Scaled Relative Graphs: A Geometric Framework for Systems

• Scaled Relative Graphs (SRG) are a geometric framework capturing incremental input-output properties by mapping gain and phase relations in the complex plane.
• They generalize classical tools like the Nyquist locus to address nonlinear, multivariable, and time-varying systems via soft and hard SRG formulations.
• SRG enables rigorous stability and robustness analysis through separation theorems that certify closed-loop stability in feedback interconnections.

Scaled Relative Graphs (SRG) provide a geometric and operator-theoretic framework for analyzing the input-output behavior, incremental gain, phase, and stability margins of both linear and nonlinear dynamical systems. The SRG perspective encodes critical system-theoretic properties in the complex plane, generalizing classical tools such as the Nyquist locus and enabling rigorous graphical stability and robustness criteria for broad classes of (potentially unbounded) operators, including those describing nonlinear feedback, multivariable, and time-varying systems.

1. Formal Definition: Soft and Hard SRGs

Let $L^n$ denote the Hilbert space of square-integrable $\mathbb{R}^n$-valued signals on $[0,\infty)$, with inner product

$$\langle u, v \rangle = \int_0^\infty u(t)^\top v(t)\,dt.$$

Given a causal operator $P: L^n \to L^n$, the soft scaled relative graph is

$$\mathrm{SRG}(P) = \bigg\{\, \gamma(\Delta u, \Delta y)\,e^{\pm j\,\theta(\Delta u, \Delta y)}\,:\, (u_i, y_i) \in G(P),\; \Delta u \neq 0,\, \Delta y \neq 0 \, \bigg\},$$

where $\gamma(u, v) = \frac{\|v\|_2}{\|u\|_2}$, $$\theta(u, v) = \arccos \frac{\langle u, v\rangle}{\|u\|_2 \|v\|_2}$$, $\Delta u = u_1 - u_2$, and $\Delta y = y_1 - y_2$.

The hard scaled relative graph is defined via truncation: $\mathbb{R}^n$0 where the truncation operator $\mathbb{R}^n$1 for $\mathbb{R}^n$2, zero otherwise; $\mathbb{R}^n$3 are computed analogously on $\mathbb{R}^n$4.

Their inverse SRGs $\mathbb{R}^n$5 are obtained by swapping input and output.

2. Incremental System Properties via SRGs

SRGs characterize incremental input-output properties directly:

• Incremental positivity: $\mathbb{R}^n$6 is incrementally positive iff

$\mathbb{R}^n$7

• Incremental passivity: $\mathbb{R}^n$8 is incrementally passive iff

$\mathbb{R}^n$9

• Strict incremental forms: If

$[0,\infty)$0

then

$[0,\infty)$1

where

$[0,\infty)$2

3. Graphical Construction and Interpretation

SRGs are constructed by evaluating all pairs of signal increments:

• For every $[0,\infty)$3, compute $[0,\infty)$4.
• The hard SRG is the union over all time truncations, giving a generally larger set.

This process results in a region of the complex plane encoding all incremental (gain, phase) behaviors. For linear time-invariant (LTI) SISO systems, the SRG collapses to the Nyquist image (modulo boundedness), while for general nonlinear or time-varying systems, it captures a large class of input-output relations.

4. Main Separation Theorems for Nonlinear Feedback Stability

Consider the positive-feedback interconnection: $[0,\infty)$5 with input–output map $[0,\infty)$6.

The Hard SRG Separation Theorem states: $[0,\infty)$7

The Soft SRG Separation Theorem: $[0,\infty)$8

$[0,\infty)$9

In essence, closed-loop stability is certified if the SRGs of the open-loop systems and their inverses remain separated in the complex plane, preventing incremental gain–phase pairs that conspire to give a unity loop gain at an unstable phase.

Unlike earlier results, these theorems do not require the “chordal” property and are applicable to potentially unbounded open-loop systems (via the hard SRG) (Chen et al., 19 Apr 2025).

5. Relationship Between Soft and Hard SRGs

The soft SRG is always contained in the closure of the hard SRG: $$\langle u, v \rangle = \int_0^\infty u(t)^\top v(t)\,dt.$$0 As the truncation window $$\langle u, v \rangle = \int_0^\infty u(t)^\top v(t)\,dt.$$1, calculations on the hard SRG recover the soft SRG as a limiting case.

Sector bounds and strict passivity/positivity regions directly translate to geometric constraints—strict SRG containment in half-planes or sectors yields quantitative gain and phase margins.

6. Illustrative Examples

LTI Integrator ($$\langle u, v \rangle = \int_0^\infty u(t)^\top v(t)\,dt.$$2):

• For any $$\langle u, v \rangle = \int_0^\infty u(t)^\top v(t)\,dt.$$3 with $$\langle u, v \rangle = \int_0^\infty u(t)^\top v(t)\,dt.$$4 and $$\langle u, v \rangle = \int_0^\infty u(t)^\top v(t)\,dt.$$5:

$$\langle u, v \rangle = \int_0^\infty u(t)^\top v(t)\,dt.$$6

i.e., the imaginary axis (soft SRG). On finite windows, positivity holds ($$\langle u, v \rangle = \int_0^\infty u(t)^\top v(t)\,dt.$$7), so the hard SRG fills the closed right half-plane minus the origin.

Static Saturation ($$\langle u, v \rangle = \int_0^\infty u(t)^\top v(t)\,dt.$$8, slope $$\langle u, v \rangle = \int_0^\infty u(t)^\top v(t)\,dt.$$9):

• Incremental gain $P: L^n \to L^n$0, positivity holds:

$P: L^n \to L^n$1

i.e., a quarter-disk sector in the right half-plane.

Computational Strategy:

• Choose representative increments (e.g., LTI: exponentials; static nonlinearity: piecewise-constant).
• Compute incremental gain and phase for each pair.
• The union yields the SRG region; for hard SRG, repeat over all time windows.

7. Generalization and Significance

These results generalize the soft SRG separation theorem for bounded systems and remove the need for a chordal over-approximation class, applying directly to unbounded or possibly unstable systems via the hard SRG. As such, the framework reconciles incremental positivity and passivity in a unified geometric language, suitable for both classical and contemporary nonlinear feedback analysis.

This approach underpins a range of stability, robustness, and controller synthesis results, providing an operationally rich geometric calculus that supports analysis of feedback phenomena beyond traditional frequency-domain limitations (Chen et al., 19 Apr 2025).