355 research outputs found
Extremal norms for positive linear inclusions
For finite-dimensional linear semigroups which leave a proper cone invariant
it is shown that irreducibility with respect to the cone implies the existence
of an extremal norm. In case the cone is simplicial a similar statement applies
to absolute norms. The semigroups under consideration may be generated by
discrete-time systems, continuous-time systems or continuous-time systems with
jumps. The existence of extremal norms is used to extend results on the
Lipschitz continuity of the joint spectral radius beyond the known case of
semigroups that are irreducible in the representation theory interpretation of
the word
Relaxed ISS Small-Gain Theorems for Discrete-Time Systems
In this paper ISS small-gain theorems for discrete-time systems are stated,
which do not require input-to-state stability (ISS) of each subsystem. This
approach weakens conservatism in ISS small-gain theory, and for the class of
exponentially ISS systems we are able to prove that the proposed relaxed
small-gain theorems are non-conservative in a sense to be made precise. The
proofs of the small-gain theorems rely on the construction of a dissipative
finite-step ISS Lyapunov function which is introduced in this work.
Furthermore, dissipative finite-step ISS Lyapunov functions, as relaxations of
ISS Lyapunov functions, are shown to be sufficient and necessary to conclude
ISS of the overall system.Comment: input-to-state stability, Lyapunov methods, small-gain conditions,
discrete-time non-linear systems, large-scale interconnection
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 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
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
Derandomized Distributed Multi-resource Allocation with Little Communication Overhead
We study a class of distributed optimization problems for multiple shared
resource allocation in Internet-connected devices. We propose a derandomized
version of an existing stochastic additive-increase and multiplicative-decrease
(AIMD) algorithm. The proposed solution uses one bit feedback signal for each
resource between the system and the Internet-connected devices and does not
require inter-device communication. Additionally, the Internet-connected
devices do not compromise their privacy and the solution does not dependent on
the number of participating devices. In the system, each Internet-connected
device has private cost functions which are strictly convex, twice continuously
differentiable and increasing. We show empirically that the long-term average
allocations of multiple shared resources converge to optimal allocations and
the system achieves minimum social cost. Furthermore, we show that the proposed
derandomized AIMD algorithm converges faster than the stochastic AIMD algorithm
and both the approaches provide approximately same solutions
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