Skip to main content
SpringerLink
Log in

Fast Control Systems: Nonlinear Approach

  • Chapter
  • First Online:
New Perspectives and Applications of Modern Control Theory

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The vector field f (resp. function f) is radially unbounded if \(x\rightarrow \infty \) implies \(\Vert f(x)\Vert \rightarrow +\infty \) (resp. \(|h(x)|\rightarrow +\infty \)).

  2. 2.

    A compact set \(\varOmega \) is strictly positively invariant for (12.13) if \(x_0\in \partial \varOmega \Rightarrow \varphi _{x_0}(t)\in \mathrm {int}(\varOmega ), t>0\).

References

  1. Adamy, J., Flemming, A.: Soft variable-structure controls: a survey. Automatica 40(11), 1821–1844 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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 

  3. Armstrong-Helouvry, B.: Control of Machines with Friction, vol. 128. Springer Science & Business Media (2012)

    Google Scholar 

  4. Bacciotti, A., Rosier, L.: Liapunov Functions and Stability in Control Theory. Springer Science & Business Media (2006)

    Google Scholar 

  5. Balakrishnan, A.: Superstability of systems. Appl. Math. Comput. 164(2), 321–326 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Bhat, S.P., Bernstein, D.S.: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751–766 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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. Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM (1990)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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. Coron, J.M., Praly, L.: Adding an integrator for the stabilization problem. Syst. Control Lett. 17(2), 89–104 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Creutz, D., Mazo Jr, M., Preda, C.: Superstability and finite time extinction for c_0-semigroups. arXiv:0907.4812 (2009)

  13. Cruz-Zavala, E., Moreno, J.A., Fridman, L.M.: Uniform robust exact differentiator. IEEE Trans. Autom. Control 56(11), 2727–2733 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Efimov, D., Perruquetti, W.: Oscillations conditions in homogenous systems. IFAC Proc. Vol. 43(14), 1379–1384 (2010)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Falkovich, G.: Fluid Mechanics: A Short Course for Physicists. Cambridge University Press (2011)

    Google Scholar 

  17. Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides: Control Systems, vol. 18. Springer Science & Business Media (2013)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Haimo, V.T.: Finite time controllers. SIAM J. Control Optim. 24(4), 760–770 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hale, J.: Functional differential equations. In: Analytic Theory of Differential Equations, pp. 9–22 (1971)

    Google Scholar 

  21. Hermes, H.: Nilpotent approximations of control systems and distributions. SIAM J. Control Optim. 24(4), 731–736 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  22. John, R.C.: Introduction to Calculus and Analysis, vol. I, XXIII 661 (1965)

    Google Scholar 

  23. Kawski, M.: Geometric homogeneity and stabilization. Nonlinear Control Syst. Des. 1995, 147 (2016)

    Google Scholar 

  24. Korobov, V.: A general approach to synthesis problem. Doklady Academii Nauk SSSR 248, 1051–1063 (1979)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Lyapunov, A.M.: The general problem of the stability of motion. Int. J. Control 55(3), 531–534 (1992)

    Article  MathSciNet  Google Scholar 

  27. Moulay, E., Perruquetti, W.: Finite time stability of differential inclusions. IMA J. Math. Control Inf. 22(4), 465–475 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Orlov, Y.: Finite time stability and robust control synthesis of uncertain switched systems. SIAM J. Control Optim. 43(4), 1253–1271 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44. Springer Science & Business Media (2012)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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 

  32. Polyakov, A.: Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans. Autom. Control 57(8), 2106–2110 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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. Polyakov, A., Efimov, D., Fridman, E., Perruquetti, W.: On homogeneous distributed parameters equations. IEEE Trans. Autom. Control 61(11), 3657–3662 (2016)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  39. Poznyak, A.S.: Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Technique, vol. 1. Elsevier, Amsterdam (2008)

    Google Scholar 

  40. Rosier, L.: Homogeneous lyapunov function for homogeneous continuous vector field. Syst. Control Lett. 19(6), 467–473 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  41. Rosier, L.: Etude de quelques problemes de stabilisation. Ph.D. thesis, School (1993)

    Google Scholar 

  42. Roxin, E.: On stability in control systems. J. Soc. Industrial Appl. Math. Ser. A: Control 3(3), 357–372 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  43. Roxin, E.: On finite stability in control systems. Rendiconti del Circolo Matematico di Palermo 15(3), 273–282 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  44. Ruzhansky, M., Sugimoto, M.: On global inversion of homogeneous maps. Bull. Math. Sci. 5(1), 13–18 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sabinina, E.: A class of non-linear degenerating parabolic equations. Doklady Akademii Nauk SSSR 143(4), 794 (1962)

    MathSciNet  MATH  Google Scholar 

  46. Tikhonov, A.N.: Systems of differential equations containing small parameters in the derivatives. Matematicheskii sbornik 73(3), 575–586 (1952)

    MathSciNet  Google Scholar 

  47. Utkin, V.I.: Sliding Modes in Control and Optimization. Springer Science & Business Media (2013)

    Google Scholar 

  48. Zubov, V.: On systems of ordinary differential equations with generalized homogenous right-hand sides. Mathematica 1, 80–88 (1958)

    Google Scholar 

  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. Zubov, V.I., Boron, L.F.: Methods of AM Lyapunov and their application. Noordhoff Groningen (1964)

    Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Inria Lille, 40 avenue Halley, 59650, Villenueve d’Ascq, France

    Andrey Polyakov

Authors
  1. Andrey Polyakov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrey Polyakov .

Editor information

Editors and Affiliations

  1. Centro de Investigaciones Económicas, Administrativas y Sociales, Instituto Politécnico Nacional, Mexico city, Mexico

    Julio B. Clempner

  2. Department of Automatic Control, Center for Research and Advanced Studies, Mexico city, Mexico

    Wen Yu

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Cite this chapter

Polyakov, A. (2018). Fast Control Systems: Nonlinear Approach. In: Clempner, J., Yu, W. (eds) New Perspectives and Applications of Modern Control Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-62464-8_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-62464-8_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-62463-1

  • Online ISBN: 978-3-319-62464-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation