Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Input to State Stability

Living reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5102-9_78-1


The notion of input to state stability (ISS) qualitatively describes stability of the mapping from initial states and inputs to internal states (and more generally outputs). This entry focuses on the definition of ISS and a discussion of equivalent characterizations.


Input to state stability Lyapunov functions Dissipation Asymptotic stability 
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  1. Angeli D, Sontag ED, Wang Y (2000a) A characterization of integral input-to-state stability. IEEE Trans Autom Control 45(6):1082–1097CrossRefMATHMathSciNetGoogle Scholar
  2. Angeli D, Sontag ED, Wang Y (2000b) Further equivalences and semiglobal versions of integral input to state stability. Dyn Control 10(2):127–149MATHMathSciNetGoogle Scholar
  3. Angeli D, Ingalls B, Sontag ED, Wang Y (2004) Separation principles for input-output and integral-input-to-state stability. SIAM J Control Optim 43(1):256–276CrossRefMATHMathSciNetGoogle Scholar
  4. Freeman RA, Kokotović PV (1996) Robust nonlinear control design, state-space and Lyapunov techniques. Birkhauser, BostonCrossRefMATHGoogle Scholar
  5. Isidori A (1999) Nonlinear control systems II. Springer, LondonCrossRefMATHGoogle Scholar
  6. Isidori A, Marconi L, Serrani A (2003) Robust autonomous guidance: an internal model-based approach. Springer, LondonCrossRefGoogle Scholar
  7. Jiang Z-P, Teel A, Praly L (1994) Small-gain theorem for ISS systems and applications. Math Control Signals Syst 7:95–120CrossRefMATHMathSciNetGoogle Scholar
  8. Khalil HK (1996) Nonlinear systems, 2nd edn. Prentice-Hall, Upper Saddle RiverGoogle Scholar
  9. Krstić M, Deng H (1998) Stabilization of uncertain nonlinear systems. Springer, LondonMATHGoogle Scholar
  10. Krstić M, Kanellakopoulos I, Kokotović PV (1995) Nonlinear and adaptive control design. Wiley, New YorkGoogle Scholar
  11. Lin Y, Sontag ED, Wang Y (1996) A smooth converse Lyapunov theorem for robust stability. SIAM J Control Optim 34(1):124–160CrossRefMATHMathSciNetGoogle Scholar
  12. Massera JL (1956) Contributions to stability theory. Ann Math 64:182–206CrossRefMATHMathSciNetGoogle Scholar
  13. Praly L, Wang Y (1996) Stabilization in spite of matched unmodelled dynamics and an equivalent definition of input-to-state stability. Math Control Signals Syst 9:1–33CrossRefMATHMathSciNetGoogle Scholar
  14. Sepulchre R, Jankovic M, Kokotović PV (1997) Constructive nonlinear control. Springer, New YorkCrossRefMATHGoogle Scholar
  15. Sontag ED (1989) Smooth stabilization implies coprime factorization. IEEE Trans Autom Control 34(4):435–443CrossRefMATHMathSciNetGoogle Scholar
  16. Sontag ED (1998) Comments on integral variants of ISS. Syst Control Lett 34(1–2):93–100CrossRefMATHMathSciNetGoogle Scholar
  17. Sontag ED (1999) Stability and stabilization: discontinuities and the effect of disturbances. In: Nonlinear analysis, differential equations and control (Montreal, 1998). NATO science series C, Mathematical and physical sciences, vol 528. Kluwer Academic, Dordrecht, pp 551–598Google Scholar
  18. Sontag ED (2006) Input to state stability: basic concepts and results. In: Nistri P, Stefani G (eds) Nonlinear and optimal control theory. Springer, Berlin, pp 163–220Google Scholar
  19. Sontag ED, Wang Y (1995) On characterizations of the input-to-state stability property. Syst Control Lett 24(5):351–359CrossRefMATHMathSciNetGoogle Scholar
  20. Sontag ED, Wang Y (1996) New characterizations of input-to-state stability. IEEE Trans Autom Control 41(9):1283–1294CrossRefMATHMathSciNetGoogle Scholar
  21. Willems JC (1976) Mechanisms for the stability and instability in feedback systems. Proc IEEE 64:24–35CrossRefMathSciNetGoogle Scholar

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© Springer-Verlag London 2013

Authors and Affiliations

  1. 1.Rutgers UniversityNew BrunswickUSA