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The Scaled Relative Graph of a Linear Operator
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The scaled relative graph (SRG) of an operator is a subset of the complex plane. It captures several salient features of an operator, such as contractiveness, and can be used to reveal the geometric nature of many of the inequality based arguments used in the convergence analyses of fixed point iterations. In this paper we show that the SRG of a linear operator can be determined from the numerical range of a closely related linear operator. Furthermore we demonstrate that the SRG of a linear operator has a range of spectral and convexity properties, and satisfies an analogue of Hildebrant's theorem.
Forward citations
Cited by 3 Pith papers
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Scaled Relative Graph Analysis of General Interconnections of SISO Nonlinear Systems
The authors introduce an extended Scaled Relative Graph that includes Nyquist encirclement data, enabling stability and L2-gain analysis of feedback interconnections with unstable linear components.
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A Dissipativity Framework for Constructing Scaled Graphs
A dissipativity-based LMI framework computes scaled graphs for LTI, reset, and piecewise-linear systems, with an exactness guarantee for normal LTI systems.
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A homotopy theorem for incremental stability
A new incremental homotopy theorem shows that feedback stability follows from an incremental gain bound along a deformation path, yielding corrected SRG separation and incremental IQC criteria.
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