Computable Characterisations of Scaled Relative Graphs of Closed Operators

TL;DR

The paper tackles computing Scaled Relative Graphs ($\mathrm{SRG}$) for closed linear operators, including unbounded ones, to support stability analysis of dynamical systems. It develops a gain-based characterisation showing that $\mathrm{cl}\,\mathrm{SRG}(\mathbf{T}) = \bigcap_{\alpha\in\mathbb{R}} \mathrm{Ann}(\mathbf{T},\alpha)$, with annuli defined by the operator's maximum and minimum gains $\bar{\sigma}(\mathbf{T})$ and $\underline{\sigma}(\mathbf{T})$, and leverages the Beltrami–Klein mapping to carry convexity arguments into the unit disc. The work provides exact, computable constructions for transfer-function induced operators on $\mathcal{L}_{2}^{m}$ and on truncations $\mathcal{L}_{2,\tau}^{m}$, including explicit frequency-domain formulas for gains and an LMI-based approach (Bounded Real Lemma) for state-space realizations. A grid-based Algorithm gracefully approximates SRGs for operator sequences, with examples illustrating how truncation can fill interior regions when poles lie on the imaginary axis. Overall, the results offer a practical toolkit for exact stability and robustness analysis of LTI and related systems using $\mathrm{SRG}$ diagrams across bounded and unbounded operator models.

Abstract

Scaled Relative Graphs (SRGs) provide a promising tool for stability and robustness analysis of multi-input-multi-output systems. In this paper, we provide tools for exact and computable constructions of the SRG for closed linear operators, based on maximum and minimum gain computations. The results are suitable for bounded and unbounded operators, and we specify how they can be used to draw SRGs for the typical operators that are used to model linear-time-invariant dynamical systems. Furthermore, for the special case of state-space models, we show how the Bounded Real Lemma can be used to construct the SRG.

Figures (5)

• Figure 1: The set $g_{\mathrm{BK}} \! \left(S\right)$ and the bounding annulus with centre point $\alpha$ on the extended complex plane (a) and under Beltrami-Klein mapping (b).
• Figure 2: Illustration of Algorithm \ref{['alg:outer']} for computation of the . The orange area shows the while the grey area shows the approximation that is given by the intersection of the annuli defined by $\{ \alpha_1, \alpha_2, \alpha_3\}$.
• Figure 3: The figure shows the of $T_1(s) = \frac{e^{-s}}{s+1}$. The hatched black area shows the of $\mathbf{T}_{T(s)}$ and the orange area shows the of $\mathbf{T}_{T(s)}^\tau$ when $\tau \rightarrow \infty$.
• Figure 4: The figure shows the of a MIMO system $T_2(s) = [ \frac{1}{s-1}, \frac{s}{s-1} ; \frac{s+1}{s+3}, \frac{1}{s+2} ]$. The of $\mathbf{T}_{T(s)}$ is the hatched black area, while the of $\mathbf{T}_{T(s)}^\tau$ when $\tau \rightarrow \infty$ is represented by the orange area.
• Figure 5: The figure shows the of $T_3(s) = 1/s$. For $\mathbf{T}_{T(s)}$ the is the extended imaginary axis (hatched black), while for $\mathbf{T}_{T(s)}^\tau$ when $\tau \rightarrow \infty$ it is the extended closed right-half-plane including the imaginary axis (orange).

Theorems & Definitions (22)

• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 1
• Definition 1
• Proposition 1
• Theorem 3