Fast Control Systems: Nonlinear Approach

  • Andrey PolyakovEmail author


This chapter treats the problem of fast control design for nonlinear systems. First, we discuss the question: which nonlinear system can be called fast? Next, we develop some tools for analysis and design of such control systems. The method generalized homogeneity is mainly utilized for these purposes. Finally, we survey possible research directions of the fast control systems.


Fast Control System Finite-time Stability Uniform Asymptotic Stability Homogeneous Norm Homogeneous Dilatation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This study is partially supported by The French National Research Agency, Grant ANR Finite4SoS (ANR 15 CE23 0007) and the Russian Federation Ministry of Education and Science, contract/grant numbers 02.G25.31.0111 and 14.Z50.31.0031.


  1. 1.
    Adamy, J., Flemming, A.: Soft variable-structure controls: a survey. Automatica 40(11), 1821–1844 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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
  3. 3.
    Armstrong-Helouvry, B.: Control of Machines with Friction, vol. 128. Springer Science & Business Media (2012)Google Scholar
  4. 4.
    Bacciotti, A., Rosier, L.: Liapunov Functions and Stability in Control Theory. Springer Science & Business Media (2006)Google Scholar
  5. 5.
    Balakrishnan, A.: Superstability of systems. Appl. Math. Comput. 164(2), 321–326 (2005)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bhat, S.P., Bernstein, D.S.: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751–766 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bhat, S.P., Bernstein, D.S.: Geometric homogeneity with applications to finite-time stability. Math. Control, Signals, Syst. (MCSS) 17(2), 101–127 (2005)Google Scholar
  8. 8.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM (1990)Google Scholar
  9. 9.
    Clarke, F.H., Ledyaev, Y.S., Stern, R.J.: Asymptotic stability and smooth lyapunov functions. J. Differ. Equ. 149(1), 69–114 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Coron, J.M., Nguyen, H.M.: Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach. Journal (2015)Google Scholar
  11. 11.
    Coron, J.M., Praly, L.: Adding an integrator for the stabilization problem. Syst. Control Lett. 17(2), 89–104 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Creutz, D., Mazo Jr, M., Preda, C.: Superstability and finite time extinction for c_0-semigroups. arXiv:0907.4812 (2009)
  13. 13.
    Cruz-Zavala, E., Moreno, J.A., Fridman, L.M.: Uniform robust exact differentiator. IEEE Trans. Autom. Control 56(11), 2727–2733 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Efimov, D., Perruquetti, W.: Oscillations conditions in homogenous systems. IFAC Proc. Vol. 43(14), 1379–1384 (2010)CrossRefGoogle Scholar
  15. 15.
    Efimov, D., Polyakov, A., Fridman, E., Perruquetti, W., Richard, J.P.: Comments on finite-time stability of time-delay systems. Automatica 50(7), 1944–1947 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Falkovich, G.: Fluid Mechanics: A Short Course for Physicists. Cambridge University Press (2011)Google Scholar
  17. 17.
    Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides: Control Systems, vol. 18. Springer Science & Business Media (2013)Google Scholar
  18. 18.
    Galaktionov, V.A., Vazquez, J.L.: Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations. Arch. Rational Mech. Anal. 129(3), 225–244 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Haimo, V.T.: Finite time controllers. SIAM J. Control Optim. 24(4), 760–770 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hale, J.: Functional differential equations. In: Analytic Theory of Differential Equations, pp. 9–22 (1971)Google Scholar
  21. 21.
    Hermes, H.: Nilpotent approximations of control systems and distributions. SIAM J. Control Optim. 24(4), 731–736 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    John, R.C.: Introduction to Calculus and Analysis, vol. I, XXIII 661 (1965)Google Scholar
  23. 23.
    Kawski, M.: Geometric homogeneity and stabilization. Nonlinear Control Syst. Des. 1995, 147 (2016)Google Scholar
  24. 24.
    Korobov, V.: A general approach to synthesis problem. Doklady Academii Nauk SSSR 248, 1051–1063 (1979)Google Scholar
  25. 25.
    Levant, A.: Homogeneity approach to high-order sliding mode design. Automatica 41(5), 823–830 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lyapunov, A.M.: The general problem of the stability of motion. Int. J. Control 55(3), 531–534 (1992)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Moulay, E., Perruquetti, W.: Finite time stability of differential inclusions. IMA J. Math. Control Inf. 22(4), 465–475 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Orlov, Y.: Finite time stability and robust control synthesis of uncertain switched systems. SIAM J. Control Optim. 43(4), 1253–1271 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44. Springer Science & Business Media (2012)Google Scholar
  30. 30.
    Perrollaz, V., Rosier, L.: Finite-time stabilization of 2x2 hyperbolic systems on tree-shaped networks. SIAM J. Control Optim. 52(1), 143–163 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Perruquetti, W., Floquet, T., Moulay, E.: Finite-time observers: application to secure communication. IEEE Trans. Autom. Control 53(1), 356–360 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Polyakov, A.: Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans. Autom. Control 57(8), 2106–2110 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Polyakov, A., Coron, J.M., Rosier, L.: On finite-time stabilization of evolution equations: a homogeneous approach. In: 2016 IEEE 55th Conference on Decision and Control (CDC), pp. 3143–3148. IEEE (2016)Google Scholar
  34. 34.
    Polyakov, A., Efimov, D., Fridman, E., Perruquetti, W.: On homogeneous distributed parameters equations. IEEE Trans. Autom. Control 61(11), 3657–3662 (2016)CrossRefzbMATHGoogle Scholar
  35. 35.
    Polyakov, A., Efimov, D., Perruquetti, W.: Finite-time and fixed-time stabilization: Implicit lyapunov function approach. Automatica 51, 332–340 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Polyakov, A., Efimov, D., Perruquetti, W., Richard, J.P.: Implicit lyapunov-krasovski functionals for stability analysis and control design of time-delay systems. IEEE Trans. Autom. Control 60(12), 3344–3349 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Polyakov, A., Fridman, L.: Stability notions and lyapunov functions for sliding mode control systems. J. Frankl. Inst. 351(4), 1831–1865 (2014)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Poznyak, A.S.: Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Technique, vol. 1. Elsevier, Amsterdam (2008)Google Scholar
  40. 40.
    Rosier, L.: Homogeneous lyapunov function for homogeneous continuous vector field. Syst. Control Lett. 19(6), 467–473 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Rosier, L.: Etude de quelques problemes de stabilisation. Ph.D. thesis, School (1993)Google Scholar
  42. 42.
    Roxin, E.: On stability in control systems. J. Soc. Industrial Appl. Math. Ser. A: Control 3(3), 357–372 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Roxin, E.: On finite stability in control systems. Rendiconti del Circolo Matematico di Palermo 15(3), 273–282 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Ruzhansky, M., Sugimoto, M.: On global inversion of homogeneous maps. Bull. Math. Sci. 5(1), 13–18 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Sabinina, E.: A class of non-linear degenerating parabolic equations. Doklady Akademii Nauk SSSR 143(4), 794 (1962)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Tikhonov, A.N.: Systems of differential equations containing small parameters in the derivatives. Matematicheskii sbornik 73(3), 575–586 (1952)MathSciNetGoogle Scholar
  47. 47.
    Utkin, V.I.: Sliding Modes in Control and Optimization. Springer Science & Business Media (2013)Google Scholar
  48. 48.
    Zubov, V.: On systems of ordinary differential equations with generalized homogenous right-hand sides. Mathematica 1, 80–88 (1958)Google Scholar
  49. 49.
    Zubov, V.: On systems of ordinary differential equations with generalized homogenous right-hand sides. izvestia vuzov. mathematica, J. 1, 80–88 (1958)Google Scholar
  50. 50.
    Zubov, V.I., Boron, L.F.: Methods of AM Lyapunov and their application. Noordhoff Groningen (1964)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Inria LilleVillenueve d’AscqFrance

Personalised recommendations