Stability results for MIMO LTI systems via Scaled Relative Graphs

TL;DR

This paper addresses the challenge of stability analysis for MIMO LTI feedback systems by proposing Scaled Relative Graphs (SRGs) as a decoupled, frequency-wise stability tool. By formulating a Generalized Feedback Stability Theorem (GFT) and specializing it to LTI systems, the authors show that stability can be certified via per-frequency SRG intersections, offering a visual and data-driven alternative to the classical Generalized Nyquist Criterion (GNC). They prove the equivalence of SRG-based conditions with the GNC under suitable chord-property and invertibility assumptions and demonstrate the method on several numerical examples, including high-order MIMO systems. The SRG approach provides intuitive insight, reduces conservative coupling, and is amenable to data-driven estimation, potentially enabling more scalable stability analysis for complex MIMO systems.

Abstract

This paper proposes a new approach for stability analysis of multi-input, multi-output (MIMO) feedback systems through Scaled Relative Graphs (SRGs). Unlike traditional methods, such as the Generalized Nyquist Criterion (GNC), which relies on a coupled analysis that requires the multiplication of models, our approach enables the evaluation of system stability in a decoupled fashion and provides an intuitive, visual representation of system behavior. Our results provide conditions for certifying the stability of feedback MIMO Linear Time-Invariant (LTI) systems.

Figures (4)

• Figure 1: Feedback interconnection between $H_1$ and $H_2$.
• Figure 2: (a) 3D plot of $\operatorname{SRG}(H_1(\textup{j}\omega))~\forall \omega\in[10^{-5},10^5]$rad/s. (b) projection of $\operatorname{SRG}(H_1(\textup{j}\omega))$ in the complex plane in yellow and the Nyquist plot in black dashed line. (c) 3D plot of $\operatorname{SRG}(H_2(\textup{j}\omega))$$\forall \omega\in[0.316,3.16]$rad/s. (d) 3D plot of $\operatorname{SRG}(H_3(\textup{j}\omega)) \forall \omega\in[10^{-1},10^1]$rad/s.
• Figure 3: $-\operatorname{SRG}(H_{4}(\textup{j}\omega))$ in green, $\operatorname{SRG}(H_2^{-1}(\textup{j}\omega))$ in orange $~\forall \omega\in[10^{-3},10^3]$rad/s. (a) SRGs 3D visualization. (b) 2D projection of the SRGs in the complex plane for $\omega\in[10^{-3},1]$ rad/s. (c) 2D projection of the SRGs in the complex plane for $\omega\in[1,10^{3}]$ rad/s. (d) Nyquist plot using $\det(I+H_{4}(\textup{j}\omega)H_2(\textup{j}\omega))$.
• Figure 4: $\operatorname{SRG}(H(\textup{j}\omega))$ in green, $-\operatorname{SRG}(H_{5}^{-1}(\textup{j}\omega))$ in orange and $-\operatorname{SRG}(H_{6}^{-1}(\textup{j}\omega))$ in yellow $~\forall \omega\in[10^{-3},10^3]$rad/s. (a) Nyquist plot of $\det(I+H(\textup{j}\omega)H_{5}(\textup{j}\omega))$ (b) Nyquist plot of $\det(I+H(\textup{j}\omega)H_{6}(\textup{j}\omega))$ (c) 3D plot of the $\operatorname{SRG}(H(\textup{j}\omega))$ and $-\operatorname{SRG}(H_{5}^{-1}(\textup{j}\omega))$ (d) 3D plot of the $\operatorname{SRG}(H(\textup{j}\omega))$ and $-\operatorname{SRG}(H_{6}^{-1}(\textup{j}\omega))$.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1: Chord Property of LTI systems
• Theorem 4
• Theorem 5