Scaled relative graphs: nonexpansive operators via 2D Euclidean geometry

A rolled-off passivity theorem

“…In this section, we briefly introduce the theory of SRGs. We give only the theory required for the proof of Theorem 1, and refer the interested reader to reference [14] for the complete theory of SRG interconnections and their use in the theory of optimization, reference [16] for the use of SRGs in systems theory and their relation to the Nyquist diagram, and references [14,16,17,33] for the computation of SRGs for particular systems.…”

“…The SRGs of interconnected systems can be determined or approximated from the SRGs of the components. For a full treatment of SRG interconnections, we refer the reader to [14], Theorems 4.1-4.5. Here, we describe what happens to the SRG under input and output scaling, summation and inversion.…”

“…We verify (µ, γ)-dissipativity for H 1 and H 2 by plotting bounding approximations of their SRGs. To do this, we use the SRG characterizations of reference [16] and the interconnection theory of reference [14]. The bounding SRG for H 1 is obtained by taking the SRGs of the saturation, relay and unit lag, given by [16, Thm.…”

“…In contrast, incremental (µ, γ)-dissipativity can be read directly from the Scaled Relative Graph (SRG) of a nonlinear system. SRGs have recently been introduced by Ryu, Hannah and Yin [14] for the study of monotone operator methods in optimization. The SRG allows incremental properties of operators, such as Lipschitz continuity and monotonicity, to be determined graphically, and leads to intuitive and rigorous proofs of convergence using geometric transformations in the Euclidean plane.…”

“…The SRG allows incremental properties of operators, such as Lipschitz continuity and monotonicity, to be determined graphically, and leads to intuitive and rigorous proofs of convergence using geometric transformations in the Euclidean plane. The SRG is particularly suited to proving tightness of bounds, and novel tightness results have been obtained [14,15]. In reference [16], the author and his colleagues showed that the SRG generalizes the Nyquist diagram of an LTI transfer function (generalized by Pates [17]), and applied SRG techniques to the study of feedback systems.…”