A Dissipativity Framework for Constructing Scaled Graphs

TL;DR

This paper introduces a dissipativity-based framework to efficiently compute scaled graphs for nonlinear systems, enhancing the graphical analysis tools for stability and control design.

Contribution

It develops a novel approach linking dissipativity, linear matrix inequalities, and integral quadratic constraints to compute scaled graphs for various systems.

Findings

Framework is exact for certain linear systems

Enables efficient computation of scaled graphs

Illustrated with multiple practical examples

Abstract

Scaled relative graphs have been originally introduced in the context of convex optimization and have recently gained attention in the control systems community for the graphical analysis of nonlinear systems. Of particular interest in stability analysis of feedback systems is the scaled graph, a special case of the scaled relative graph. In many ways, scaled graphs can be seen as a generalization of the classical Nyquist plot for linear time-invariant systems, and facilitate a powerful graphical tool for analyzing nonlinear feedback systems. In their current formulation, however, scaled graphs require characterizing the input-output behaviour of a system for an uncountable number of inputs. This poses a practical bottleneck in obtaining the scaled graph of a nonlinear system, and currently limits its use. This paper presents a framework grounded in dissipativity for efficiently computing the scaled graph of several important classes of systems, including multivariable linear time-invariant systems, impulsive systems, and piecewise linear systems. The proposed approach leverages novel connections between linear matrix inequalities, integral quadratic constraints, and scaled graphs, and is shown to be exact for specific linear time-invariant systems. The results are accompanied by several examples illustrating the potential and effectiveness of the presented framework.