Sean Phillips

Results on incremental stability for a class of hybrid systems

Incremental stability is the notion that the distance between every pair of solutions to the system has stable behavior and approaches zero asymptotically. This paper introduces this notion for a class of hybrid systems. In particular, we define incremental stability as well as incremental partial stability, and study their properties. The approach used to derive our results consists of recasting the incremental stability problem as a set stabilization problem, for which the tools for asymptotic stability of hybrid systems are applicable. In particular, we propose an auxiliary hybrid system to study the stability of the diagonal set, which relates to incremental stability of the original system. The proposed notions are illustrated in examples throughout the paper.

Robust Asymptotic Stability of Desynchronization in Impulse-Coupled Oscillators

The property of desynchronization in an all-to-all network of homogeneous impulse-coupled oscillators is studied. Each impulse-coupled oscillator is modeled as a hybrid system with a single timer state that self-resets to zero when it reaches a threshold, at which event all other impulse-coupled oscillators adjust their timers following a common reset law. In this setting, desynchronization is considered as each impulse-coupled oscillators timer having equal separation between successive resets. We show that for the considered model, desynchronization is an asymptotically stable property. For this purpose, we recast desynchronization as a set stabilization problem and employ Lyapunov stability tools for hybrid systems. Furthermore, several perturbations are considered showing that desynchronization is a robust property. Perturbations on the continuous and discrete dynamics are considered. Numerical results are presented to illustrate the main contributions.

On the synchronization of two impulsive oscillators under communication constraints

The problem of establishing synchronization of a class of impulsive oscillators is considered. Each impulsive oscillator is modeled as a hybrid system that self resets to zero when its state reaches a threshold and is externally reset via an impulsive update law when information from other oscillators is received. At every reset to zero, the oscillators broadcast a packet and select a different communication channel. The mechanism resembles that of a firefly model in which the external resets correspond to flashes of the fireflies. Oscillators can only receive information when they are on the same channel that a packet was broadcast. This mechanism leads to the possibility of information loss. Under such a communication constraint, we show that the coupled impulsive oscillators (almost globally) synchronize their channel selections. To establish this result, we model the interconnection of oscillators as a hybrid system and apply recently developed Lyapunov stability tools. Numerical simulations are included to illustrate the results.

Results on the asymptotic stability properties of desynchronization in impulse-coupled oscillators

The property of desynchronization in impulse-coupled oscillators is studied. Each impulsive oscillator is modeled as a hybrid system with a single timer state that self-resets to zero when it reaches a threshold, at which point any other impulsive oscillator adjusts their timers following a common law. This law dictates the reaction to an external reset. In this setting, desynchronization is considered as timers having equal separation among each other and between successive resets. We show that, for the considered model, desynchronization is an (almost global) asymptotic stability property, which, due to the regularity properties of the hybrid systems, is robust to small perturbations. To establish this result, we recast desynchronization as a set stabilization problem and employ Lyapunov stability tools for hybrid systems. The results are illustrated in examples and simulations.

Robust Distributed Estimation for Linear Systems Under Intermittent Information

We provide a comprehensive solution to the estimation problem of the state for a linear time-invariant system in a distributed fashion over networks that allow only intermittent information transmission. By attaching to each node an observer that employs information received from its neighbors triggered by asynchronous communication events, we propose a distributed state observer that guarantees global exponential stability of the zero estimation error set. The design of parameters is formulated as linear matrix inequalities. A thorough robustness analysis of the proposed observer to unmodeled dynamics, unknown communication times, as well as measurement and communication noise characterized in terms of input-to-state stability is presented. These properties of the proposed observer are shown analytically and validated numerically.

Basic properties and characterizations of incremental stability prioritizing flow time for a class of hybrid systems

Abstract This paper introduces incremental stability notions for a class of hybrid dynamical systems given in terms of differential equations and difference equations with state constraints. The specific class of hybrid systems considered are those that do not have consecutive jumps nor Zeno behavior. The notion of incremental asymptotic stability is used to describe the behavior of the distance between every pair of solutions to the system having stable behavior (incremental stability) and approaching zero asymptotically (incremental attractivity). A version of this notion that is uniform (in hybrid time) with respect to initial conditions is also introduced. These notions prioritize flow time and are illustrated in examples. Basic properties of the class of systems are considered and those implied by the new notions are revealed. An equivalence characterization of the uniform notion is provided in terms of a K L -function. Moreover, sufficient and necessary conditions under which asymptotic stability implies the new incremental notions are provided. We consider the case when the original hybrid system has an asymptotically stable compact set and also the case when an auxiliary hybrid system, which has twice the dimension of the original system, has a diagonal-like set asymptotically stable.

A framework for modeling and analysis of dynamical properties of spiking neurons

A hybrid systems framework for modeling and analysis of robust stability of spiking neurons is proposed. The framework is developed for a population of n interconnected neurons. Several well-known neuron models are studied within the framework, including both excitatory and inhibitory simplified Hodgkin-Huxley, Hopf, and SNIPER models. For each model, we characterize the sets that the solutions to each system converge to. Using Lyapunov stability tools for hybrid systems, the stability properties for each case are established. An external stimuli is introduced to the simplified Hodgkin-Huxley model to achieve a global asymptotic stability property. Due to the regularity properties of the data of the hybrid models considered, the asserted stability properties are robust to small perturbations. Simulations provide insight on the results and the capabilities of the proposed framework.

Robust synchronization of interconnected linear systems over intermittent communication networks

The property of synchronization of multiple linear time-invariant systems connected over a network with stochastically-driven isolated communication events is studied. We propose a solution to the problem of designing a feedback controller that, using information obtained over such networks, asymptotically drives the values of their states to synchronization and renders such a condition Lyapunov stable. To solve this problem, we propose a controller with hybrid dynamics, namely, the controller exhibits continuous dynamics between communication events and, at such events, has variables that jump. Due to the additional continuous and discrete dynamics inherent to the networked systems and communication structure, we use a hybrid systems framework to model the closed-loop system and design the controller. The problem of synchronization is then recast as a compact set stabilization problem and, by employing Lyapunov stability tools for hybrid systems, sufficient conditions for asymptotic stability of the synchronization set are provided. Furthermore, we show that the synchronization property is robust to a class of perturbations on the transmitted data. Numerical examples illustrating the main results are included.

Robust synchronization of two linear systems over intermittent communication networks

The property of synchronization between two identical linear time-invariant (LTI) systems connected through a network with stochastically-driven isolated communication events is studied. More precisely, the goal is to design feedback controllers that, using information obtained over such networks, asymptotically drive the values of their state to synchronization and render the synchronization condition Lyapunov stable. To solve this problem, we propose a dynamic controller with hybrid dynamics, namely, the controller exhibits continuous dynamics between communication events while it has variables that jump at such events. Due to the additional continuous and discrete dynamics inherent in the networked systems and communication structure, we utilize a hybrid systems framework to model the closed-loop system. The problem of synchronization is then recast as a set stabilization problem and, by utilizing recent Lyapunov stability tools for hybrid systems, sufficient conditions for asymptotic stability of the synchronization set are provided for two network topologies: a cascade (unidirectional) network and a feedback (bidirectional) communication network with independent transmission instances. Furthermore, we study the robustness of synchronization by considering a class of perturbations on the transmitted data. Numerical examples are provided.