Advertisement

An ISS small gain theorem for general networks

  • Sergey Dashkovskiy
  • Björn S. Rüffer
  • Fabian R. Wirth
Original Article

Abstract

We provide a generalized version of the nonlinear small gain theorem for the case of more than two coupled input-to-state stable systems. For this result the interconnection gains are described in a nonlinear gain matrix, and the small gain condition requires bounds on the image of this gain matrix. The condition may be interpreted as a nonlinear generalization of the requirement that the spectral radius of the gain matrix is less than 1. We give some interpretations of the condition in special cases covering two subsystems, linear gains, linear systems and an associated lower-dimensional discrete time dynamical system.

Keywords

Interconnected systems Input-to-state stability Small gain theorem Large-scale systems Monotone maps 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Angeli D, Sontag E (2004) Interconnections of monotone systems with steady-state characteristics. In: Optimal control, stabilization and nonsmooth analysis, Lecture Notes in Control and Inform. Sci., vol. 301. Springer, Berlin, pp 135–154Google Scholar
  2. 2.
    Angeli D, De Leenheer P and Sontag ED (2004). A small-gain theorem for almost global convergence of monotone systems. Syst Control Lett 52(5): 407–414 MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Berman A and Plemmons RJ (1979). Nonnegative matrices in the mathematical sciences. Academic [Harcourt Brace Jovanovich Publishers], New York MATHGoogle Scholar
  4. 4.
    Dashkovskiy S, Rüffer BS, Wirth FR (2006a) Construction of ISS Lyapunov functions for networks. Tech. Rep. 06-06, Zentrum für Technomathematik, University of Bremen, Technical ReportGoogle Scholar
  5. 5.
    Dashkovskiy S, Rüffer BS, Wirth FR (2006b) Discrete time monotone systems: Criteria for global asymptotic stability and applications. In: Proceedings of the 17th international symposium on mathematical theory of networks and systems (MTNS), Kyoto, Japan, pp 89–97Google Scholar
  6. 6.
    Dashkovskiy S, Rüffer BS, Wirth FR (2006c) Explicit ISS Lyapunov functions for networks (in preparation)Google Scholar
  7. 7.
    Dashkovskiy S, Rüffer BS, Wirth FR (2006d) An ISS Lyapunov function for networks of ISS systems. In: Proceedings of the 17th international symposium on mathematical theory of networks and systems (MTNS), Kyoto, Japan, pp 77–82Google Scholar
  8. 8.
    Enciso GA and Sontag ED (2006). Global attractivity, I/O monotone small-gain theorems and biological delay systems. Discrete Contin Dyn Syst 14(3): 549–578 MATHMathSciNetGoogle Scholar
  9. 9.
    Grüne L (2002). Input-to-state dynamical stability and its Lyapunov function characterization. IEEE Trans Automat Control 47(9): 1499–1504 CrossRefMathSciNetGoogle Scholar
  10. 10.
    Hahn W (1967). Stability of motion. Springer, New York MATHGoogle Scholar
  11. 11.
    Horvath CD and Lassonde M (1997). Intersection of sets with n-connected unions. Proc Am Math Soc 125(4): 1209–1214 MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Jiang ZP and Wang Y (2001). Input-to-state stability for discrete-time nonlinear systems. Automatica J IFAC 37(6): 857–869 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Jiang ZP, Teel AR and Praly L (1994). Small-gain theorem for ISS systems and applications. Math Control Signals Syst 7(2): 95–120 MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Jiang ZP, Mareels IMY and Wang Y (1996). A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems. Automatica J IFAC 32(8): 1211–1215 MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Jiang ZP, Lin Y and Wang Y (2004). Nonlinear small-gain theorems for discrete-time feedback systems and applications. Automatica J IFAC 40(12): 2129–2136 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Karow M, Hinrichsen D and Pritchard AJ (2006). Interconnected systems with uncertain couplings: Explicit formulae for mu-values, spectral value sets, and stability radii. SIAM J Control Optim 45(3): 856–884 CrossRefMathSciNetGoogle Scholar
  17. 17.
    Laila DS and Nešić D (2003). Discrete-time Lyapunov-based small-gain theorem for parameterized interconnected ISS systems. IEEE Trans Automat Control 48(10): 1783–1788 CrossRefMathSciNetGoogle Scholar
  18. 18.
    Lancaster P and Tismenetsky M (1985). The theory of matrices 2nd edn. Academic, Orlando MATHGoogle Scholar
  19. 19.
    Lur YY (2005). On the asymptotic stability of nonnegative matrices in max algebra. Linear Algebra Appl 407: 149–161 MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Michel AN and Miller RK (1977). Qualitative analysis of large scale dynamical systems. Academic [Harcourt Brace Jovanovich Publishers], New York MATHGoogle Scholar
  21. 21.
    Potrykus HG, Allgöwer F, Qin SJ (2003) The character of an idempotent-analytic nonlinear small gain theorem. In: Positive systems (Rome, 2003), Lecture Notes in Control and Inform. Sci., vol 294. Springer, Berlin, pp 361–368Google Scholar
  22. 22.
    Rouche N, Habets P and Laloy M (1977). Stability theory by Liapunov’s direct method. Springer, New York MATHGoogle Scholar
  23. 23.
    Šiljak DD (1979). Large-scale dynamic systems, North-Holland series in system science and engineering, vol.~3. North-Holland Publishing Co., New York Google Scholar
  24. 24.
    Sontag E and Teel A (1995). Changing supply functions in input/state stable systems. IEEE Trans Automat Control 40(8): 1476–1478 MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Sontag ED (1989). Smooth stabilization implies coprime factorization. IEEE Trans Automat Control 34(4): 435–443 MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Sontag ED (2001) The ISS philosophy as a unifying framework for stability-like behavior. In: Nonlinear control in the year 2000, vol. 2 (Paris), Lecture Notes in Control and Inform. Sci., vol 259. Springer, London, pp 443–467Google Scholar
  27. 27.
    Sontag ED and Wang Y (1996). New characterizations of input-to-state stability. IEEE Trans Automat Control 41(9): 1283–1294 MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Szász G (1963) Introduction to lattice theory. Third revised and enlarged edition. MS revised by R. Wiegandt; translated by B. Balkay and G. Tóth, Academic, New YorkGoogle Scholar
  29. 29.
    Teel AR (1996). A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Trans Automat Control 41(9): 1256–1270 MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Teel AR (1998). Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem. IEEE Trans Automat Control 43(7): 960–964 MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Teel AR (2005) Input-to-state stability and the nonlinear small gain theorem, private communication, 2005Google Scholar
  32. 32.
    Vidyasagar M (1981) Input-output analysis of large-scale interconnected systems, Lecture Notes in Control and Information Sciences, vol. 29. Springer, BerlinGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2007

Authors and Affiliations

  • Sergey Dashkovskiy
    • 1
  • Björn S. Rüffer
    • 1
  • Fabian R. Wirth
    • 2
  1. 1.Universität Bremen, Zentrum für TechnomathematikBremenGermany
  2. 2.The Hamilton InstituteNUI MaynoothMaynoothIreland

Personalised recommendations