Fundamentals of Nonlinear Systems

  • Zhiyong ChenEmail author
  • Jie Huang
Part of the Advanced Textbooks in Control and Signal Processing book series (C&SP)


In this chapter, we review some fundamental concepts and properties of nonlinear control systems that will be referred to in the subsequent chapters.


  1. 1.
    Rudin W (1976) Principles of mathematical analysis, 3rd edn. McGraw-Hill, New YorkzbMATHGoogle Scholar
  2. 2.
    Khalil H (2002) Nonlinear systems. Prentice Hall, New JerseyzbMATHGoogle Scholar
  3. 3.
    Chen Z, Huang J (2005) Robust input-to-state stability and small gain theorem for nonlinear systems containing time-varying uncertainty. Advanced robust and adaptive control-theory and applications. Springer, New YorkGoogle Scholar
  4. 4.
    Isidori A (1995) Nonlinear control systems, 3rd edn. Springer, New YorkCrossRefzbMATHGoogle Scholar
  5. 5.
    Isidori A (1999) Nonlinear control systems, vol II. Springer, New YorkzbMATHGoogle Scholar
  6. 6.
    Krstic M, Kanellakopoulos I, Kokotović P (1995) Nonlinear and adaptive control design. Wiley, New YorkGoogle Scholar
  7. 7.
    Nijmeijer H, van der Schaft AJ (1990) Nonlinear dynamical control systems. Springer, New YorkCrossRefzbMATHGoogle Scholar
  8. 8.
    Slotine JJ, Li W (1991) Applied nonlinear control. Prentice Hall, Englewood CliffszbMATHGoogle Scholar
  9. 9.
    Ilchmann A (1993) Non-identifier-based high-gain adaptive control. Springer, BerlinzbMATHGoogle Scholar
  10. 10.
    Mareels I, Polderman JW (1996) Adaptive systems: an introduction. Birkhäuser, BostonCrossRefGoogle Scholar
  11. 11.
    Marino R, Tomei P (1995) Nonlinear control design: geometric, adaptive and robust. Prentice Hall, Englewood CliffszbMATHGoogle Scholar
  12. 12.
    Narendra KS, Annaswamy AM (1989) Stable adaptive systems. Printice-Hall, Englewood CliffszbMATHGoogle Scholar
  13. 13.
    LaSalle JP (1968) Stability theory for ordinary differential equations. J Diff Equat 4:57–65CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Yoshizawa T (1966) Stability theory by Lyapunov’s second method. The Mathematical Society of Japan, TokyoGoogle Scholar
  15. 15.
    Boyd S, Sastry S (1983) On parameter convergence in adaptive control. Syst Control Lett 3:311–319CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Liu L, Chen Z, Huang J (2009) Parameter convergence and minimal internal model with an adaptive output regulation problem. Automatica 45:1306–1311CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Ortega R, Fradkov A (1993) Asymptotic stability of a class of adaptive systems. Int J Adapt Control Signal Process 7:255–260CrossRefzbMATHGoogle Scholar
  18. 18.
    Yuan JS-C, Wonham WM (1977) Probing signals for model reference identification. IEEE Trans Autom Control 22:530–538CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Loria A, Panteley E (2002) Uniform exponential stability of linear time-varying systems: revisited. Syst Control Lett 47:13–24CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Sontag ED (1989) Smooth stabilization implies coprime factorization. IEEE Trans Autom Control 34:435–443CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Sontag ED (1990) Further facts about input to state stabilization. IEEE Trans Autom Control 34:473–476CrossRefMathSciNetGoogle Scholar
  22. 22.
    Sontag ED (1995) On the input-to-state stability property. Int J Control 1:24–36zbMATHGoogle Scholar
  23. 23.
    Sontag ED, Wang Y (1996) New characterizations of input-to-state stability. IEEE Trans Autom Control 41:1283–1294CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Sontag ED, Wang Y (1999) Notions of input to output stability. Syst Control Lett 38:351–359CrossRefMathSciNetGoogle Scholar
  25. 25.
    Sontag ED, Wang Y (2001) Lyapunov characterizations of input to output stability. SIAM J Control Optim 39:226–249CrossRefGoogle Scholar
  26. 26.
    Sontag ED, Teel A (1995) Changing supply function in input/state stable systems. IEEE Trans Autom Control 40:1476–1478CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Ilchmann A, Ryan EP (1994) Universal \(\lambda \)-tracking for nonlinearly perturbed systems in the presence of noise. Automatica 30:337–346CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Ryan EP (1994) A nonlinear universal servomechanism. IEEE Trans Autom Control 39:753–761CrossRefzbMATHGoogle Scholar
  29. 29.
    Ye XD, Huang J (2003) Decentralized adaptive output regulation for a class of large-scale nonlinear systems. IEEE Trans Autom Control 48:276–281CrossRefMathSciNetGoogle Scholar
  30. 30.
    Jiang ZP, Mareels I (1997) A small-gain control method for nonlinear cascaded systems with dynamic uncertainties. IEEE Trans Autom Control 42:292–308CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Jiang ZP, Mareels I, Wang Y (1996) A Lyapunov formulation of the nonlinear small gain theorem for interconnected ISS systems. Automatica 32:1211–1215CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Jiang ZP, Teel AR, Praly L (1994) Small-gain theorem for ISS systems and applications. Math Control Signals Systems 7:95–120CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Chen Z, Huang J (2005) A simplified small gain theorem for time-varying nonlinear systems. IEEE Trans Autom Control 50:1904–1908CrossRefGoogle Scholar
  34. 34.
    Angeli D, Sontag ED, Wang Y (2000) A characterization of integral input-to-state stability. IEEE Trans Autom Control 45(6):1082–1097CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.School of Electrical Engineering and Computer ScienceUniversity of NewcastleCallaghanAustralia
  2. 2.Department of Mechanical and Automation EngineeringThe Chinese University of Hong KongShatinChina

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