2(1), 16–23 (2008) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Slotine, J.J.E., Li, W.: Applied Nonlinear Control. Prentice Hall, New Jersey (1991) zbMATHGoogle Scholar
  18. 18.
    Khan, M.Kh., Spurgeon, S.K.: Robust MIMO water level control in interconnected twin-tanks using second order sliding mode control. Control Eng. Pract. 14, 375–386 (2006) CrossRefGoogle Scholar
  19. 19.
    Almutairi, N.B., Zribi, M.: Sliding mode control of coupled tanks. Mechatronics 16, 427–441 (2006) CrossRefGoogle Scholar
  20. 20.
    Yau, H.T., Chen, C.L.: Chattering-free fuzzy sliding-mode control strategy for uncertain chaotic systems. Chaos Solitons Fractals 30, 709–718 (2006) CrossRefGoogle Scholar
  21. 21.
    Shahnazi, R., Shanechi, H., Pariz, N.: Position control of induction and DC servomotors: a novel adaptive fuzzy PI sliding mode control. In: Power Engineering Society General Meeting, pp. 1–9 (2006) Google Scholar
  22. 22.
    Wang, T., Tong, Sh.Ch.: Fuzzy sliding mode control for nonlinear systems. Int. Conf. Mach. Learn. Cybern. 2, 839–844 (2004) Google Scholar
  23. 23.
    Moshiri, B., Jalili-Kharaajoo, M., Besharati, F.: Application of fuzzy sliding mode based on genetic algorithms to control of robotic manipulators. In: Emerging Technologies and Factory Automation, vol. 2, pp. 169–172 (2003) Google Scholar
  24. 24.
    Khoei, A., Hadidi, Kh., Khorasani, M.R., Amirkhanzadeh, R.: Fuzzy level control of a tank with optimum valve movement. Fuzzy Sets Syst. 150, 507–523 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Poulsen, N.K., Kouvaritakis, B., Cannon, M.: Nonlinear constrained predictive control applied to a coupled-tanks apparatus. IEE Proc. Control Theory Appl. 148, 17–24 (2001) CrossRefGoogle Scholar
  26. 26.
    Delavari, H., Ranjbar, A.: Robust intelligent control of coupled tanks. In: WSEAS International Conferences, pp. 1–6, Istanbul (2007) Google Scholar
  27. 27.
    Delavari, H., Ranjbar, A.: Genetic-based fuzzy sliding mode control of an interconnected twin-tanks. In: IEEE Region 8 EUROCON 2007 Conference, pp. 714–719, Poland (2007) Google Scholar
  28. 28.
    Alli, H., Yakut, O.: Fuzzy sliding-mode control of structures. Eng. Struct. 27, 277–284 (2005) CrossRefGoogle Scholar
  29. 29.
    Liang, C.Y., Su, J.P.: A new approach to the design of a fuzzy sliding mode controller. Fuzzy Sets Syst. 139, 111–124 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Hung, L.Ch., Lin, H.P., Chung, H.Y.: Design of self-tuning fuzzy sliding mode control for TORA system. Expert Syst. Appl. 32, 201–212 (2007) CrossRefGoogle Scholar
  31. 31.
    Yau, H.T., Chen, C.L.: Chattering-free fuzzy sliding-mode control strategy for uncertain chaotic systems. Chaos Solitons Fractals 30, 709–718 (2006) CrossRefGoogle Scholar
  32. 32.
    Hosein Nia, S.H., Ranjbar, A.N., Ganji, D.D., Soltani, H., Ghasemi, J.: Maintaining the stability of nonlinear differential equations by the enhancement of HPM. Phys. Lett. A 372, 2855–2861 (2008) CrossRefGoogle Scholar
  33. 33.
    Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.A.: Equilibrium points, stability and numerical solutions of fractional order predator–prey and rabies models. J. Math. Anal. Appl. 325(1), 542–553 (2007) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • H. Delavari
    • 1
  • A. N. Ranjbar
    • 2
  • R. Ghaderi
    • 1
  • S. Momani
    • 3
  1. 1.Faculty of Electrical and Computer EngineeringBabol (Noushirvani) University of TechnologyBabolIran
  2. 2.Faculty of EngineeringGolestan UniversityGorganIran
  3. 3.Department of MathematicsMutah UniversityAl-KarakJordan

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