An ISS small gain theorem for general networks
- 557 Downloads
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.
KeywordsInterconnected systems Input-to-state stability Small gain theorem Large-scale systems Monotone maps
Unable to display preview. Download preview PDF.
- 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
- 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.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.Dashkovskiy S, Rüffer BS, Wirth FR (2006c) Explicit ISS Lyapunov functions for networks (in preparation)Google Scholar
- 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
- 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
- 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
- 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
- 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
- 31.Teel AR (2005) Input-to-state stability and the nonlinear small gain theorem, private communication, 2005Google Scholar
- 32.Vidyasagar M (1981) Input-output analysis of large-scale interconnected systems, Lecture Notes in Control and Information Sciences, vol. 29. Springer, BerlinGoogle Scholar