Nonlinear Dynamics

, Volume 80, Issue 3, pp 1463–1481 | Cite as

Nonlinear augmented observer design and application to quadrotor aircraft

  • Xinhua Wang
  • Bijan Shirinzadeh
Original Paper


This paper presents a nonlinear augmented observer, and it is applied to an underactuated quadrotor aircraft. Not only the proposed augmented observer can estimate the velocity from the position measurement, but also the uncertainties can be obtained. The robustness in time domain and in frequency domain is analyzed, respectively. The merits of the presented augmented observer include its synchronous estimation of velocity and uncertainties, finite-time stability, ease of parameters selection, and sufficient stochastic noise rejection. Moreover, a simple control law based on the augmented observer is designed to stabilize the flight dynamics. Simulation results illustrate the effectiveness of the proposed method.


Nonlinear augmented observer Underactuated Quadrotor aircraft Uncertainties Finite-time stability  Noise rejection 



This research is supported in part by Australian Research Council (ARC) Discovery (Grant Nos. DP 0986814, DP 110104970), ARC linkage infrastructure, Equipment and Facilities (Grant Nos. LE 0347024, LE 0668508).


  1. 1.
    Altug, E., Ostrowski, J.P., Mahony, R.: Control of a quadrotor helicopter using visual feedback. In: IEEE International Conference on Robotics & Automation, ICRA’02, Washington, DC, 11–15 May, pp. 72–77 (2002)Google Scholar
  2. 2.
    Lee, D., Lim, H., Kim, H.J., Kim, Y., Seong, K.J.: Adaptive image-based visual servoing for an underactuated quadrotor system. J. Guid. Control Dyn. 35(4), 1335–1353 (2012)CrossRefGoogle Scholar
  3. 3.
    Ryan, T., Kim, H.J.: LMI-based gain synthesis for simple robust quadrotor Control. IEEE Trans. Autom. Sci. Eng. 10(4), 1173–1178 (2013)CrossRefGoogle Scholar
  4. 4.
    Besnard, H., Shtessel, Y.B., Landrum, B.: Quadrotor vehicle control via sliding mode controller driven by sliding mode disturbance observer. J. Frankl. Inst. 349(2), 658–684 (2012)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Lee, D.J., Franchi, A., Son, H.I., Ha, C., Bulthoff, H.H., Giordano, P.R.: Semiautonomous haptic teleoperation control architecture of multiple unmanned aerial vehicles. IEEE/ASME Trans. Mechatron. 18(4), 1334–1345 (2013)CrossRefGoogle Scholar
  6. 6.
    Kendoul, F., Lara, D., Fantoni-Coichot, I., Lozano, R.: Real-time nonlinear embedded control for an autonomous quadrotor helicopter. J. Guid. Control Dyn. 30(4), 1049–1061 (2007)CrossRefGoogle Scholar
  7. 7.
    Salazar-Cruz, S., Escareno, J., Lara, D., Lozano, R.: Embedded control system for a four-rotor UAV. Int. J. Adapt. Control Signal Process. 21(2–3), 189–204 (2007)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Hoffmann, G., Huang, H., Waslander, S., Tomlin, C.: Quadrotor helicopter flight dynamics and control: theory and experiment. In: AIAA Guidance, Navigation and Control Conference and Exhibit, Hiltion Head, South Carolina, 20–23 August, pp. 1670–1689 (2007)Google Scholar
  9. 9.
    Grzonka, S., Grisetti, G., Burgard, W.: Towards a navigation system for autonomous indoor flying. In: IEEE International Conference on Robotics and Automation, Kobe, 12–17 May, pp. 2878–2883 (2009)Google Scholar
  10. 10.
    Hamela, T., Mahony, R.: Image based visual servo control for a class of aerial robotic system. Automatica 43(11), 1976–1983 (2007)Google Scholar
  11. 11.
    Guenard, N., Hamel, T., Mahony, R.: A practical visual servo control for an unmanned aerial vehicle. IEEE Trans. Robot. 24(2), 331–340 (2008)CrossRefGoogle Scholar
  12. 12.
    Kendoul, F., Fantoni, I., Nonami, K.: Optic flow-based vision system for autonomous 3d localization and control of small aerial vehicles. Robot. Auton. Syst. 57(6–7), 591–602 (2009)CrossRefGoogle Scholar
  13. 13.
    Valenti, M., Bethke, B., Fiore, G., How, J.P., Feron, E.: Indoor Multi-vehicle flight testbed for fault detection, isolation, and recovery. In: AIAA Guidance, Navigation, and Control Conference (GNC), Keystone, CO, (AIAA-2006-6200), pp. 1–18 (2006)Google Scholar
  14. 14.
    How, J.P., McGrew, J., Frank, A., Hines, G.: Autonomous aircraft flight control for constrained environments. In: IEEE International Conference on Robotics and Automation, ICRA 2008, Pasadena, CA, 19–23 May, pp. 2213–2214 (2008)Google Scholar
  15. 15.
    Achtelik, M., Bachrach, A., He, R., Prentice, S., Roy, N.: Stereo vision and laser odometry for autonomous helicopters in GPS-denied indoor environments. In: Proceedings of the SPIE, April, vol. 7332, pp. 733219-1–733219-10 (2009)Google Scholar
  16. 16.
    Chamseddine, A., Zhang, Y., Rabbath, C.A., Join, C., Theilliol, D.: Flatness-based trajectory planning/replanning for a quadrotor unmanned aerial vehicle. IEEE Trans. Aerosp. Electron. Syst. 48(4), 2832–2848 (2012)CrossRefGoogle Scholar
  17. 17.
    Carrillo, L.R., Dzul, A., Lozano, R.: Hovering quad-rotor control: a comparison of nonlinear controllers using visual feedback. IEEE Trans. Aerosp. Electron. Syst. 48(4), 3159–3170 (2012)CrossRefGoogle Scholar
  18. 18.
    Atassi, A.N., Khalil, H.K.: Separation results for the stabilization of nonlinear systems using different high-gain observer designs. Syst. Control Lett. 39(3), 183–191 (2000)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Levant, A.: High-order sliding modes, differentiation and output-feedback control. Int. J. Control 76(9–10), 924–941 (2003)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Wang, X., Chen, Z., Yang, G.: Finite-time-convergent differentiator based on singular perturbation technique. IEEE Trans. Autom. Control 52(9), 1731–1737 (2007)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Wang, X., Shirinzadeh, B.: Rapid-convergent nonlinear differentiator. Mech. Syst. Signal Process. 28(4), 414–431 (2012)CrossRefGoogle Scholar
  22. 22.
    Wang, X., Shirinzadeh, B.: Nonlinear continuous integral-derivative observer. Nonlinear Dyn. 77(3), 793–806 (2014)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Luque-Vega, L., Castillo-Toledo, B., Loukianov, A.G.: Robust block second order sliding mode control for a quadrotor. J. Frankl. Inst. 349(2), 719–739 (2012)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Patel, A.R., Patel, M.A., Vyas, D.R.: Modeling and analysis of quadrotor using sliding mode control. In: 2012 44th Southeastern Symposium on System Theory (SSST), Jacksonville, FL, 11–13 March, pp. 111–114 (2012)Google Scholar
  25. 25.
    Hwang, C.L.: Hybrid neural network under-actuated sliding-mode control for trajectory tracking of quad-rotor unmanned aerial vehicle. In: The 2012 International Joint Conference on Neural Networks (IJCNN), Brisbane, QLD, 10–15 June, pp. 1–8 (2012)Google Scholar
  26. 26.
    Yeh, F.K.: Attitude controller design of mini-unmanned aerial vehicles using fuzzy sliding-mode control degraded by white noise interference. IET Control Theory Appl. 6(9), 1205–1212 (2012)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Wang, L.X.: Fuzzy systems are universal approximations. In: Proceedings of the IEEE International Conference on Fuzzy Systems, San Diego, 8–12 March, pp. 1163–1170 (1992)Google Scholar
  28. 28.
    Park, J., Sandberg, I.W.: Universal approximation using radial-basis-function networks. Neural Comput. 3, 246–257 (1991)CrossRefGoogle Scholar
  29. 29.
    Bhat, S.P., Bernstein, D.S.: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751–766 (2000)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Haimo, V.T.: Finite time controllers. SIAM J. Control Optim. 24(4), 760–771 (1986)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Bhat, S.P., Bernstein, D.S.: Geometric homogeneity with applications to finite-time stability. Math. Control Signals Syst. 17(2), 101–127 (2005)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Li, S., Du, H., Lin, X.: Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics. Automatica 47(8), 1706–1712 (2011)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Sun, H., Li, S., Sun, C.: Finite time integral sliding mode control of hypersonic vehicles. Nonlinear Dyn. 73(1–2), 229–244 (2013)CrossRefMATHGoogle Scholar
  34. 34.
    Hu, Q., Li, B., Zhang, A.: Robust finite-time control allocation in spacecraft attitude stabilization under actuator misalignment. Nonlinear Dyn. 73(1–2), 53–71 (2013)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Aghababa, M.P., Aghababa, H.P.: Chaos suppression of rotational machine systems via finite-time control method. Nonlinear Dyn. 69(4), 1881–1888 (2012)Google Scholar
  36. 36.
    Guo, Z., Huang, L.: Global exponential convergence and global convergence in finite time of non-autonomous discontinuous neural networks. Nonlinear Dyn. 58(1–2), 349–359 (2009)Google Scholar
  37. 37.
    Kahlil, H.K.: Nonlinear Systems, 3rd edn. Prentice-Hall, Englewood Cliffs (2002)Google Scholar
  38. 38.
    Slotine, J.J.E., Li, W.: Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs (1991)MATHGoogle Scholar
  39. 39.
    Munkres, J.R.: Topology a First Course. Prentice-Hall, Englewood Cliffs (1975)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Robotics and Mechatronics Research Laboratory, Department of Mechanical and Aerospace EngineeringMonash UniversityMelbourneAustralia

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