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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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 135–154
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
Berman A and Plemmons RJ (1979). Nonnegative matrices in the mathematical sciences. Academic [Harcourt Brace Jovanovich Publishers], New York
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 Report
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 89–97
Dashkovskiy S, Rüffer BS, Wirth FR (2006c) Explicit ISS Lyapunov functions for networks (in preparation)
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 77–82
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
Grüne L (2002). Input-to-state dynamical stability and its Lyapunov function characterization. IEEE Trans Automat Control 47(9): 1499–1504
Hahn W (1967). Stability of motion. Springer, New York
Horvath CD and Lassonde M (1997). Intersection of sets with n-connected unions. Proc Am Math Soc 125(4): 1209–1214
Jiang ZP and Wang Y (2001). Input-to-state stability for discrete-time nonlinear systems. Automatica J IFAC 37(6): 857–869
Jiang ZP, Teel AR and Praly L (1994). Small-gain theorem for ISS systems and applications. Math Control Signals Syst 7(2): 95–120
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
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
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
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
Lancaster P and Tismenetsky M (1985). The theory of matrices 2nd edn. Academic, Orlando
Lur YY (2005). On the asymptotic stability of nonnegative matrices in max algebra. Linear Algebra Appl 407: 149–161
Michel AN and Miller RK (1977). Qualitative analysis of large scale dynamical systems. Academic [Harcourt Brace Jovanovich Publishers], New York
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 361–368
Rouche N, Habets P and Laloy M (1977). Stability theory by Liapunov’s direct method. Springer, New York
Šiljak DD (1979). Large-scale dynamic systems, North-Holland series in system science and engineering, vol.~3. North-Holland Publishing Co., New York
Sontag E and Teel A (1995). Changing supply functions in input/state stable systems. IEEE Trans Automat Control 40(8): 1476–1478
Sontag ED (1989). Smooth stabilization implies coprime factorization. IEEE Trans Automat Control 34(4): 435–443
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 443–467
Sontag ED and Wang Y (1996). New characterizations of input-to-state stability. IEEE Trans Automat Control 41(9): 1283–1294
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 York
Teel AR (1996). A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Trans Automat Control 41(9): 1256–1270
Teel AR (1998). Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem. IEEE Trans Automat Control 43(7): 960–964
Teel AR (2005) Input-to-state stability and the nonlinear small gain theorem, private communication, 2005
Vidyasagar M (1981) Input-output analysis of large-scale interconnected systems, Lecture Notes in Control and Information Sciences, vol. 29. Springer, Berlin
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dashkovskiy, S., Rüffer, B.S. & Wirth, F.R. An ISS small gain theorem for general networks. Math. Control Signals Syst. 19, 93–122 (2007). https://doi.org/10.1007/s00498-007-0014-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00498-007-0014-8