Discontinuous Integral Control for Systems with Relative Degree Two

  • Jaime A. MorenoEmail author


For systems with relative degree two we propose a homogeneous controller capable of tracking a smooth but unknown reference signal, despite of a Lipschitz continuous perturbation, and by means of a continuous control signal. The proposed control scheme consists of two terms: (i) a continuous and homogeneous state feedback, and (ii) a discontinuous integral term. The state feedback term aims at stabilizing (in finite time) the closed loop while the (discontinuous) integral term estimates the perturbation and the unknown reference signal in finite time and provides for perfect compensation in closed loop. By adding a continuous and homogeneous observer we complete an output feedback scheme, when not all states are available for measurement. The global finite time stability of the closed loop and its insensitivity with respect to matched Lipschitz continuous perturbations is proved in detail using a smooth and homogeneous Lyapunov function.



The author would like to thank the financial support from PAPIIT-UNAM (Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica), projects IN113614 and IN113617; Fondo de Colaboración II-FI UNAM, Project IISGBAS-100-2015; CONACyT (Consejo Nacional de Ciencia y Tecnología), project 241171.


  1. 1.
    Andrieu, V., Praly, L., Astolfi, A.: Homogeneous approximation, recursive observer design, and output feedback. SIAM J. Control Optim. 47(4), 1814–1850 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bacciotti, A., Rosier, L.: Liapunov functions and stability in control theory. Springer Science & Business Media (2006)Google Scholar
  3. 3.
    Bernuau, E., Efimov, D., Perruquetti, W., Polyakov, A.: On an extension of homogeneity notion for differential inclusions. In: Control conference (ECC), 2013 European, pp. 2204–2209. IEEE (2013)Google Scholar
  4. 4.
    Bernuau, E., Efimov, D., Perruquetti, W., Polyakov, A.: On homogeneity and its application in sliding mode control. J. Frankl. Inst. 351(4), 1866–1901 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bernuau, E., Polyakov, A., Efimov, D., Perruquetti, W.: Verification of iss, iiss and ioss properties applying weighted homogeneity. Syst. Control Lett. 62(12), 1159–1167 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bhat, S.P., Bernstein, D.S.: Geometric homogeneity with applications to finite-time stability. Math. Control Signal Syst. (MCSS) 17(2), 101–127 (2005)Google Scholar
  7. 7.
    Cruz-Zavala, E., Moreno, J.A.: Lyapunov functions for continuous and discontinuous differentiators. IFAC-Pap. OnLine 49(18), 660–665 (2016)Google Scholar
  8. 8.
    Cruz-Zavala, E., Moreno, J.A.: Homogeneous high order sliding mode design: A lyapunov approach. Automatica 80, 232–238 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cruz-Zavala E, Moreno J.A..: Lyapunov approach to higher-order sliding mode design. Control Robot. Sens. 3–28 (2016)Google Scholar
  10. 10.
    Deimling, K.: Multivalued Differential Equations, vol. 1. Walter de Gruyter, Berlin (1992)Google Scholar
  11. 11.
    Ding, S., Levant, A., Li, S.: Simple homogeneous sliding-mode controller. Automatica 67, 22–32 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Filippov, A.F.: Differential Equations With Discontinuous Right-hand Side. American Mathematical Society, 191–231 (1988)Google Scholar
  13. 13.
    Fridman, L., Levant, A.: Sliding mode control in engineering. In: High-order sliding modes, Marcel Dekker, USA (2002)Google Scholar
  14. 14.
    Fridman, L., Moreno, J.A., Bandyopadhyay, B., Kamal, S., Chalanga, A.: Continuous nested algorithms: The fifth generation of sliding mode controllers. In: Recent Advances in Sliding Modes: From Control to Intelligent Mechatronics, pp. 5–35. Springer, Berlin (2015)Google Scholar
  15. 15.
    Hahn, W.: Stability Of Motion, vol. 138. Springer, Berlin (1967)Google Scholar
  16. 16.
    Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1952)Google Scholar
  17. 17.
    Hermes, H.: Homogeneous coordinates and continuous asymptotically stabilizing feedback controls. Differ. Equ. Stab. Control 109 249–260 (1991)Google Scholar
  18. 18.
    Hestenes, M.R.: Calculus Of Variations And Optimal Control Theory. John Wiley and Sons Inc, Gauthier-Villars (1966)zbMATHGoogle Scholar
  19. 19.
    Isidori, A.: Nonlinear control systems II. Springer, London (1999)CrossRefzbMATHGoogle Scholar
  20. 20.
    Isidori, A.: Nonlinear Control Systems. Springer Science and Business Media (2013)Google Scholar
  21. 21.
    Kamal, S., Moreno, J.A., Chalanga, A., Bandyopadhyay, B., Fridman, L.M.: Continuous terminal sliding-mode controller. Automatica 69, 308–314 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Khalil, H.K.: Noninear Systems. Prentice-Hall, New Jersey (1996)Google Scholar
  23. 23.
    Levant, A.: Sliding order and sliding accuracy in sliding mode control. Int. J. Control 58(6), 1247–1263 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Levant, A.: Robust exact differentiation via sliding mode technique. Automatica 34(3), 379–384 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Levant, A.: Universal single-input-single-output (siso) sliding-mode controllers with finite-time convergence. IEEE Trans. Autom. Control 46(9), 1447–1451 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Levant, A.: Higher-order sliding modes, differentiation and output-feedback control. Int. J. Control 76(9–10), 924–941 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Levant, A.: Homogeneity approach to high-order sliding mode design. Automatica 41(5), 823–830 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Levant, A.: Principles of 2-sliding mode design. Automatica 43(4), 576–586 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Levant, A., Livne, M.: Weighted homogeneity and robustness of sliding mode control. Automatica 72, 186–193 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Moreno, J.A.: A lyapunov approach to output feedback control using second-order sliding modes. IMA J. Math. Control Inf. 291–308 (2012)Google Scholar
  31. 31.
    Moreno, J.A.: Discontinuous integral control for mechanical systems. In: Variable structure systems (VSS), 2016 14th International workshop, 142–147. IEEE, Newyork (2016)Google Scholar
  32. 32.
    Moreno, J.A., Osorio, M.: Strict lyapunov functions for the super-twisting algorithm. IEEE Trans. Autom. Control 57(4), 1035–1040 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Nakamura, H., Yamashita, Y., Nishitani, H.: Smooth lyapunov functions for homogeneous differential inclusions. In: SICE 2002. Proceedings of the 41st SICE Annual Conference, vol. 3, pp. 1974–1979. IEEE, Newyork (2002)Google Scholar
  34. 34.
    Nakamura, N., Nakamura, H., Yamashita, Y., Nishitani, H.: Homogeneous stabilization for input affine homogeneous systems. IEEE Trans. Autom. Control 54(9), 2271–2275 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Orlov, Y.: Finite time stability of homogeneous switched systems. In: Decision and Control. Proceedings of the 42nd IEEE Conference, vol. 4, pp. 4271–4276. IEEE, Newyork (2003)Google Scholar
  36. 36.
    Orlov, Y.V.: Discontinuous Systems: Lyapunov Analysis And Robust Synthesis Under Uncertainty Conditions. Springer Science and Business Media (2008)Google Scholar
  37. 37.
    Rosier, L.: Homogeneous lyapunov function for homogeneous continuous vector field. Syst. Control Lett. 19(6), 467–473 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Shtessel, Y., Edwards, C., Fridman, L., Levant, A.: Sliding Mode Control And Observation. Springer, Berlin (2014)Google Scholar
  39. 39.
    Torres-González, V., Fridman, L.M., Moreno, J.A.: Continuous twisting algorithm. In: Decision and control (CDC), IEEE 54th annual conference on 2015, pp. 5397–5401. IEEE (2015)Google Scholar
  40. 40.
    Utkin, V., Guldner, J., Shi, J.: Sliding Mode Control In Electro-mechanical Systems, vol. 34. CRC press (2009)Google Scholar
  41. 41.
    Utkin, V.I.: Sliding Modes In Control And Optimization. Springer Science and Business Media (2013)Google Scholar
  42. 42.
    Zamora, C.A., Moreno, J.A., Kamal, S.: Control integral dis-continuo para sistemas mecanicos. In: Congreso Anual de Asociaci6n Mexicana de Control Automatico (2013)Google Scholar
  43. 43.
    Zubov, V.I., Boron, L.F.: Methods of AM Lyapunov And Their Application. Noordhoff Groningen (1964)Google Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Instituto de IngenieríaUniversidad Nacional Autónoma de Mexico (UNAM)CoyoacánMexico

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