Analysis of Non-Square Nonlinear MIMO Systems using Scaled Relative Graphs

TL;DR

This paper introduces a new SRG analysis method for non-square nonlinear MIMO systems, enabling stability and performance analysis in more general settings.

Contribution

It develops an embedding-based SRG analysis framework for non-square operators, extending previous square-only results and providing practical stability theorems.

Findings

Generalized SRG interconnection rules for restricted input spaces

Derived stability theorems ensuring causality and $L_2$-gain bounds

Demonstrated advantages on nonlinear system examples

Abstract

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems. There have been recent efforts to generalize SRG analysis to Multiple-Input Multiple-Output (MIMO) systems. However, these attempts yielded only results for square systems, due to the inherent Hilbert space structure of the SRG. In this paper, we develop an SRG analysis method that accommodates non-square operators. The key element is the embedding of operators to a space of operators acting on a common Hilbert space, while restricting the input space to the original input dimension, to avoid conservatism. We generalize SRG interconnection rules to restricted input spaces and develop stability theorems to guarantee causality, well-posedness and (incremental) $L_2$-gain bounds for the overall interconnection. We show utilization of the proposed theoretical concepts on the analysis of nonlinear systems in a Linear Fractional Representation (LFR) form, which is a rather general class of systems, and the LFR is directly utilizable for control. Moreover, we provide formulas for the computation of MIMO SRGs of stable LTI operators and diagonal and non-square static nonlinear operators. Finally, we demonstrate the advantages of our embedding approach on several examples.