A singular perturbation approach to saturated controller design with application to bounded stabilization of wing rock phenomenon

Original Paper


This paper proposes a singular perturbation-based approach to design saturated controllers for a class of nonlinear systems and shows its effectiveness in suppressing wing rock phenomenon in aircraft. Approximate dynamic inversion technique is exploited to design a high-gain dynamic controller which enforces time-scale separation in the closed-loop system. The convergence analysis of closed-loop system is done using the framework of contraction theory. Explicit tracking error bounds are derived both in the presence and absence of disturbances in system dynamics. The derived bounds are valid beyond a restrictive range of perturbation parameter, unlike Lyapunov-based approaches.


Singular perturbation Contraction theory Saturated controller Wing rock Approximate dynamic inversion 


  1. 1.
    Angeli, D.: A lyapunov approach to incremental stability properties. IEEE Trans. Autom. Control 47(3), 410–421 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chakrabortty, A., Arcak, M.: Time-scale separation redesigns for stabilization and performance recovery of uncertain nonlinear systems. Automatica 45(1), 34–44 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chen, G., Moiola, J.L., Wang, H.O.: Effects of the bogie and body inertia on the nonlinear wheel-set hunting recognized by the hopf bifurcation theory. Int. J. Bifurc. Chaos 10(3), 511–548 (2000)CrossRefGoogle Scholar
  4. 4.
    Del Vecchio, D., Slotine, J.J.E.: A contraction theory approach to singularly perturbed systems. IEEE Trans. Autom. Control 58(3), 752–757 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Forni, F., Sepulchre, R.: A differential lyapunov framework for contraction analysis. IEEE Trans. Autom. Control 59(3), 614–628 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Guglieri, G., Quagliotti, F.: Analytical and experimental analysis of wing rock. Nonlinear Dyn. 24(2), 129–146 (2001)CrossRefMATHGoogle Scholar
  7. 7.
    Hovakimyan, E.L.N., Cao, C.: Adaptive dynamic inversion via time-scale separation. In: Proceedings of 45th IEEE Conference Decision and Control, pp. 1075–1080. San Diego (2006)Google Scholar
  8. 8.
    Hovakimyan, E.L.N., Sasane, A.: Dynamic inversion for nonaffine-in-control systems via time-scale separation. Part i. J. Dyn. Control Syst. 13(4), 451–465 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hovakimyan, N., Lavretsky, E., Cao, C.: Dynamic inversion for multivariable non-affine-in-control systems via time-scale separation. Int. J. Control 81(12), 1960–1967 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hsu, C.-H., Lan, C.E.: Theory of wing rock. J. Aircr. 22(10), 920–924 (1985)CrossRefGoogle Scholar
  11. 11.
    Hu, T., Lin, Z.: Control Systems with Actuator Saturation: Analysis and Design. Springer, Berlin (2001)CrossRefMATHGoogle Scholar
  12. 12.
    Jouffroy, J., Lottin, J.: On the use of contraction theory for the design of nonlinear observers for ocean vehicles. Proc. Am. Control Conf. 4, 2647–2652 (2002)Google Scholar
  13. 13.
    Kaliora, G., Astolfi, A.: Nonlinear control of feedforward systems with bounded signals. IEEE Trans. Autom. Control. 49, 1975–1990 (2004)CrossRefMATHGoogle Scholar
  14. 14.
    Khalil, H.: Nonlinear Systems, 3rd edn. Prentice Hall, New Jersy (2002)MATHGoogle Scholar
  15. 15.
    Kokotovic, P.V., O’Reilly, J., Khalil, H.K.: Singular Perturbation Methods in Control: Analysis and Design. Academic Press Inc, Orlando (1986)MATHGoogle Scholar
  16. 16.
    Kuperman, A., Zhong, Q.-C.: Ude-based linear robust control for a class of nonlinear systems with application to wing rock motion stabilization. Nonlinear Dyn. 81(1–2), 789–799 (2015)CrossRefGoogle Scholar
  17. 17.
    Lohmiller, W., Slotine, J.-J.E.: On contraction analysis for non-linear systems. Automatica 34(6), 683–696 (1998)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Luo, J., Lan, C.E.: Control of wing-rock motion of slender delta wings. J. Guidance Control Dyn. 16(2), 225–231 (1993)CrossRefGoogle Scholar
  19. 19.
    Nayfeh, A., Elzebda, J., Mook, D.: Analytical study of the subsonic wing-rock phenomenon for slender delta wings. J. Aircr. 26(9), 805–809 (1989)CrossRefGoogle Scholar
  20. 20.
    Rayguru, M.M., Kar, I.N.: Contraction based stabilization of approximate feedback linearizable systems. In: Proceedings of European Control Conference, pp. 587–592. (2015)Google Scholar
  21. 21.
    Rayguru, M.M., Kar, I.N.: Contraction-based stabilisation of nonlinear singularly perturbed systems and application to high gain feedback. Int. J. Control 90, 1778–1792 (2016).  https://doi.org/10.1080/00207179.2016.1221139 MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Saberi, A., Khalil, H.: Stabilization and regulation of nonlinear singularly perturbed systems-composite control. IEEE Trans. Autom. Control 30(8), 739–747 (1985)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Sepulchre, R.: Slow peaking and low-gain designs for global stabilization of nonlinear systems. IEEE Trans. Autom. Control. 45, 453–461 (2000)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Serajian, R.: Parameters changing influence with different lateral stiffnesses on nonlinear analysis of hunting behavior of a bogie. J. Measurements Eng. 1(4), 195–206 (2013)Google Scholar
  25. 25.
    Sharma, B., Kar, I.: Adaptive control of wing rock system in uncertain environment using contraction theory. In: American Control Conference, pp. 2963–2968. IEEE (2008)Google Scholar
  26. 26.
    Sharma, B.B., Kar, I.N.: Adaptive control of wing rock system in uncertain environment using contraction theory. In: Proceedings of American Control Conference, pp. 2963–2968. (2008)Google Scholar
  27. 27.
    Sharma, B.B., Kar, I.N.: Observer-based synchronization scheme for a class of chaotic systems using contraction theory. Nonlinear Dyn. 63(3), 429–445 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Sontag, E.D.: Contractive systems with inputs. In: Willems, J., Hara, S., Ohta, Y., Fujioka, H. (eds.) Perspectives in Mathematical System Theory, Control, and Signal Processing, Series Lecture Notes in Control and Information Sciences, pp. 217–228. Springer, Berlin (2010)Google Scholar
  29. 29.
    Teel, A.R.: Global stabilization and restricted tracking for multiple integrators with bounded controls. Syst. Control Lett. 18, 165–171 (1992)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Teel, A.R.: A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Trans. Autom. Control 41, 1256–1270 (1996)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Teo, J., How, J.P., Lavretsky, E.: On approximate dynamic inversion and proportional-integral control. In 2009 American Control Conference, pp. 1592–1597. IEEE (2009)Google Scholar
  32. 32.
    Teo, J., How, J.P.: Equivalence between approximate dynamic inversion and proportional-integral control. In: Proceedings of 47th IEEE Conference Decision and Control, pp. 2179–2183. Cancun, Mexico (2008)Google Scholar
  33. 33.
    Teo, J., How, J.P., Lavretsky, E.: Proportional-integral controllers for minimum-phase nonaffine-in-control systems. IEEE Trans. Autom. Control 55(6), 1477–1483 (2010)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Wang, W., Slotine, J.-J.E.: On partial contraction analysis for coupled nonlinear oscillators. Biol. Cybern. 92(1), 38–53 (2005)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Yang, C., Sun, J., Ma, X.: Stabilization bound of singularly perturbed systems subject to actuator saturation. Automatica 49(2), 457–462 (2013)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Younesian, D., Jafari, A.A., Serajian, R.: Bifurcation control: theories, methods, and applications. Int. J. Auto Eng. 3(4), 186–196 (2011)Google Scholar
  37. 37.
    Zamani, M., Tabuada, P.: Backstepping design for incremental stability. IEEE Trans. Autom. Control 56(9), 2184–2189 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Electrical EngineeringIndian Institute of Technology DelhiNew DelhiIndia

Personalised recommendations