Rotated Scaled Relative Graph (SRG)

• Rotated SRG is a phase-referenced variant of the standard Scaled Relative Graph that re-aligns gain and phase representations along an arbitrary axis, mitigating the conjugate-pair artifact.
• It achieves this by applying a rotation (e^(jθ) ⋅ SRG(e^(–jθ)A)) which preserves essential geometric properties like hyperbolic and Euclidean convexity, thereby refining stability criteria in MIMO systems.
• The method underpins advanced feedback stability analysis, leveraging Davis–Wielandt shell projections and SDP-based tomographic computations to improve conditions for cyclic MIMO feedback loops.

Searching arXiv for the most relevant papers on rotated Scaled Relative Graphs and related SRG developments. A rotated Scaled Relative Graph (SRG) is a phase-referenced variant of the Scaled Relative Graph in which the symmetry axis of the graphical representation is moved from the real axis to a prescribed direction $e^{j\theta}\mathbb R$. In current literature, the term refers primarily to the one-parameter families $\mathrm{SRG}_\theta$ and $\theta$-symmetric SRG introduced for complex-valued MIMO LTI analysis, especially as mixed gain–phase projections of the Davis–Wielandt shell (Zhang et al., 26 Jul 2025, Yang et al., 8 Oct 2025). More broadly, earlier SRG work already treated rotation as intrinsic to the standard SRG through its polar encoding of gain and angle, even when no separate rotated object was named (Pates, 2021, Chaffey et al., 2021).

1. Standard SRG foundation

The classical SRG is defined for an operator $R:\mathcal L\to\mathcal L$ on a Hilbert space by comparing differences of two input–output pairs. With

$$\angle(u,y):=\cos^{-1}\!\left(\frac{\Re\langle u,y\rangle}{\|u\|\|y\|}\right)\in[0,\pi],$$

the pointwise scaled relative graph associated with $(u_1,u_2)$ is

$$z_R(u_1,u_2):= \left\{ \frac{\|y_1-y_2\|}{\|u_1-u_2\|} e^{\pm j\angle(u_1-u_2,\;y_1-y_2)} \;\middle|\; y_1\in R(u_1),\;y_2\in R(u_2) \right\},$$

and the SRG over $\mathcal U\subseteq\mathcal L$ is

$$\operatorname{SRG}_{\mathcal U}(R):=\bigcup_{u_1,u_2\in\mathcal U} z_R(u_1,u_2).$$

Its modulus encodes incremental gain, and its argument encodes phase-like information (Krebbekx et al., 2024).

For linear operators, the same geometry appears in the simpler form

$$\mathrm{SRG}(C) = \left\{ \frac{\|Cx\|}{\|x\|}\exp\{\pm j\,\angle(x,Cx)\} :\;0\neq x\in\mathbb{C}^n \right\},$$

which is symmetric with respect to the real axis because of the explicit $\mathrm{SRG}_\theta$0 construction (Yang et al., 8 Oct 2025). Earlier SRG literature emphasized that this already represents operator action as a complex scaling together with a rotation angle; for real matrices the full SRG is the upper-half image and its conjugate reflection, and for general linear operators the graph is the union of “polar representations” of input–output pairs (Huang et al., 2019, Pates, 2021).

The link to classical frequency-domain analysis is that for an LTI system the SRG in the closed upper half-plane is the $\mathrm{SRG}_\theta$1-convex hull of the Nyquist plot in the same half-plane,

$\mathrm{SRG}_\theta$2

so the Nyquist diagram is recovered as a special case, while the SRG extends to nonlinear and multivalued operators (Krebbekx et al., 2024, Chaffey et al., 2022).

2. Rotation as an intrinsic geometric feature

Before explicit rotated variants were formalized, SRG theory already contained two rotation-related ideas. First, the factor $\mathrm{SRG}_\theta$3 makes the construction inherently rotation-aware: the SRG point is a gain multiplied by a phase term determined by the angle between input and output increments. Second, the geometry of SRGs is naturally described through hyperbolic transforms such as the Beltrami–Klein map, which converts SRG geometry into convex numerical-range geometry (Pates, 2021).

For a bounded linear operator $\mathrm{SRG}_\theta$4, one such transformed representation is given through

$\mathrm{SRG}_\theta$5

with inverse-type map $\mathrm{SRG}_\theta$6, and the paper states the central identity

$\mathrm{SRG}_\theta$7

Under this transform, $\mathrm{SRG}_\theta$8 becomes Euclidean convex, while the original SRG is hyperbolically convex (Pates, 2021). Closely related work on normal matrices described the SRG as a hyperbolic arc-edge polygon generated by eigenvalues in the upper half-plane, again making the “rotation + scaling” interpretation explicit (Huang et al., 2019).

This earlier viewpoint is significant because later rotated SRGs do not discard the standard definition; rather, they replace the real-axis symmetry built into the standard complex SRG by a symmetry about an arbitrary phase axis. The rotated SRG is therefore best understood as a re-referencing of an already rotation-sensitive object, not as a departure from SRG geometry.

3. Explicit rotated variants: $\mathrm{SRG}_\theta$9-SRG and $\theta$0-symmetric SRG

The modern, explicit notion of a rotated SRG appears in two closely related formulations for complex matrices.

Variant	Defining relation	Geometric feature
Standard SRG	$\theta$1	Symmetric about real axis
Rotated SRG / $\theta$2-SRG	$\theta$3	Symmetric about $\theta$4
$\theta$5-symmetric SRG	$\theta$6	Uses explicit phase reference $\theta$7

The $\theta$8-symmetric construction introduces the rotated angle

$\theta$9

and defines

$R:\mathcal L\to\mathcal L$0

The key identity is

$R:\mathcal L\to\mathcal L$1

so the rotated SRG is obtained by rotating the matrix action by $R:\mathcal L\to\mathcal L$2, taking the standard SRG, and rotating the result back (Yang et al., 8 Oct 2025). The same basic definition is adopted in the Davis–Wielandt-shell framework, where the rotated SRG is denoted $R:\mathcal L\to\mathcal L$3 and used as a one-parameter family of mixed gain–phase projections (Zhang et al., 26 Jul 2025).

A central motivation is removal of the conjugate-pair artifact of the standard complex SRG. For a scalar $R:\mathcal L\to\mathcal L$4, the standard SRG gives $R:\mathcal L\to\mathcal L$5, whereas the $R:\mathcal L\to\mathcal L$6-symmetric SRG can reduce exactly to the scalar itself: for $R:\mathcal L\to\mathcal L$7, choosing $R:\mathcal L\to\mathcal L$8 yields

$R:\mathcal L\to\mathcal L$9

This exact scalar reduction is presented as a decisive advantage when the rotated SRG is used as a MIMO extension of Nyquist geometry (Yang et al., 8 Oct 2025).

4. Davis–Wielandt-shell interpretation

The main conceptual setting of the rotated SRG is the Davis–Wielandt (DW) shell. For a matrix $$\angle(u,y):=\cos^{-1}\!\left(\frac{\Re\langle u,y\rangle}{\|u\|\|y\|}\right)\in[0,\pi],$$0,

$$\angle(u,y):=\cos^{-1}\!\left(\frac{\Re\langle u,y\rangle}{\|u\|\|y\|}\right)\in[0,\pi],$$1

This is a three-dimensional geometric object in $$\angle(u,y):=\cos^{-1}\!\left(\frac{\Re\langle u,y\rangle}{\|u\|\|y\|}\right)\in[0,\pi],$$2 that jointly encodes numerical-range information and gain information (Zhang et al., 26 Jul 2025).

Within that framework, classical graphical objects appear as projections or “shadows” of the DW shell. The rotated SRG is obtained by rotating the viewing direction before taking the SRG-type shadow. The paper states that $$\angle(u,y):=\cos^{-1}\!\left(\frac{\Re\langle u,y\rangle}{\|u\|\|y\|}\right)\in[0,\pi],$$3 is produced by a $$\angle(u,y):=\cos^{-1}\!\left(\frac{\Re\langle u,y\rangle}{\|u\|\|y\|}\right)\in[0,\pi],$$4-projection of the DW shell followed by a paraboloidal projection to the complex plane, and uses this to place the numerical range, normalized numerical range, signed SRG, gain measures, and phase measures within one common geometry (Zhang et al., 26 Jul 2025).

This interpretation matters because it isolates the source of conservatism. The DW shell is the most informative geometric object in the hierarchy; every reduction from three dimensions to two dimensions or one dimension loses information. Within the class of two-dimensional graphical conditions for bi-component feedback loops, the rotated SRG is presented as the least conservative condition currently available (Zhang et al., 26 Jul 2025). The reason is not that it contains more data than the DW shell, but that the phase reference $$\angle(u,y):=\cos^{-1}\!\left(\frac{\Re\langle u,y\rangle}{\|u\|\|y\|}\right)\in[0,\pi],$$5 is optimized before projection, so the two-dimensional shadow is better aligned with the actual feedback geometry.

5. Product properties and feedback stability criteria

The principal use of rotated SRGs is graphical stability analysis of feedback interconnections. Two families of results are especially prominent.

For standard negative feedback of stable transfer matrices $$\angle(u,y):=\cos^{-1}\!\left(\frac{\Re\langle u,y\rangle}{\|u\|\|y\|}\right)\in[0,\pi],$$6, the exact DW-shell condition is

$$\angle(u,y):=\cos^{-1}\!\left(\frac{\Re\langle u,y\rangle}{\|u\|\|y\|}\right)\in[0,\pi],$$7

checked frequencywise. The corresponding rotated-SRG sufficient condition is that, for each $$\angle(u,y):=\cos^{-1}\!\left(\frac{\Re\langle u,y\rangle}{\|u\|\|y\|}\right)\in[0,\pi],$$8, there exists $$\angle(u,y):=\cos^{-1}\!\left(\frac{\Re\langle u,y\rangle}{\|u\|\|y\|}\right)\in[0,\pi],$$9 such that

$(u_1,u_2)$0

This is the main rotated-SRG separation condition for bi-component feedback loops (Zhang et al., 26 Jul 2025).

For cyclic interconnections of cascaded stable MIMO systems $(u_1,u_2)$1, the $(u_1,u_2)$2-symmetric SRG is equipped with a submultiplicative rule: $(u_1,u_2)$3 provided one factor satisfies the corresponding arc property. This yields the frequencywise stability condition

$(u_1,u_2)$4

with at least $(u_1,u_2)$5 of the sets satisfying the relevant arc properties (Yang et al., 8 Oct 2025).

These results are structurally important because ordinary graph separation is not adequate for cyclic products. The rotated SRG restores a usable product geometry by combining gain and refined phase in one two-dimensional set. In the scalar case this removes the spurious conjugate duplication that can make product tests fail. One example uses

$(u_1,u_2)$6

with $(u_1,u_2)$7 at $(u_1,u_2)$8. The standard SRG gives $(u_1,u_2)$9 and $$z_R(u_1,u_2):= \left\{ \frac{\|y_1-y_2\|}{\|u_1-u_2\|} e^{\pm j\angle(u_1-u_2,\;y_1-y_2)} \;\middle|\; y_1\in R(u_1),\;y_2\in R(u_2) \right\},$$0, so the product contains $$z_R(u_1,u_2):= \left\{ \frac{\|y_1-y_2\|}{\|u_1-u_2\|} e^{\pm j\angle(u_1-u_2,\;y_1-y_2)} \;\middle|\; y_1\in R(u_1),\;y_2\in R(u_2) \right\},$$1; with matched phase choices in the $$z_R(u_1,u_2):= \left\{ \frac{\|y_1-y_2\|}{\|u_1-u_2\|} e^{\pm j\angle(u_1-u_2,\;y_1-y_2)} \;\middle|\; y_1\in R(u_1),\;y_2\in R(u_2) \right\},$$2-symmetric SRG, each scalar collapses to itself and the product criterion succeeds (Yang et al., 8 Oct 2025).

6. Applications, computation, and related terminology

The rotated SRG has been developed primarily for MIMO LTI stability analysis, but its meaning is clarified by comparison with adjacent SRG developments. In nonlinear and operator-theoretic papers preceding the explicit $$z_R(u_1,u_2):= \left\{ \frac{\|y_1-y_2\|}{\|u_1-u_2\|} e^{\pm j\angle(u_1-u_2,\;y_1-y_2)} \;\middle|\; y_1\in R(u_1),\;y_2\in R(u_2) \right\},$$3-rotated constructions, authors frequently stated that no separate “rotated SRG” was defined because the standard SRG already uses a rotated complex encoding through $$z_R(u_1,u_2):= \left\{ \frac{\|y_1-y_2\|}{\|u_1-u_2\|} e^{\pm j\angle(u_1-u_2,\;y_1-y_2)} \;\middle|\; y_1\in R(u_1),\;y_2\in R(u_2) \right\},$$4 (Chaffey et al., 2022, Quan et al., 2024). This older terminology remains relevant: the explicit rotated SRG is a refinement of phase reference, not a replacement of the original gain–angle encoding.

Computation has been addressed directly. The DW-shell framework proposes an SDP-based tomography algorithm for plotting $$z_R(u_1,u_2):= \left\{ \frac{\|y_1-y_2\|}{\|u_1-u_2\|} e^{\pm j\angle(u_1-u_2,\;y_1-y_2)} \;\middle|\; y_1\in R(u_1),\;y_2\in R(u_2) \right\},$$5-SRGs by slicing the DW shell with $$z_R(u_1,u_2):= \left\{ \frac{\|y_1-y_2\|}{\|u_1-u_2\|} e^{\pm j\angle(u_1-u_2,\;y_1-y_2)} \;\middle|\; y_1\in R(u_1),\;y_2\in R(u_2) \right\},$$6-oriented hyperplanes, solving lossless semidefinite programs for the slice endpoints, and mapping those endpoints to the complex plane (Zhang et al., 26 Jul 2025). The $$z_R(u_1,u_2):= \left\{ \frac{\|y_1-y_2\|}{\|u_1-u_2\|} e^{\pm j\angle(u_1-u_2,\;y_1-y_2)} \;\middle|\; y_1\in R(u_1),\;y_2\in R(u_2) \right\},$$7-symmetric paper also introduces arc-hull and annular-sector over-approximations, with

$$z_R(u_1,u_2):= \left\{ \frac{\|y_1-y_2\|}{\|u_1-u_2\|} e^{\pm j\angle(u_1-u_2,\;y_1-y_2)} \;\middle|\; y_1\in R(u_1),\;y_2\in R(u_2) \right\},$$8

providing simpler but more conservative graphical tests (Yang et al., 8 Oct 2025).

A distinct but related rotational theme appears in applications where physical coordinate changes should not alter the graphical certificate. In power-electronics-dominated grids, standard SRG analysis is invariant under dq-frame rotations: $$z_R(u_1,u_2):= \left\{ \frac{\|y_1-y_2\|}{\|u_1-u_2\|} e^{\pm j\angle(u_1-u_2,\;y_1-y_2)} \;\middle|\; y_1\in R(u_1),\;y_2\in R(u_2) \right\},$$9 and specifically

$\mathcal U\subseteq\mathcal L$0

This invariance concerns coordinate rotation of the underlying model, not the phase-reference rotation of $\mathcal U\subseteq\mathcal L$1, but both exploit the same complex-plane gain–phase geometry (Baron-Prada et al., 22 Jan 2026).

Further generalizations extend the rotational idea beyond Hilbert-space phase. In normed spaces, directional SRGs replace the inner-product angle by left and right directional angles induced by compatible regular pairings, so that continuous Euclidean rotation is replaced by sign-pattern transitions in $\mathcal U\subseteq\mathcal L$2 and facet transitions in $\mathcal U\subseteq\mathcal L$3 (Padoan, 2 Apr 2026). This suggests that the rotated SRG is one member of a broader family of phase-sensitive operator graphs whose exact geometric meaning depends on the ambient space and on the chosen symmetry or reference structure.

In that broader context, the rotated SRG occupies a specific position: it is the explicitly phase-referenced, two-dimensional gain–phase shadow of an operator, devised to sharpen feedback separation conditions while preserving graphical tractability. Its distinguishing features are the tunable reference angle $\mathcal U\subseteq\mathcal L$4, exact scalar reduction, compatibility with product/cascade analysis, and its role as the least conservative currently available two-dimensional graphical condition for bi-component MIMO feedback loops (Zhang et al., 26 Jul 2025, Yang et al., 8 Oct 2025).