17,881 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
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
On a small-gain approach to distributed event-triggered control
In this paper the problem of stabilizing large-scale systems by distributed
controllers, where the controllers exchange information via a shared limited
communication medium is addressed. Event-triggered sampling schemes are
proposed, where each system decides when to transmit new information across the
network based on the crossing of some error thresholds. Stability of the
interconnected large-scale system is inferred by applying a generalized
small-gain theorem. Two variations of the event-triggered controllers which
prevent the occurrence of the Zeno phenomenon are also discussed.Comment: 30 pages, 9 figure
Asymptotic amplitudes and cauchy gains: A small-gain principle and an application to inhibitory biological feedback
The notions of asymptotic amplitude for signals, and Cauchy gain for
input/output systems, and an associated small-gain principle, are introduced.
These concepts allow the consideration of systems with multiple, and possibly
feedback-dependent, steady states. A Lyapunov-like characterization allows the
computation of gains for state-space systems, and the formulation of sufficient
conditions insuring the lack of oscillations and chaotic behaviors in a wide
variety of cascades and feedback loops. An application in biology (MAPK
signaling) is worked out in detail.Comment: Updates and replaces math.OC/0112021 See
http://www.math.rutgers.edu/~sontag/ for related wor
Asymptotic stability equals exponential stability, and ISS equals finite energy gain---if you twist your eyes
In this paper we show that uniformly global asymptotic stability for a family
of ordinary differential equations is equivalent to uniformly global
exponential stability under a suitable nonlinear change of variables. The same
is shown for input-to-state stability and input-to-state exponential stability,
and for input-to-state exponential stability and a nonlinear
estimate.Comment: 14 pages, several references added, remarks section added, clarified
constructio
- …