Nonlinear Dynamics

, Volume 93, Issue 4, pp 2543–2563 | Cite as

Fixed-time sliding mode attitude tracking control for a submarine-launched missile with multiple disturbances

  • Liang Zhang
  • Changzhu WeiEmail author
  • Liang Jing
  • Naigang Cui
Original Paper


This paper studies a novel adaptive fixed-time sliding mode attitude tracking control for a submarine-launched missile, which is affected by sea winds, sea waves, ocean currents and other disturbances during the water-exit process. Firstly, the nonlinear water-exit dynamic model of the submarine-launched missile is established, and then it is transformed into a simple second-order attitude tracking system. Subsequently, a novel non-singular fixed-time fast terminal sliding mode surface (NFFTSMS) with fixed-time convergence is presented, and the pre-established settling time is also developed. Moreover, a novel adaptive non-singular fixed-time fast terminal sliding mode control (ANFFTSMC) is presented by employing a fixed-time disturbance observer, a fixed-time differentiator and the proposed NFFTSMS. Closed-loop stability of the proposed controller is proved by utilizing the Lyapunov methodology. Finally, numerical simulations including two typical launch trajectories of the missile are carried out to demonstrate the strong robustness of the proposed control scheme.


Submarine-launched missile Attitude tracking control Non-singular fixed-time fast terminal sliding mode control Multiple disturbances Fixed-time disturbance observer Fixed-time differentiator 



The authors would like to thank the financial supports by the National Nature Science Fund of China (Grant No. 61403100), the open Fund of National Defense Key Discipline Laboratory of Micro-Spacecraft Technology (Grant No. HIT.KLOF.MST.201704), and the Fundamental Research Funds for the Central Universities (Grant No. HIT.NSRIF.2015.037).

Compliance with ethical standards

Conflict of interests

The authors declare that there is no conflict of interest regarding the publication of this paper.


  1. 1.
    Wang, Y.J.: Research on the attitude control and trajectory optimization in the process of water-exit for the submarine-launched missile (Master Dissertation). Harbin Institute of Technology, Harbin, China (2017)Google Scholar
  2. 2.
    Qi, Q., Chen, Q.G., Zhou, Y., Wang, H.Y., Zhou, H.M.: Submarine-launched cruise missile ejecting launch simulation and research. In: 2011 International Conference on Electronic and Mechanical Engineering and Information Technology, pp. 4542–4545. Harbin, China (2011)Google Scholar
  3. 3.
    Zhang, Z.X.: Dynamics modeling and simulation of water-exit course of small submarine-launched missile under wave disturbance. J. Natl. Univ. Def. Technol. 37(6), 91–95 (2015)Google Scholar
  4. 4.
    Yang, J., Feng, J.F., Li, Y.L., Liu, A., Hu, J.H., Ma, Z.C.: Water-exit process modeling and added-mass calculation of the submarine-launched missile. Pol. Marit. Res. 24, 152–164 (2017)CrossRefGoogle Scholar
  5. 5.
    Gao, Q.S.: The underwater vehicle motion control and research of visual simulation technology. Harbin Engineering University (Master Dissertation). Harbin, China (2011)Google Scholar
  6. 6.
    Lian, Y.Q., Tian, B., Wang, S.Z.: The simulation of submarine-launched missile out-water movement based on MTALBA/Simulink. Appl. Mech. Mater. 182–183, 1328–1332 (2012)CrossRefGoogle Scholar
  7. 7.
    Wang, S.Z., Wang, H.P., Yang, M., Wang, L.P.: Simulation on three dimensional water-exit trajectory of submarine launched missile. Adv. Mater. Res. 791–793, 1069–1072 (2013)CrossRefGoogle Scholar
  8. 8.
    Xu, X.Q., Tian, B., Li, B.S.: The model and simulation of submarine to surface missile underwater trajectory. J. Proj. Rockets Missiles Guid. 30(5), 149–152 (2010)Google Scholar
  9. 9.
    Bai, Y.L.: Research on the dynamics and nonlinear control of the submarine-launched missile in multimedia environment (Ph.D. Dissertation). Harbin Institute of Technology. Harbin, China (2013)Google Scholar
  10. 10.
    Ge, D.H., Zhu, H., Cai, P., Liu, H.G., Huang, X.: Attitude sliding mode control of deep-sea submerged buoy based on feedback linearization. Fire Control Command Control 41(1), 16–18 (2016)Google Scholar
  11. 11.
    Zou, A.M., Ruiter, A.H.J.D., Kumar, K.D.: Finite-time attitude tracking control for rigid spacecraft with control input constraints. IET Control Theory Appl. 11(7), 931–940 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Wu, Y.J., Zuo, J.X., Sun, L.H.: Smooth backstepping sliding mode control for missile attitude system based on parameters online adjusting and estimating for square of disturbance upper bound, Proc. Inst. Mech. Eng. Part G. J. Aerosp. Eng. (2017). Google Scholar
  13. 13.
    Jitpattanakul, A., Pukdeboon, C.: Adaptive output feedback integral sliding mode attitude tracking control of spacecraft without unwinding. Adv. Mech. Eng. 9(7), 1–16 (2017)CrossRefzbMATHGoogle Scholar
  14. 14.
    Lan, Q., Qian, C., Li, S.: Finite-time disturbance observer design and attitude tracking control of a rigid spacecraft. J. Dyn. Syst. Meas. Control 139(6), 1–12 (2017)CrossRefGoogle Scholar
  15. 15.
    Zong, Q., Zhang, X.Y., Shao, S.K., Tian, B.L., Liu, W.J.: Disturbance observer-based fault-tolerant attitude tracking control for rigid spacecraft with finite-time convergence. Proc. Inst. Mech. Eng. Part G. J. Aerosp. Eng. (2017). Google Scholar
  16. 16.
    Lee, D.: Nonlinear disturbance observer-based robust control of attitude tracking of rigid spacecraft. Nonlinear Dyn. 88(2), 1317–1328 (2017)CrossRefzbMATHGoogle Scholar
  17. 17.
    Wang, Z., Wu, Z.: Nonlinear attitude control scheme with disturbance observer for flexible spacecrafts. Nonlinear Dyn. 81(1–2), 257–264 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Liu, S.Y., Liu, Y.C., Wang, N.: Nonlinear disturbance observer-based backstepping finite-time sliding mode tacking control of underwater vehicles with system uncertainties and external disturbances. Nonlinear Dyn. 88(1), 465–476 (2017)CrossRefzbMATHGoogle Scholar
  19. 19.
    Utkin, V., Guldner, J., Shi, J.: Sliding Mode Control in Electro-Mechanical Systems. CRC Press, Boca Raton (2009)CrossRefGoogle Scholar
  20. 20.
    Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers, Dordrecht (1988)CrossRefGoogle Scholar
  21. 21.
    Roxin, E.: On finite stability in control systems. Rend. Circ. Mat. Palermo. 15(3), 273–283 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Drakunov, S., Dogruel, M., Ozguner, U.: Sliding mode control in hybrid systems. In: Proceedings of the 1993 IEEE International Symposium on Intelligent Control, pp. 186–189, Chicago, USA (1993)Google Scholar
  23. 23.
    Moulay, E., Perruquetti, W.: Finite time stability of differential inclusions. IMA J. Math. Control Inf. 22(4), 465–475 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Defoort, M., Floquet, T., Kokosy, A., Perruquetti, W.: A novel higher order sliding mode control scheme. Syst. Control Lett. 58(2), 102–108 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Bacciotti, A., Rosier, L.: Lyapunov Functions and Stability in Control Theory, 2nd edn. Springer, Berlin (2005)CrossRefzbMATHGoogle Scholar
  26. 26.
    Huang, Y., Jia, Y.: Adaptive fixed-time relative position tracking and attitude synchronization control for non-cooperative target spacecraft fly-around mission. J. Franklin Inst. 354(18), 8461–8489 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Basin, M., Panathula, C.B., Shtessel, Y.: Multivariable continuous fixed-time second-order sliding mode control: design and convergence time estimation. IET Control Theory Appl. 11(8), 1104–1111 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Jiang, B.Y., Hu, Q.L., Friswell, M.: Fixed-time attitude control for rigid spacecraft with actuator saturation and faults. IEEE Trans. Control Syst. Technol. 24(5), 1892–1898 (2016)CrossRefGoogle Scholar
  29. 29.
    Ning, B.D., Han, Q.L., Zuo, Z.Y., Jin, J., Zheng, J.C.: Collective behaviors of mobile robots beyond the nearest neighbor rules with switching topology. IEEE Trans. Cybern. (2017). Google Scholar
  30. 30.
    Ning, B.D., Han, Q.L., Zuo, Z.Y.: Distributed optimization for multiagent systems: an edge-based fixed-time consensus approach. IEEE Trans. Cybern. (2017). Google Scholar
  31. 31.
    Ni, J.K., Liu, L., Liu, C.X., Hu, X.Y.: Fractional order fixed-time nonsingular terminal sliding mode synchronization and control of fractional order chaotic systems. Nonlinear Dyn. 89(3), 2065–2083 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ni, J.K., Liu, L., Liu, C.X., Liu, J.: Fixed-time leader-following consensus for second-order multiagent systems with input delay. IEEE Trans. Ind. Electron. 64(11), 8635–8646 (2017)CrossRefGoogle Scholar
  33. 33.
    Ni, J.K., Liu, L., Liu, C.X., Hu, X.Y., Shen, T.S.: Fixed-time dynamic surface high-order sliding mode control for chaotic oscillation in power system. Nonlinear Dyn. 86(1), 401–420 (2016)CrossRefzbMATHGoogle Scholar
  34. 34.
    Andrieu, A., Praly, L., Astolfi, A.: Homogeneous approximation, recursive observer and output feedback. SIAM J. Control Optim. 47, 1814–1850 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Cruz-Zavala, E., Moreno, J., Fridman, L.: Uniform robust exact differentiator. IEEE Trans. Autom. Control 56(11), 2727–2733 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Polyakov, A.: Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans. Autom. Control 57(8), 2106–2110 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Polyakov, A., Fridman, L.: Stability notions and Lyapunov functions for sliding mode control systems. J. Franklin Inst. 351(4), 1831–1865 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Polyakov, A., Efimov, D., Perruquetti, W.: Finite-time and fixed-time stabilization: implicit Lyapunov function approach. Automatica 51, 332–340 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Bhat, S., Bernstein, D.: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751–766 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Li, H.F., Li, C.D., Huang, T.W., Quyang, D.Q.: Fixed-time stability and stabilization of impulsive dynamical systems. J. Franklin Inst. 354(18), 8626–8644 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Huang, Y., Jia, Y.M.: Fixed-time consensus tracking control for second-order multi-agent systems with bounded input uncertainties via NFFTSM. IET Control Theory Appl. 11(16), 2900–2909 (2017)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Chen, K.J.: Launch Vehicle Flight Dynamics and Guidance. National Defense Industry Press, Beijing (2014)Google Scholar
  43. 43.
    Yan, W.S.: Torpedo Navigation Mechnics. Northwestern Polytechnical University Press, Xian (2005)Google Scholar
  44. 44.
    Ni, J., Liu, L., Chen, M., Liu, C.: Fixed-time disturbance observer design for Brunovsky system IEEE Trans. Circuits Syst. II Exp. Briefs (2017). Google Scholar
  45. 45.
    Qian, C., Lin, W.: A Continuous feedback approach to global strong stabilization of nonlinear systems. IEEE Trans. Autom. Control 46(7), 1061–1079 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Polyakov, A.: Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans. Autom. Control 57(8), 2106–2110 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Basin, M., Yu, P., Shtessel, Y.: Finite- and fixed-time differentiators utilizing HOSM techniques. IET Control Theory Appl. 11(8), 1144–1152 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of AstronauticsHarbin Institute of TechnologyHarbinChina

Personalised recommendations