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Homogeneous Sliding Modes in Noisy Environments

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Emerging Trends in Sliding Mode Control

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 318))

Abstract

One of the main achievements of the high-order sliding mode control (HOSMC) theory is the standardized output-feedback regulation based on the robust high-order differentiation. The method employs universal HOSM controllers valid for any relative degree combined with standard HOSM differentiators. In this chapter, we present recently developed new universal controllers and filtering differentiators and demonstrate their output-feedback application in the presence of large sampling noises.

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References

  1. Andrieu, V., Praly, L., Astolfi, A.: Homogeneous approximation, recursive observer design, and output feedback. SIAM J. Control Optim. 47(4), 1814–1850 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Angulo, M.T., Fridman, L., Moreno, J.A.: Output-feedback finite-time stabilization of disturbed feedback linearizable nonlinear systems. Automatica 49(9), 2767–2773 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Angulo, M.T., Moreno, J.A., Fridman, L.M.: Robust exact uniformly convergent arbitrary order differentiator. Automatica 49(8), 2489–2495 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Apaza-Perez, W.A., Fridman, L., Moreno, J.A.: Higher order sliding-mode observers with scaled dissipative stabilisers. Int. J. Control 91(11), 2511–2523 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bacciotti, A., Rosier, L.: Liapunov Functions and Stability in Control Theory. Springer, London (2005)

    Book  MATH  Google Scholar 

  7. Barbot, J.-P., Levant, A., Livne, M., Lunz, D.: Discrete differentiators based on sliding modes. Automatica 112 (2020)

    Google Scholar 

  8. Barbot, J.-P., Saadaoui, H., Djemai, M., Manamanni, N.: Nonlinear observer for autonomous switching systems with jumps. Nonlinear Anal. Hybrid Syst. 1(4), 537–547 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bartolini, G.: Chattering phenomena in discontinuous control systems. Int. J. Syst. Sci. 20, 2471–2481 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bartolini, G., Ferrara, A., Usai, E.: Chattering avoidance by second-order sliding mode control. IEEE Trans. Autom. Control 43(2), 241–246 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bartolini, G., Pisano, A., Punta, E., Usai, E.: A survey of applications of second-order sliding mode control to mechanical systems. IEEE Trans. Autom. Control 76(9/10), 875–892 (2003)

    MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bhat, S.P., Bernstein, D.S.: Geometric homogeneity with applications to finite-time stability. Math. Control Signals Syst. 17(2), 101–127 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Boiko, I., Fridman, L., Pisano, A., Usai, E.: Analysis of chattering in systems with second-order sliding modes. IEEE Trans. Autom. Control 52(11), 2085–2102 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chalanga, A., Kamal, S., Fridman, L.M., Bandyopadhyay, B., Moreno, J.A.: Implementation of super-twisting control: super-twisting and higher order sliding-mode observer-based approaches. IEEE Trans. Ind. Electron. 63(6), 3677–3685 (2016)

    Article  Google Scholar 

  17. Clarke, F.H., Ledayev, Y.S., Stern, R.J.: Asymptotic stability and smooth Lyapunov functions. J. Differ. Equ. 149(1), 69–114 (1998)

    Article  MathSciNet  Google Scholar 

  18. Cruz-Zavala, E., Moreno, J.A.: Lyapunov approach to higher-order sliding mode design. In: Recent Trends in Sliding Mode Control, pp. 3–28. Institution of Engineering and Technology IET (2016)

    Google Scholar 

  19. Cruz-Zavala, E., Moreno, J.A.: Homogeneous high order sliding mode design: a Lyapunov approach. Automatica 80, 232–238 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cruz-Zavala, E., Moreno, J.A.: Levant’s arbitrary order exact differentiator: a Lyapunov approach. IEEE Trans. Autom. Control 64(7), 3034–39 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cruz-Zavala, E., Moreno, J.A., Fridman, L.: Uniform robust exact differentiator. IEEE Trans. Autom. Control 56(11), 2727–2733 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cruz-Zavala, E., Moreno, J.A., Fridman, L.: Uniform sliding mode controllers and uniform sliding surfaces. IMA J. Math. Control. Inf. 29(4), 491–505 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Defoort, M., Floquet, T., Kokosy, A., Perruquetti, W.: A novel higher order sliding mode control scheme. Syst. Control Lett. 58(2), 102–108 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ding, S.H., Levant, A., Li, S.H.: Simple homogeneous sliding-mode controller. Automatica 67(5), 22–32 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dinuzzo, F., Ferrara, A.: Higher order sliding mode controllers with optimal reaching. IEEE Trans. Autom. Control 54(9), 2126–2136 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Dvir, Y., Efimov, D., Levant, A., Polyakov, A., Perruquetti, W.: Acceleration of finite-time stable homogeneous systems. Int. J. Robust Nonlinear Control 28(5), 1757–1777 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Edwards, C., Spurgeon, S.K.: Sliding Mode Control: Theory And Applications. Taylor & Francis, London (1998)

    Book  MATH  Google Scholar 

  28. Efimov, D., Levant, A., Polyakov, A., Perruquetti, W.: Discretization of asymptotically stable homogeneous systems by explicit and implicit Euler methods. In: 55th IEEE Conference on Decision and Control, CDC’2016, Las-Vegas, December 12–14 (2016)

    Google Scholar 

  29. Efimov, D.V., Fridman, L.: A hybrid robust non-homogeneous finite-time differentiator. IEEE Trans. Autom. Control 56(5), 1213–1219 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Feng, Y., Yu, X., Man, Z.: Non-singular terminal sliding mode control of rigid manipulators. Automatica 38(12), 2159–2167 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic Publishers, Dordrecht (1988)

    Book  MATH  Google Scholar 

  32. Floquet, T., Barbot, J.-P.: Super twisting algorithm-based step-by-step sliding mode observers for nonlinear systems with unknown inputs. Int. J. Syst. Sci. 38(10), 803–815 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Floquet, T., Barbot, J.P., Perruquetti, W.: Higher-order sliding mode stabilization for a class of nonholonomic perturbed systems. Automatica 39(6), 1077–1083 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Fridman, L.: Chattering analysis in sliding mode systems with inertial sensors. Int. J. Control 76(9/10), 906–912 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. 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 International Publishing (2015)

    Google Scholar 

  36. Galias, Z., Yu, X.: Euler’s discretization of single input sliding-mode control systems. IEEE Trans. Autom. Control 52(9), 1726–1730 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Hanan, A., Jbara, A., Levant, A.: New homogeneous controllers and differentiators. In: Variable-Structure Systems and Sliding-Mode Control, pp. 3–28. Springer (2020)

    Google Scholar 

  38. Hanan, A., Jbara, A., Levant, A.: Homogeneous output-feedback control. In: Proceedings of the 21th IFAC World Congress, Berlin, 2020, accepted for presentation (2020)

    Google Scholar 

  39. Harmouche, M., Laghrouche, S., Chitour, Y., Hamerlain, M.: Stabilisation of perturbed chains of integrators using Lyapunov-based homogeneous controllers. Int. J. Control 90(12), 2631–2640 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  40. Isidori, A.: Nonlinear Control Systems I. Springer, New York (1995)

    Google Scholar 

  41. Kawski, M.: Homogeneous stabilizing feedback laws. Control Theory Adv. Technol. 6, 497–516 (1990)

    MathSciNet  Google Scholar 

  42. Koch, S., Reichhartinger, M., Horn, M., Fridman, L.: Discrete-time implementation of homogeneous differentiators. IEEE Trans. Autom. Control 65(2), 757–762 (2020)

    Article  MATH  Google Scholar 

  43. Laghrouche, S., Plestan, F., Glumineau, A.: Higher order sliding mode control based on integral sliding mode. Automatica 43(3), 531–537 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  44. Levant, A.: Sliding order and sliding accuracy in sliding mode control. Int. J. Control 58(6), 1247–1263 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  45. Levant, A.: Robust exact differentiation via sliding mode technique. Automatica 34(3), 379–384 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  46. Levant, A.: Higher order sliding modes, differentiation and output-feedback control. Int. J. Control 76(9/10), 924–941 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  47. Levant, A.: Homogeneity approach to high-order sliding mode design. Automatica 41(5), 823–830 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  48. Levant, A.: Quasi-continuous high-order sliding-mode controllers. IEEE Trans. Autom. Control 50(11), 1812–1816 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  49. Levant, A.: Chattering analysis. IEEE Trans. Autom. Control 55(6), 1380–1389 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  50. Levant, A.: On fixed and finite time stability in sliding mode control. In: Proceedings of the 52 IEEE Conference on Decision and Control, Florence, Italy, December 10–13, 2013 (2013)

    Google Scholar 

  51. Levant, A.: Non-Lyapunov homogeneous SISO control design. In: 56th Annual IEEE Conference on Decision and Control (CDC) 2017, pp. 6652–6657 (2017)

    Google Scholar 

  52. Levant, A.: Filtering differentiators and observers. In: 15th International Workshop on Variable Structure Systems (VSS), 2018, pp. 174–179 (2018)

    Google Scholar 

  53. Levant, A.: Sliding mode control: finite-time observation and regulation. In: Baillieul, J., Samad, T. (eds.) Encyclopedia of Systems and Control. Springer, London (2021). https://doi.org/10.1007/978-1-4471-5102-9_100069-1

  54. Levant, A., Efimov, D., Polyakov, A., Perruquetti, W.: Stability and robustness of homogeneous differential inclusions. In: Proceedings of the 55th IEEE Conference on Decision and Control, Las-Vegas, December 12–14, 2016 (2016)

    Google Scholar 

  55. Levant, A., Livne, M.: Exact differentiation of signals with unbounded higher derivatives. IEEE Trans. Autom. Control 57(4), 1076–1080 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  56. Levant, A., Livne, M.: Uncertain disturbances’ attenuation by homogeneous MIMO sliding mode control and its discretization. IET Control Theory Appl. 9(4), 515–525 (2015)

    Article  MathSciNet  Google Scholar 

  57. Levant, A., Livne, M.: Weighted homogeneity and robustness of sliding mode control. Automatica 72(10), 186–193 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  58. Levant, A., Livne, M.: Globally convergent differentiators with variable gains. Int. J. Control 91(9), 1994–2008 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  59. Levant, A., Livne, M.: Robust exact filtering differentiators. Eur. J. Control (2019)

    Google Scholar 

  60. Levant, A., Livne, M., Yu, X.: Sliding-mode-based differentiation and its application. IFAC-PapersOnLine 50(1), 1699–1704 (2017)

    Article  Google Scholar 

  61. Levant, A., Yu, X.: Sliding-mode-based differentiation and filtering. IEEE Trans. Autom. Control 63(9), 3061–3067 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  62. Livne, M., Levant, A.: Proper discretization of homogeneous differentiators. Automatica 50, 2007–2014 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  63. Man, Z., Paplinski, A.P., Wu, H.: A robust MIMO terminal sliding mode control scheme for rigid robotic manipulators. IEEE Trans. Autom. Control 39(12), 2464–2469 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  64. Moreno, J.A.: Levant’s arbitrary order differentiator with varying gain. In: Proceedings of the 20th IFAC World Congress, Toulouse, July 9–14, France, 2017 (2017)

    Google Scholar 

  65. Moreno, J.A., Osorio, M.: Strict Lyapunov functions for the super-twisting algorithm. IEEE Trans. Autom. Control 57(4), 1035–1040 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  66. Perruquetti, W., Floquet, T., Moulay, E.: Finite-time observers: application to secure communication. IEEE Trans. Autom. Control 53(1), 356–360 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  67. Pisano, A., Usai, E.: Sliding mode control: a survey with applications in math. Math. Comput. Simul. 81(5), 954–979 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  68. Plestan, F., Glumineau, A., Laghrouche, S.: A new algorithm for high-order sliding mode control. Int. J. Robust Nonlinear Control 18(4/5), 441–453 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  69. Polyakov, A.: Sliding mode control design using canonical homogeneous norm. Int. J. Robust Nonlinear Control 29(3), 682–701 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  70. Polyakov, A., Efimov, D., Perruquetti, W.: Finite-time and fixed-time stabilization: Implicit Lyapunov function approach. Automatica 51(1), 332–340 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  71. Polyakov, A., Efimov, D., Perruquetti, W.: Robust stabilization of MIMO systems in finite/fixed time. Int. J. Robust Nonlinear Control 26(1), 69–90 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  72. Polyakov, A., Fridman, L.: Stability notions and Lyapunov functions for sliding mode control systems. J. Frankl. Inst. 351(4), 1831–1865 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  73. Reichhartinger, M., Spurgeon, S.: An arbitrary-order differentiator design paradigm with adaptive gains. Int. J. Control 91(9), 2028–2042 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  74. Sanchez, T., Moreno, J.A., Fridman, L.M.: Output feedback continuous twisting algorithm. Automatica 96, 298–305 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  75. Shtessel, Y., Edwards, C., Fridman, L., Levant, A.: Sliding Mode Control and Observation. Birkhauser (2014)

    Google Scholar 

  76. Shtessel, Y.B., Shkolnikov, I.A.: Aeronautical and space vehicle control in dynamic sliding manifolds. Int. J. Control 76(9/10), 1000–1017 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  77. Utkin, V.I.: Sliding Modes in Control and Optimization. Springer, Berlin, Germany (1992)

    Book  MATH  Google Scholar 

  78. Utkin, V.I.: Sliding mode control design principles and applications to electric drives. IEEE Trans. Ind. Electron. 40(1), 23–36 (1993)

    Article  Google Scholar 

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Hanan, A., Jbara, A., Levant, A. (2021). Homogeneous Sliding Modes in Noisy Environments. In: Mehta, A., Bandyopadhyay, B. (eds) Emerging Trends in Sliding Mode Control. Studies in Systems, Decision and Control, vol 318. Springer, Singapore. https://doi.org/10.1007/978-981-15-8613-2_1

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