# Scaled relative graphs for system analysis

@article{Chaffey2021ScaledRG, title={Scaled relative graphs for system analysis}, author={Thomas Chaffey and Fulvio Forni and Rodolphe Sepulchre}, journal={2021 60th IEEE Conference on Decision and Control (CDC)}, year={2021}, pages={3166-3172} }

Scaled relative graphs were recently introduced to analyze the convergence of optimization algorithms using two dimensional Euclidean geometry. In this paper, we connect scaled relative graphs to the classical theory of input/output systems. It is shown that the Nyquist diagram of an LTI system on L2 is the convex hull of its scaled relative graph under a particular change of coordinates. The SRG may be used to visualize approximations of static nonlinearities such as the describing function…

## 5 Citations

### The Scaled Relative Graph of a Linear Operator

- Mathematics
- 2021

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…

### Convergence Analyses of Davis-Yin Splitting via Scaled Relative Graphs

- Computer Science
- 2022

This work formalizes an SRG theory for the DYS operator and uses it to obtain tighter contraction factors and to study the linear rates of convergence of operator splitting methods.

### The Singular Angle of Nonlinear Systems

- MathematicsArXiv
- 2021

The proposed system singular angle, based on the angle between $\mathcal{L}_2$-signals, describes an upper bound for the "rotating effect" from the system input to output signals, different from the recently appeared nonlinear system phase.

### Non-Euclidean Monotone Operator Theory with Applications to Recurrent Neural Networks

- Mathematics, Computer ScienceArXiv
- 2022

It is demonstrated that casting the computation as a suitable operator splitting problem improves convergence rates and several classic iterative methods for computing zeros of monotone operators are directly applicable in the non-Euclidean framework.

### Graphical Nonlinear System Analysis

- MathematicsArXiv
- 2021

This work uses the recently introduced concept of a Scaled Relative Graph (SRG) to develop a graphical analysis of input-output properties of feedback systems and measures important robustness indicators of nonlinear feedback systems.

## References

SHOWING 1-10 OF 22 REFERENCES

### Scaled relative graphs: nonexpansive operators via 2D Euclidean geometry

- MathematicsMath. Program.
- 2022

Many iterative methods in applied mathematics can be thought of as fixed-point iterations, and such algorithms are usually analyzed analytically, with inequalities. In this paper, we present a…

### Scaled Relative Graph of Normal Matrices

- MathematicsArXiv
- 2020

This work further study the SRG of linear operators and characterize theSRG of block-diagonal and normal matrices and views the SRGs as a generalization of the spectrum to multi-valued nonlinear operators.

### Tight coefficients of averaged operators via scaled relative graph

- MathematicsJournal of Mathematical Analysis and Applications
- 2020

### Phase of Nonlinear Systems

- MathematicsArXiv
- 2020

The proposed nonlinear system phase, serving as a counterpart of $\mathcal{L}_2$-gain, quantifies the passivity and is highly related to the dissipativity, which possesses a nice physical interpretation which quantifying the tradeoff between the real energy and reactive energy.

### System analysis via integral quadratic constraints

- MathematicsProceedings of 1994 33rd IEEE Conference on Decision and Control
- 1994

A stability theorem for systems described by IQCs is presented that covers classical passivity/dissipativity arguments but simplifies the use of multipliers and the treatment of causality.

### The gap metric: Robustness of stabilization of feedback systems

- Mathematics
- 1985

In this paper we introduce the gap metric to study the robustness of the stability of feedback systems which may employ not necessarily stable open-loop systems. We elaborate on the computational…

### Frequency response functions and Bode plots for nonlinear convergent systems

- MathematicsProceedings of the 45th IEEE Conference on Decision and Control
- 2006

This paper extends frequency response functions defined for linear systems to nonlinear convergent systems, which give rise to non linear Bode plots, which serve as a graphical tool for performance analysis of nonlinear Convergent systems in the frequency domain.

### Higher-order sinusoidal input describing functions for the analysis of non-linear systems with harmonic responses

- Engineering
- 2006

### An improved frequency time domain stability criterion for autonomous continuous systems

- Computer ScienceIEEE Transactions on Automatic Control
- 1967

A sufficient condtion given for the asymptotic stability of a system having a single monotonic nonlinearity with slope confined to 0 and a transfer function which corresponds to a nonzero time function for t, resulting in Z(j\omega) multiplier whose phase angle is capable of varying from +90° to -90° any desired number of times.

### L2 Gain And Passivity Techniques In Nonlinear Control

- Computer Science
- 2016

L2 gain and passivity techniques in nonlinear control is downloaded for free to help people who are facing with some harmful virus inside their desktop computer.