Scaled Graph Bounding Techniques for Reset Systems

TL;DR

This paper introduces scaled graph bounding techniques for reset systems, leveraging quadratic dissipativity to analyze their behavior and reveal fundamental limitations of existing approximation methods.

Contribution

It develops over-bounding techniques for scaled graphs of reset systems and connects quadratic dissipativity to piecewise quadratic storage functions.

Findings

Established a link between quadratic dissipativity and scaled graphs.

Revealed fundamental limitations of quadratic-based scaled graph approximations.

Provided new insights into the accuracy of reset system analysis methods.

Abstract

Reset systems can overcome fundamental limitations of linear time-invariant control. The recently introduced notion of scaled (relative) graphs provides a promising framework for developing graphical analysis and design tools for reset systems, in line with widely adopted loopshaping methods for linear systems. The aim of this paper is to derive techniques for over-bounding the scaled graph of reset systems, and obtain insights in their accuracy. We exploit connections between quadratic dissipativity and scaled graphs to recast the over-bounding problem as the search for piecewise quadratic storage functions. Using specific sampling techniques, we reveal a fundamental limitation of general scaled graph approximation methods that are based on quadratic dissipativity.