Analysis of Non-Square Nonlinear MIMO Systems using Scaled Relative Graphs
This work provides a novel method for analyzing non-square MIMO nonlinear systems, which is a known bottleneck in SRG theory, enabling broader applicability in control and systems analysis.
The paper extends Scaled Relative Graph (SRG) analysis to non-square MIMO nonlinear systems by embedding operators into a common Hilbert space while restricting the input space to avoid conservatism. It generalizes SRG interconnection rules and stability theorems, demonstrating advantages on examples.
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.