Computing Scaled Relative Graphs of Discrete-Time LTI Systems: A Frequency-Domain Approach

arXiv:2609.38272 · eess.SY, cs.SY · Submitted 2026-09-29 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "Computing Scaled Relative Graphs of Discrete-Time LTI Systems".

Dev: The scaled relative graph (SRG) analysis provides a powerful geometric tool for characterizing operators, and this paper characterizes the SRG closure of causal,

Rosa: First, who's behind it and why it matters.

Paper summary: Rosa: So we've been talking about this paper, "Computing Scaled Relative Graphs of Discrete-Time LTI Systems: A Frequency-Domain Approach," and to recap, its core thesis is that it provides a way to characterize the SRG closure for causal, stable square discrete-time linear time-invariant systems by relating it directly to the convex hull of transformed frequency-response matrices.

Dev: Dev agrees with that summary, noting that this characterization is significant because it allows for model-based computation of this closure right from a state-space realization using frequency evaluations and planar convex hulls, which lets us skip solving linear matrix inequalities for gain bounds.

Taro: Taro sees the significance in how it bypasses those LMI solutions, suggesting that this is a useful computational shortcut when we need to check stability properties quickly.

Rosa: It seems like the paper is making a big claim by showing that the Beltrami–Klein image of this closure equals the convex hull of these transformed frequency-response matrices, which for real-coefficient SISO systems yields the hyperbolic convex hull of the discrete-time Nyquist locus. Rosa is thinking about how impactful this connection to classical control theory might be for our work.

Dev: That connection is interesting because it grounds a complex operator property in a known geometric shape, but Dev also sees the technical hurdle regarding the Hardy space restriction imposed by one-sided inputs, which excludes persistent sinusoids.

Taro: Taro agrees that dealing with that Hardy space restriction is key, as it shows how points in frequency-wise SRGs are actually limits of operator SRG points generated by admissible inputs, which is a crucial piece of the puzzle for understanding system behavior under those constraints.

Rosa: So, the paper isn't just stating a result; it's providing a specific mechanism showing *how* those frequency-wise SRG points arise as limits from admissible inputs, which helps us understand the underlying dynamics better. Rosa is wondering if this mechanistic insight helps when we apply it to more complex physical systems with messy, non-ideal dynamics.

Dev: From an engineering standpoint, that mechanistic detail is useful because it explains the structure of the operator SRG point generation process, which informs how we might design input sequences that are computationally feasible for our control loops.

Taro: Taro pushes on whether this structural understanding helps when we have to contend with non-ideal dynamics; if the underlying mechanism is understood, perhaps we can build a more robust framework that handles those deviations better than just relying on idealized inputs.

Rosa: Exactly, so the paper suggests that understanding this mechanism allows us to move beyond simply applying pre-defined methods and instead builds a system description that is more inherently descriptive of its actual operational limits. Rosa is curious about how this level of detail translates into practical safety assurances for field operations.

Dev: And Dev wants to ensure we keep bringing the engineering reality back; if we are building something that needs high-fidelity stability checks, he's asking if this model-based computation offers a reliable path toward meeting our required loop rates without introducing unacceptable latency.

Taro: Taro brings up the autonomy angle again; he is interested in whether this geometric characterization can help define safety boundaries when systems encounter unpredictable external factors or misbehave, which is where real-world resilience truly matters.

Conclusion: Rosa: We've covered how this paper characterizes the SRG closure using frequency-domain tools, linking it to the convex hull of transformed frequency-response matrices, which for SISO systems relates to the Nyquist locus, but now we need to discuss what this actually means in simpler terms regarding its implications and where it might lead.

Dev: Dev is looking at the paper's authors and title again, considering how these results translate into real-world application; he wants a plain-language explanation of what this characterization offers for practical deployment.

Taro: Taro pushes on the broader impact of this work, pushing on what it means for the autonomy research community in terms of defining new standards or frameworks for system analysis.

Rosa: The implication is that we get a robust, model-based tool to determine stability properties using geometry derived from frequency response data instead of solving iterative matrix inequalities, which might change how we approach verification in complex systems. Rosa is thinking about the practical shift in verification workflows.

Dev: That shift means less reliance on solving those kinds of optimization problems for gain bounds and more on geometric constructions, but Dev is concerned about the computational cost if the geometric construction itself becomes too slow for fast-loop requirements.

Taro: Taro wants to know what this means for the autonomy research community in terms of establishing a new standard; he's interested in whether this approach provides a new framework that other researchers can use to analyze system stability more effectively.

Rosa: The paper offers a tool that formalizes how we compute the SRG closure through geometry, which is significant because it gives us an established way to verify the system's stability properties based on frequency response data. Rosa is thinking about how this new verification pathway could be used across various domains in robotics and beyond.

Dev: So Dev wants a practical summary of what this means for deployment—less LMI solving, but he still needs reassurance that the geometric construction will actually keep up with our required real-time loop rates without introducing unacceptable latency.

Taro: Taro is interested in whether this provides a new framework that other researchers can use to analyze system stability more effectively, pushing on the broader impact for autonomy research in terms of defining new standards.

Rosa: Rosa concludes that the main point is that this work gives us a systematic geometric way to verify stability properties using frequency response data, which could fundamentally alter how we think about system verification across different fields.

Dev: Dev summarizes his view by emphasizing the practical trade-off between computational cost and loop rate; he wants to make sure that we have a clear understanding of whether this geometric computation offers a reliable path toward meeting our required real-time performance targets.

Taro: Taro wraps up by stressing the importance of this for autonomy research, framing it as a new framework for system analysis that other researchers can adopt to analyze stability more effectively in complex autonomous systems.

Guoming Shi, Fredy Ruiz

Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano

eess.SY, cs.SY

Submitted: 2026-09-29

Updated: 2026-09-29

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 83/100

The gist: The scaled relative graph (SRG) analysis provides a powerful geometric tool for characterizing operators, and this paper characterizes the SRG closure of causal, stable square discrete-time linear

Key concepts

Scaled Relative Graph (SRG) Closure: The SRG closure is a geometric tool used to characterize operators related to discrete-time LTI systems. The paper shows this closure can be found by taking the convex hull of transformed frequency-response matrices, which simplifies system analysis.
Frequency Response Matrix ($G_d( heta)$): This matrix is derived from the system's transfer function evaluated at specific frequencies ($ heta$). It represents how the system responds to sinusoidal inputs at different frequencies, and its numerical range is a key component in defining the SRG closure.
Beltrami–Klein Image: This geometric mapping relates points in one space (like the SRG closure) to another (like a convex hull). The paper uses this map to show that the SRG closure corresponds exactly to the Beltrami–Klein image of the convex hull of frequency-wise numerical ranges.

Terminology

Verification — 4× in this paper · explained in 2 episodes
Core — 2× in this paper · explained in 2 episodes
Fidelity — 1× in this paper · explained in 2 episodes
Hilbert Space — 1× in this paper · explained in 2 episodes

Summary

The scaled relative graph (SRG) analysis provides a powerful geometric tool for characterizing operators, and this paper characterizes the SRG closure of causal, stable square discrete-time linear time-invariant (LTI) systems by relating it to the convex hull of transformed frequency-response matrices. This characterization is significant because it enables model-based computation of the SRG closure directly from a state-space realization using frequency evaluations and planar convex hulls, bypassing the need to solve linear matrix inequalities for gain bounds.

The gist

The Beltrami–Klein image of the operator SRG closure equals the convex hull of the numerical ranges of the transformed frequency-response matrices.

System Characterization and Mathematical Framework

The paper considers a square LTI system described by state-space equations (1) and (2), assuming real coefficients and Schur stability, with an induced causal convolution operator Td mapping from the Hardy space of one-sided square-summable sequences, denoted as the input domain. The key mathematical objects are the transfer matrix, defined as G(z) = C(zI − A)−1B + D, and its frequency response Gd(θ) = G(e jθ). The analysis is restricted to inputs in the Hardy space H2, which excludes persistent sinusoids. To address this restriction, the paper constructs finitely supported input sequences that realize the required limiting quadratic-form weights.

The Main Result: SRG Closure Characterization

Theorem 1 establishes the core result:

the closure of its SRG satisfies fbk(cl SRG(Td)) = co [θ∈[0,2π) W(ΦG(θ)).

This is equivalent to characterizing the closure via the inverse Beltrami–Klein map: cl SRG(Td) = gbk (co [θ∈[0,2π) W(ΦG(θ))]. The proof proceeds by showing that points in the SRG closure are weighted averages of points from each frequency-wise numerical range. Specifically, it demonstrates that for any point generated by an admissible input sequence, its Beltrami–Klein image is a weighted average of the images of the frequency-wise numerical ranges W(ΦG(θ)), where the weights are proportional to the squared norm of the Hardy space boundary values.

Geometric Interpretation and SISO Connection

For a SISO system, Theorem 1 yields a direct connection to classical control theory: This yields the hyperbolic convex hull of the discrete-time Nyquist locus. This means that for real-coefficient single-input single-output systems, the SRG closure is equivalent to the hyperbolic convex hull of the discrete-time Nyquist locus. Corollary 1 further specifies this relationship: fbk(cl SRG(Td)) = co[fbk(Gd(θ)): θ ∈ [0, 2π)]. This implies that the closure depends only on the set of frequency-wise SRGs and not on the specific frequency at which each point is attained.

Computational Advantages and Applications

The paper highlights a computational advantage of this approach over traditional methods: This enables model-based computation of the closure directly from a state-space realization through frequency-response evaluations, numerical ranges, and planar convex hulls, without solving linear matrix inequalities. The method is shown to be preferable when a model is available. Furthermore, for MIMO systems, the paper notes that while each frequency-wise SRG may be a region rather than a point set, the Beltrami–Klein image of their union is already convex, allowing Theorem 1 to apply directly: the union equals cl SRG(Td); accordingly, it coincides with the LMI-based SRG in Fig. 3b. This demonstrates that frequency-domain computation is efficient for model-based analysis.

Conclusion and Further Implications

The characterization proves that the operator SRG closure is obtained by convexifying, in the Beltrami–Klein disk, the frequency-wise SRGs used in [11]. The final result shows that cl fbk(SRG(Td)) = KG, where KG is the convex hull of all frequency-wise numerical ranges. This provides a robust method for determining system stability and operator properties using geometric tools derived from frequency response data. Remark 2 further states that the numerical range W(ΦG(θ)) is exactly the Beltrami–Klein image of the SRG of the constant matrix Gd(θ), confirming that Theorem 1 effectively states that SRG(Gd(θ)) ⊆ cl SRG(Td) for every θ ∈ [0, 2π). The paper concludes by showing how this geometric characterization relates to the classical Nyquist locus for SISO systems.

How it works

The proof of Theorem 1 involves relating the SRG point generated by an admissible input sequence to a weighted average of points from the frequency-wise numerical ranges.

Improvements for AI systems

Based on the provided scientific paper, here are the specific improvements to AI systems that can be made by applying its core concepts:

The fundamental improvement is a shift from traditional, often computationally intensive LMI-based stability/performance analysis towards a more efficient, frequency-domain geometric approach for analyzing discrete-time Linear Time-Invariant (LTI) systems.

Here are the specific improvements and what the improved AI system can do:

Improve Stability Analysis Efficiency:
Improve Model-Based Controller Design Robustness:
Enable Direct SRG Characterization from Data:
Facilitate Complex MIMO System Performance Prediction:

Improving Stability Analysis Efficiency (via Theorem 1):

The paper establishes that the closure of the Scaled Relative Graph (SRG) for a stable, square discrete-time LTI system is equal to the Beltrami–Klein convexification of its frequency-wise SRGs, which simplifies to the hyperbolic convex hull of the Nyquist locus.

An improved AI system can perform stability analysis by:

The system does not need to repeatedly solve complex Linear Matrix Inequalities (LMIs) for gain bounds or dissipativity constraints across multiple frequencies. Instead, it can compute the required set (the SRG closure) directly from a given state-space realization using only:

a. Frequency-response evaluations of the transfer matrix, and

b. The numerical ranges of these frequency-response matrices.

This drastically reduces computational complexity for determining system stability boundaries compared to LMI solvers, especially for high-order MIMO systems where LMI computation time scales poorly with the number of states and frequencies.

Improving Model-Based Controller Design Robustness:

By leveraging the characterization in Theorem 1, an AI system can design controllers that are guaranteed to respect the stability constraints derived from the SRG closure without needing to explicitly solve a complex set of matrix inequalities for every frequency point.

The improved AI system can:

a. Compute the exact, convex hull region (the hyperbolic convex hull of the Nyquist locus) in the complex plane that defines all achievable gain/phase combinations for a stable system.

b. Use this geometric constraint to design controllers whose closed-loop transfer functions are guaranteed to lie within this geometrically defined stability region, ensuring robust performance bounds derived from frequency-domain geometry rather than iterative optimization.

Enabling Direct SRG Characterization from Data:

The paper provides a framework for computing the SRG closure from a model's frequency response without solving LMIs. Furthermore, Theorem 1 proves that the closure is generated by convex combinations of the individual frequency-wise SRGs (i.e., points generated by admissible inputs).

An improved AI system can:

a. Analyze experimental or system identification data (which yields frequency-response matrices) and directly construct the SRG closure set using the derived geometric formulas, bypassing intermediate LMI steps entirely.

b. Detect if a system configuration is unstable or marginally stable by checking if the desired operating point lies outside this geometrically characterized closure, providing a rapid diagnostic tool for system health.

Facilitating Complex MIMO System Performance Prediction:

For MIMO systems (as shown in Example C), the paper highlights that while individual frequency-wise SRGs are regions, their union's Beltrami–Klein image is often convex, allowing Theorem 1 to state that the union equals the closure.

An improved AI system can:

a. Predict and visualize the achievable gain/phase region for complex MIMO plants based on their frequency response data.

b. Distinguish between frequency-wise analysis (which only captures inputs at single frequencies) and the full SRG closure (which captures convex combinations of inputs across multiple frequencies), allowing engineers to understand how combining different operational modes affects overall system performance bounds.

Abstract

The scaled relative graph (SRG) represents the joint gain and phase properties of an input--output operator in the complex plane. This paper characterizes the SRG closure of causal, stable, square discrete-time linear time-invariant (LTI) systems on one-sided square-summable sequences. The Beltrami--Klein image of this closure equals the convex hull of the numerical ranges of the transformed frequency-response matrices. For real-coefficient single-input single-output systems, this yields the hyperbolic convex hull of the discrete-time Nyquist locus. The proof addresses the Hardy-space restriction imposed by one-sided inputs, which excludes persistent sinusoids, by replacing them with explicitly constructed finite-duration sinusoids. This construction shows how points in frequency-wise SRGs arise as limits of operator SRG points generated by admissible inputs. The characterization enables model-based computation of the closure from a state-space realization using frequency-response evaluations, numerical ranges, and planar convex hulls, without solving linear matrix inequalities.