A Dissipativity Framework for Constructing Scaled Graphs

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.