Stochastic impulsive systems driven by renewal processes

Abstract — Necessary and sufficient conditions are provided for stochastic stability and mean exponential stability of impulsive systems with jumps triggered by a renewal process, that is, the intervals between jumps are independent and identically distributed. The conditions for stochastic stability can be efficiently tested in terms of the feasibility of a set of LMIs or in terms of an algebraic test. The relation between the different stability notions for this class of systems is also discussed. The results are illustrated through their application to the stability analysis of networked control systems. We present two benchmark examples for which one can guarantee stability for inter-sampling times roughly twice as large as in a previous paper. I

Control of Impulsive Renewal Systems: Application to Direct Design in Networked Control

Abstract — We consider the control of impulsive systems with jumps triggered by a renewal process, that is, the intervals between jumps are independent and identically distributed. The control action and output measurement are assumed to take place only at jump times. Necessary and sufficient conditions, in the form of LMIs, are given for mean square stabilizability and detectability for the class of systems considered. An infinite horizon quadratic optimal control problem is solved, under appropriate stabilizability and detectability properties. The class of impulsive renewal systems is shown to be especially suited to model networked control systems utilizing CSMA-type protocols, with stochastic intervals between transmissions and packet drops. In this setting, the analysis and synthesis tools mentioned above are used to (i) prove that for an emulationbased design, stability of the closed-loop is preserved if the distribution of the intervals between transmissions assigns high probability to fast sampling (ii) illustrate through a benchmark example the potential advantages of controller direct-design over an emulation-based design. I

Lyapunov Conditions for Input-to-State Stability of Impulsive Systems

This paper introduces appropriate concepts of input-to-state stability (ISS) and integral-ISS for impulsive systems, i.e., dynamical systems that evolve according to ordinary differential equations most of the time, but occasionally exhibit discontinuities (or impulses). We provide a set of Lyapunov-based sufficient conditions for establishing these ISS properties. When the continuous dynamics are ISS but the discrete dynamics that govern the impulses are not, the impulses should not occur too frequently, which is formalized in terms of an average dwell-time (ADT) condition. Conversely, when the impulse dynamics are ISS but the continuous dynamics are not, there must not be overly long intervals between impulses, which is formalized in terms of a novel reverse ADT condition. We also investigate the cases where (i) both the continuous and discrete dynamics are ISS and (ii) one of these is ISS and the other only marginally stable for the zero input, while sharing a common Lyapunov function. In the former case we obtain a stronger notion of ISS, for which a necessary and sufficient Lyapunov characterization is available. The use of the tools developed herein is illustrated through examples from a Micro-Electro-Mechanical System (MEMS) oscillator and a problem of remote estimation over a communication network

An in-reachability based classification of invariant synchrony patterns in weighted coupled cell networks

This paper presents an in-reachability based classification of invariant synchrony patterns in Coupled Cell Networks (CCNs). These patterns are encoded through partitions on the set of cells, whose subsets of synchronized cells are called colors. We study the influence of the structure of the network in the qualitative behavior of invariant synchrony sets, in particular, with respect to the different types of (cumulative) in-neighborhoods and the in-reachability sets. This motivates the proposed approach to classify the partitions into the categories of strong, rooted and weak, according to how their colors are related with respect to the connectivity structure of the network. Furthermore, we show how this classification system acts under the partition join ($\vee$) operation, which gives us the synchrony pattern that corresponds to the intersection of synchrony sets.Comment: 48 pages, 19 figures, 3 table