Small gain theorems for large scale systems and construction of ISS Lyapunov functions

We consider interconnections of n nonlinear subsystems in the input-to-state stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. Under a small gain assumption on the monotone operator induced by the gain matrix, a locally Lipschitz continuous ISS Lyapunov function is obtained constructively for the entire network by appropriately scaling the individual Lyapunov functions for the subsystems. The results are obtained in a general formulation of ISS, the cases of summation, maximization and separation with respect to external gains are obtained as corollaries.Comment: provisionally accepted by SIAM Journal on Control and Optimizatio

Small-gain stability theorems for positive Lur'e inclusions

Stability results are presented for a class of differential and difference inclusions, so-called positive Lur{\textquoteright}e inclusions which arise, for example, as the feedback interconnection of a linear positive system with a positive set-valued static nonlinearity. We formulate sufficient conditions in terms of weighted one-norms, reminiscent of the small-gain condition, which ensure that the zero equilibrium enjoys various global stability properties, including asymptotic and exponential stability. We also consider input-to-state stability, familiar from nonlinear control theory, in the context of forced positive Lur{\textquoteright}e inclusions. Typical for the study of positive systems, our analysis benefits from comparison arguments and linear Lyapunov functions. The theory is illustrated with examples

From convergent dynamics to incremental stability

Abstract-This paper advocates that the convergent systems property and incremental stability are two intimately related though different properties. Sufficient conditions for the convergent systems property usually rely upon first showing that a system is incrementally stable, as e.g. in the celebrated Demidovich condition. However, in the current paper it is shown that incremental stability itself does not imply the convergence property, or vice versa. Moreover, characterizations of both properties in terms of Lyapunov functions are given. Based on these characterizations, it is established that the convergence property implies incremental stability for systems evolving on compact sets, and also when a suitable uniformity condition is satisfied

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 Razumikhin approach for the incremental stability of delayed nonlinear systems

International audienceThis paper provides sufficient conditions for the incremental stability of time-delayed nonlinear systems. It relies on the Razumikhin-Lyapunov approach, which consists in invoking small-gain arguments by treating the delayed state as a feedback perturbation. The results are valid for multiple delays, as well as bounded time-varying delays. We provide conditions under which the limit solution of a time-delayed nonlinear system in response to a periodic (resp. constant) input is itself periodic and of the same period (resp. constant). As an illustration, a specific focus is given on a class of delayed Lur'e systems