An impulsive dynamical systems framework for reset control systems

TL;DR

This paper applies impulsive dynamical systems theory to analyze fundamental properties of reset control systems, establishing conditions for solution existence, uniqueness, and sensitivity, with implications for system robustness and noise response.

Contribution

It introduces a framework based on impulsive dynamical systems to analyze reset control systems, providing new conditions for well-posedness and sensitivity analysis.

Findings

Reset control systems with strictly proper plants have no Zeno solutions.

Full and partial reset schemes produce well-posed reset instants.

The paper offers conditions for continuous dependence on initial conditions.

Abstract

Impulsive dynamical systems is a well-established area of dynamical systems theory, and it is used in this work to analyze several basic properties of reset control systems: existence and uniqueness of solutions, and continuous dependence on the initial condition (well-posedness). The work scope is about reset control systems with a linear and time-invariant base system, and a zero-crossing resetting law. A necessary and sufficient condition for existence and uniqueness of solutions, based on the well-posedness of reset instants, is developed. As a result, it is shown that reset control systems (with strictly proper plants) do no have Zeno solutions. It is also shown that full reset and partial reset (with a special structure) always produce well-posed reset instants. Moreover, a definition of continuous dependence on the initial condition is developed, and also a sufficient condition for reset control systems to satisfy that property. Finally, this property is used to analyze sensitivity of reset control systems to sensor noise. This work also includes a number of illustrative examples motivating the key concepts and main results.