28 research outputs found
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
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
Monotone dynamische Systeme, Graphen und Stabilität großer gekoppelter Systeme
For a class of monotone operators T on the positive orthant of n-dimensional Euclidean space we introduce the concept of decay sets. These consist of points x satisfying T(x)<x. Considering the induced dynamical system x(k 1)=T(x(k)), we establish results relating stability properties of the origin, order conditions on T, and topological properties of decay sets. In particular, we construct paths in the decay sets and derive a quasi-invertibility result of the operator (Id-T).These results are applied to derive generalized small-gain type conditions for the input-to-state stability(ISS) of large-scale interconnections of (individually input-to-state stable) control systems: The interconnection topology together with the ISS gains yields a monotone operator with an inherent graph structure. We provide trajectory estimate based small-gain theorems and also construct ISS Lyapunov functions for the composite system. It is also shown how an algorithm due to Eaves can be used to numerically verify the small-gain condition