Scaled Relative Graphs in Nonlinear Systems

• Scaled Relative Graphs (SRGs) are subsets of the complex plane that represent an operator’s incremental input-output behavior through normalized gain ratios and phase differences.
• They extend classical Nyquist and Bode methods to nonlinear operators, enabling frequency-dependent gain bounds and generalized loop transfer analysis.
• SRGs support an interconnection calculus that links properties such as monotonicity, passivity, and stability via geometric containment in the complex plane.

Scaled Relative Graphs (SRGs) are subsets of the complex plane that encode the input–output behavior of an operator through incremental gain and incremental angle. In the nonlinear systems literature, they function as a graphical frequency-domain method that extends Nyquist- and Bode-style reasoning from LTI transfer functions to incrementally stable nonlinear operators. In particular, restricting SRGs to selected input spaces yields frequency-dependent gain bounds and supports nonlinear generalizations of loop transfer, sensitivity, and bandwidth, while related developments connect SRGs to monotonicity, passivity, IQCs, matrix spectra, and feedback stability (Krebbekx et al., 2 Apr 2025, Chaffey et al., 2021).

1. Definition and geometric encoding

Let $\mathcal{L}$ be a Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and norm $\|\cdot\|$. For $u,y\in\mathcal{L}$, the angle is defined by

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

Given an operator $R$ and two inputs $u_1,u_2$, the associated complex pair is

$$z_R(u_1,u_2):= \left\{ \frac{\|Ru_1-Ru_2\|}{\|u_1-u_2\|} e^{\pm j\angle(u_1-u_2,\;Ru_1-Ru_2)} \right\},$$

and the SRG is the union of all such points over admissible input pairs. In this encoding, $|z|$ is the ratio of incremental output and input norms, while $\arg(z)$ records the angle between input and output differences (Krebbekx et al., 2 Apr 2025).

The terminology is literal. The graph is “relative” because it is built from differences $\langle\cdot,\cdot\rangle$0 and $\langle\cdot,\cdot\rangle$1, and “scaled” because the complex number is normalized by the dimensionless gain factor $\langle\cdot,\cdot\rangle$2. For linear maps, the definition simplifies to evaluation on a single direction $\langle\cdot,\cdot\rangle$3: $\langle\cdot,\cdot\rangle$4 with the full SRG obtained by adding the complex conjugates. In that linear setting, the modulus is a directional amplification factor and the argument is a turning angle between $\langle\cdot,\cdot\rangle$5 and $\langle\cdot,\cdot\rangle$6 (Huang et al., 2019).

A fundamental performance connection is that the radius of the SRG equals the incremental induced $\langle\cdot,\cdot\rangle$7-norm. Geometrically, the smallest disk centered at the origin that contains the SRG has radius equal to the incremental $\langle\cdot,\cdot\rangle$8-gain of the operator (Krebbekx et al., 2 Apr 2025).

2. Relation to linear frequency-domain theory

For LTI systems on $\langle\cdot,\cdot\rangle$9, SRGs are directly tied to classical frequency-response geometry. One formulation states that the Nyquist diagram of an LTI system is the convex hull of its SRG under a particular change of coordinates; equivalently, in the upper half-plane, the SRG is the hyperbolic-convex hull of the Nyquist locus. Under the Beltrami–Klein map, this hyperbolic convexity becomes ordinary Euclidean convexity, so the SRG can be viewed as a filled Nyquist region rather than a mere curve (Chaffey et al., 2021).

This relation is especially transparent for structured linear operators. For real normal matrices, the positive SRG is exactly the arc-edge polygon $\|\cdot\|$0, namely the hyperbolic convex hull of the eigenvalues in the upper half-plane. For symmetric matrices, this reduces to an explicit disk formula built from the ordered real eigenvalues. In this sense, the SRG acts as a geometric generalization of the spectrum: it contains the spectrum for linear operators and extends spectral-style visualization beyond eigen-directions (Huang et al., 2019).

The same linear correspondence underlies later nonlinear frequency-domain constructions. In the nonlinear Bode framework, the restriction of the SRG to selected input classes is designed so that the resulting frequency-dependent gain bounds remain compatible with familiar LTI notions of loop transfer, sensitivity, and bandwidth (Krebbekx et al., 2 Apr 2025).

3. Calculus and operator classes

A central reason SRGs are useful is that they admit an interconnection calculus. For operators $\|\cdot\|$1 on a Hilbert space and $\|\cdot\|$2,

$\|\cdot\|$3

where inversion is Möbius inversion $\|\cdot\|$4. With the appropriate chord or arc property, analogous set operations describe sums and compositions, so algebraic manipulations of systems become geometric manipulations in $\|\cdot\|$5 (Krebbekx et al., 2 Apr 2025).

This calculus is effective because many operator classes are SRG-full: membership in the class is equivalent to containment of the SRG in a corresponding complex region. For monotone operators, the region is the closed right half-plane; for $\|\cdot\|$6-strongly monotone operators, it is the shifted half-plane $\|\cdot\|$7. More generally, semimonotone classes $\|\cdot\|$8 correspond to disks or complements of disks centered on the real axis, and angle-bounded operators correspond to symmetric wedges about the positive real axis. The geometric containment condition is not merely illustrative; it is equivalent to the defining inequality when the class is SRG-full (Quan et al., 2024).

An analogous picture persists in normed spaces once the Hilbert inner product is replaced by a compatible regular pairing. In that setting, asymmetry of the pairing produces left and right directional angles and hence directional SRGs, but the main containment principles survive: Lipschitz continuity is a disk bound $\|\cdot\|$9, one-sided $u,y\in\mathcal{L}$0-Lipschitzness is a left half-plane bound $u,y\in\mathcal{L}$1, strong monotonicity is a right half-plane bound $u,y\in\mathcal{L}$2, and cocoercivity is a disk tangent at the origin (Padoan, 2 Apr 2026).

4. Soft, hard, extended, and directional variants

A major refinement of the theory is the distinction between soft and hard SRGs. The soft SRG is built from $u,y\in\mathcal{L}$3 trajectories on the infinite horizon and naturally encodes properties such as incremental positivity. The hard SRG is built from truncated $u,y\in\mathcal{L}$4 trajectories over all finite horizons and naturally encodes properties such as incremental passivity. In graphical terms, incremental positivity is equivalent to containment of the soft SRG in the closed right half-plane, whereas incremental passivity is equivalent to containment of the hard SRG in the closed right half-plane. Separation theorems formulated in terms of soft or hard SRGs then yield nonlinear feedback-stability results, including cases with unbounded open-loop operators (Chen et al., 19 Apr 2025).

This distinction matters because naive soft-SRG reasoning can fail for unstable LTI components. A later SISO analysis identified a pitfall: standard SRG calculus may suggest finite gain on a restricted $u,y\in\mathcal{L}$5 domain even when Nyquist shows that the full feedback system is unstable. To repair this, that work introduced an extended SRG

$u,y\in\mathcal{L}$6

where $u,y\in\mathcal{L}$7 is the h-convex hull of the Nyquist locus and $u,y\in\mathcal{L}$8 is defined through winding-number information. The extended SRG preserves the familiar calculus under scaling, addition, inversion, and product, while reincorporating Nyquist encirclement data into SRG-based stability analysis (Krebbekx et al., 21 Jul 2025).

The hard-SRG program subsequently gave an exact computational method for square LTI systems, including unstable systems and systems with integrators. In that framework, the hard SRG is represented as an intersection of disks, with radii determined by maximum and minimum singular-value calculations of shifted transfer matrices. In SISO, the hard SRG coincides with the extended SRG; in square MIMO, it yields a Nyquist-like criterion without requiring explicit winding-number calculations (Krebbekx et al., 21 Nov 2025).

Directional variants provide another axis of generalization. In normed spaces, regular pairings induce left and right directional SRGs rather than a single symmetric object. This changes the geometry—polyhedral in $u,y\in\mathcal{L}$9 or $$\angle(u,y):=\cos^{-1}\frac{\Re\langle u,y\rangle}{\|u\|\,\|y\|}\in[0,\pi].$$0, rather than rotational as in $$\angle(u,y):=\cos^{-1}\frac{\Re\langle u,y\rangle}{\|u\|\,\|y\|}\in[0,\pi].$$1—but preserves the containment-based interpretation of contraction, monotonicity, and cocoercivity (Padoan, 2 Apr 2026).

5. Nonlinear Bode diagrams and bandwidth

The paper “Nonlinear Bandwidth and Bode Diagrams based on Scaled Relative Graphs” develops SRG restrictions to particular input spaces in order to compute frequency-dependent gain bounds for incrementally stable nonlinear systems. This yields a nonlinear generalization of the Bode diagram in which sinusoidal, harmonic, and subharmonic inputs are treated separately. Applied to nonlinear loop transfer and sensitivity, the same construction defines open-loop and closed-loop bandwidth notions that are explicitly stated to be compatible with the LTI definitions. The paper illustrates the procedure on a DC motor with a parasitic nonlinearity and verifies the resulting bounds in simulation (Krebbekx et al., 2 Apr 2025).

A later development adds amplitude as a second independent axis. For Lur’e systems, SRGs are restricted simultaneously in frequency and in input energy content, and Sobolev estimates are used to convert energy restrictions into amplitude restrictions. The result is a three-dimensional nonlinear Bode diagram plotting $$\angle(u,y):=\cos^{-1}\frac{\Re\langle u,y\rangle}{\|u\|\,\|y\|}\in[0,\pi].$$2-gain as a function of both frequency and energy content. Two limiting statements anchor the construction to classical theory: in the zero-energy limit, the LTI Bode diagram is recovered; in the infinite-energy, zero-frequency limit, the global $$\angle(u,y):=\cos^{-1}\frac{\Re\langle u,y\rangle}{\|u\|\,\|y\|}\in[0,\pi].$$3-gain is recovered (Krebbekx et al., 10 Mar 2026).

These constructions clarify that SRGs are not only static geometric certificates. By restricting the admissible signal class, the same complex-plane object can support frequency-wise gain estimates, amplitude-dependent performance bounds, and bandwidth notions that are meaningful for nonlinear closed loops.

6. Applications, generalizations, and limitations

SRGs have been applied in circuit theory, optimization-adjacent operator analysis, reinforcement learning, model reduction, and power systems. In nonlinear circuit analysis, semimonotone and angle-bounded SRG regions were used to classify nonmonotone devices, including the Ebers–Moll transistor and tunnel diode, and to justify convergence of Chambolle–Pock iterations for common-emitter amplifier models with nonsmooth and multivalued elements (Quan et al., 2024). In circuit model reduction, SRGs represent port behaviors of nonlinear one-ports, and the SRG of the error relation between a full series/parallel chain and a truncation yields an $$\angle(u,y):=\cos^{-1}\frac{\Re\langle u,y\rangle}{\|u\|\,\|y\|}\in[0,\pi].$$4-gain bound on the approximation error while preserving incremental positivity (Chaffey et al., 2022).

For MIMO LTI feedback, SRG separation has been shown to be equivalent, under stated assumptions, to the sufficient generalized Nyquist condition $$\angle(u,y):=\cos^{-1}\frac{\Re\langle u,y\rangle}{\|u\|\,\|y\|}\in[0,\pi].$$5 for all $$\angle(u,y):=\cos^{-1}\frac{\Re\langle u,y\rangle}{\|u\|\,\|y\|}\in[0,\pi].$$6, but in a decoupled form that compares $$\angle(u,y):=\cos^{-1}\frac{\Re\langle u,y\rangle}{\|u\|\,\|y\|}\in[0,\pi].$$7 and $$\angle(u,y):=\cos^{-1}\frac{\Re\langle u,y\rangle}{\|u\|\,\|y\|}\in[0,\pi].$$8 rather than analyzing the product $$\angle(u,y):=\cos^{-1}\frac{\Re\langle u,y\rangle}{\|u\|\,\|y\|}\in[0,\pi].$$9 directly (Baron-Prada et al., 17 Mar 2025). A further refinement, the $R$0-symmetric SRG, replaces the default symmetry about the real axis by symmetry about a line $R$1, restores exact reduction to the scalar itself in the scalar case, and establishes a submultiplicative product rule that is tailored to cascades and cyclic interconnections (Yang et al., 8 Oct 2025).

The framework has also been extended beyond single operators. For pairs $R$2, the relative SRG $R$3 compares increments in the output of $R$4 to increments in the output of $R$5, rather than to input increments. This supports geometric formulations of paired monotonicity conditions of the form

$R$6

which become the region $R$7 in the complex plane. The paper emphasizes nonlinear resolvents, linear operators composed with monotone mappings, and circuit examples including NPN transistors (Quan et al., 25 Nov 2025).

Operating-point dependence is another active direction. In converter-dominated power systems, SRG analysis has been extended to LPV admittances that depend affinely on operating points. There the centralized small-signal stability test is decomposed into decentralized, frequency-wise geometric tests, and each converter obtains a feasible stability region expressed as linear inequalities in its parameter space (Baron-Prada et al., 14 Mar 2026).

Several limitations recur across the literature. Exact SRG characterizations are available only for specific classes—such as normal matrices, stable square LTI systems, or hard SRGs of square LTI systems—while general nonlinear and high-dimensional cases often require outer approximations. In normed spaces, the geometry depends on the chosen regular pairing, so left and right SRGs can differ. Most importantly, the literature now treats it as a substantive misconception that soft SRGs alone suffice for unstable LTI feedback analysis; in such settings, hard or extended SRGs are required if Nyquist encirclement information is to be represented correctly (Krebbekx et al., 21 Jul 2025, Krebbekx et al., 21 Nov 2025).