363 research outputs found

    Extremal norms for positive linear inclusions

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    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

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    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

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    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

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    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 H∞H_\infty 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

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    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
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