An ISS Small-Gain Theorem for General Networks

We provide a generalized version of the nonlinear small-gain theorem for the case of more than two coupled input-to-state stable (ISS) 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 one. We give some interpretations of the condition in special cases covering two subsystems, linear gains, linear systems and an associated artificial dynamical system.Comment: 26 pages, 3 figures, submitted to Mathematics of Control, Signals, and Systems (MCSS

A generalisation of the nonlinear small-gain theorem for systems with abstract initial conditions

We consider the development of a general nonlinear small-gain theorem for systems with abstract initial conditions. Systems are defined in a set theoretic manner from input-output pairs on a doubly infinite time axis, and a general construction of the initial conditions (i.e. a state at time zero) is given in terms of an equivalence class of trajectories on the negative time axis. By using this formulation, an ISS-type nonlinear small-gain theorem is established with complete disconnection between the stability property and the existence, uniqueness properties. We provide an illustrative example

Revisiting the iISS small-gain theorem through transient plus ISS small-gain regulation

International audienceRecently, the small-gain theorem for input-to-state stable (ISS) systems has been extended to the class of integral input-to-state stable (iISS) systems. Feedback connections of two iISS systems are robustly stable with respect to disturbance if an extended small-gain condition is satisfied. It has been proved that at least one of the two iISS subsystems needs to be ISS for guaranteeing globally asymptotic stability and iISS of the overall system. Making use of this necessary condition for the stability, this paper gives a new interpretation to the iISS small gain theorem as transient plus ISS small-gain regulation. The observation provides useful information for designing and analyzing nonlinear control systems based on the iISS small-gain theorem

A Small-Gain Theorem with Applications to Input/Output Systems, Incremental Stability, Detectability, and Interconnections

A general ISS-type small-gain result is presented. It specializes to a small-gain theorem for ISS operators, and it also recovers the classical statement for ISS systems in state-space form. In addition, we highlight applications to incrementally stable systems, detectable systems, and to interconnections of stable systems.Comment: 16 pages, no figure

A small-gain theorem for motone systems with multivalued input-state characteristics

We provide a small-gain theorem for feedback interconnections of monotone input-output systems with multi-valued input-state characteristics. This extends a small-gain theorem of Angeli and Sontag for monotone systems with singleton-valued characteristics. We prove our theorem using Thieme\u27s convergence theory for asymptotically autonomous systems. We also provide an illustrative example. © 2006 IEEE