Computing the Hard Scaled Relative Graph of LTI Systems

2025/11/21 by Krebbekx, Julius P. J., Baron-Prada, Eder, Tóth, Roland +1
#FOS: Electrical engineering #FOS: Mathematics #Optimization and Control (math.OC) #Systems and Control (eess.SY) #electronic engineering #information engineering

paper · doi:10.48550/arxiv.2511.17297

Abstract

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems, where Linear Time-Invariant (LTI) systems are the fundamental building block. To analyze feedback loops with unstable LTI components, the hard SRG is required, since it aptly captures the input/output behavior on the extended L₂ space. In this paper, we develop a systematic computational method to exactly compute the hard SRG of LTI systems, which may be unstable and contain integrators. We also study its connection to the Nyquist criterion, including the multivariable case, and demonstrate our method on several examples.

Citations

• Graphical Analysis of Nonlinear Multivariable Feedback Systems
• Scaled Relative Graph Analysis of General Interconnections of SISO Nonlinear Systems
• A Dissipativity Framework for Constructing Scaled Graphs
• Reset Controller Analysis and Design for Unstable Linear Plants using Scaled Relative Graphs
• On phase in scaled graphs
• Soft and Hard Scaled Relative Graphs for Nonlinear Feedback Stability
• Scaled Relative Graph Analysis of Lur'e Systems and the Generalized Circle Criterion
• The Scaled Relative Graph of a Linear Operator
• Scaled relative graphs: nonexpansive operators via 2D Euclidean geometry

Related

• Scaled Relative Graph Analysis of General Interconnections of SISO Nonlinear Systems
• Scaled Relative Graph Analysis of Lur'e Systems and the Generalized Circle Criterion
• Scaled relative graphs for system analysis
• Soft and Hard Scaled Relative Graphs for Nonlinear Feedback Stability
• Computable Characterisations of Scaled Relative Graphs of Closed Operators