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

Guoming Shi, Fredy Ruiz

arXiv:2609.38272v1eess.SYcs.SY

TL;DR

The paper seeks a frequency-domain characterization of the SRG closure for stable discrete-time LTI systems, against prior discrete-time methods using LMIs. It constructs admissible finite-duration inputs and proves that the closure’s Beltrami–Klein image is the convex hull of numerical ranges of transformed frequency-response matrices. For real-coefficient SISO systems, this yields the hyperbolic convex hull of the discrete-time Nyquist locus and supports model-based computation from a state-space realization.

  • Problem

    For stable discrete-time LTI systems, an explicit frequency-response description of the SRG closure is sought alongside prior discrete-time SRG methods based on LMIs.

    Source: §I · p. 1

  • Method

    The proof constructs finitely supported inputs to realize limiting quadratic-form weights and characterizes the closure through transformed frequency-response matrices and their numerical ranges.

    Source: §I · p. 1

  • Results

    The closure’s Beltrami–Klein image equals the convex hull of numerical ranges of transformed frequency-response matrices, yielding the hyperbolic convex hull of the Nyquist locus for real-coefficient SISO systems.

    Source: §I · p. 1, §III · p. 4

  • Takeaways & Limitations

    A state-space model enables computation of the closure using frequency-response evaluations, numerical ranges, and planar convex hulls instead of repeated LMI solves.

    Source: §I · p. 1, §V · p. 6

    Identifying SRGs from frequency-response data remains future work.

    Source: §V · p. 6

Abstract

from arXiv · show

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.

I. INTRODUCTION

The introduction places SRGs at the intersection of gain-and-phase analysis, control, and numerical-range methods. It presents a frequency-domain characterization of discrete-time SRG closures and explains how admissible finite-duration inputs address the one-sided signal-space restriction.

I. INTRODUCTION

  • SRGs unite gain and phase in the complex plane and have been applied to control analysis, optimization convergence, and nonlinear feedback analysis.Related work also links SRGs to dissipativity, numerical ranges, and integral quadratic constraints.Source: §I · p. 1
  • Prior work connected linear-operator SRGs to numerical ranges through the Beltrami–Klein map and developed discrete-time LMI-based methods for SRGs.The cited discrete-time formulation uses the full space of one-sided square-summable sequences in the stable case considered here.Source: §I · p. 1
  • Finitely supported inputs constructed from increasingly long truncated sinusoids address the Hardy-space restriction and realize limiting quadratic-form weights.Persistent sinusoids are not square summable, so they are inadmissible inputs.Source: §I · p. 1
  • The main result identifies the Beltrami–Klein image of the operator SRG closure with the convex hull of numerical ranges of transformed frequency-response matrices.For real-coefficient SISO systems, this gives the hyperbolic convex hull of the discrete-time Nyquist locus.Source: §I · p. 1
  • The characterization enables model-based computation from a state-space realization using frequency-response evaluations, numerical ranges, and planar convex hulls.This avoids repeatedly solving LMIs for gain bounds.Source: §I · p. 1
  • The paper proceeds from notation and SRG definitions to the closure characterization, its SISO Nyquist interpretation, numerical examples, and conclusions.The numerical examples include two filter models and two MIMO plants.Source: §I · p. 1

A. Notation

The notation section fixes the scalar, matrix, and set conventions used in the paper.

A. Notation

  • F denotes either the real field R or the complex field C, and N0 denotes the nonnegative integers.The imaginary unit is j, with j^2 = −1.Source: A. Notation · p. 1
  • For complex quantities the paper uses an overbar for conjugation, ℜ and ℑ for real and imaginary parts, and |z| for modulus; A⊤ and A∗ denote transpose and conjugate transpose.The identity matrix is I_n, or I when its dimension is clear.Source: A. Notation · p. 1
  • For square matrices, ρ(A) denotes spectral radius; for Hermitian P, P ≻ 0 means positive definite and P^1/2 is its positive definite square root.C+ denotes the closed upper half-plane, while cl S and co S denote the closure and convex hull of S.Source: A. Notation · p. 1

B. Signal Spaces

The signal-space setup uses one-sided square-summable sequences and their Hardy-space boundary values. Parseval’s identity relates sequence inner products to frequency-domain integrals, while the Hardy-space restriction excludes some L2 boundary functions.

B. Signal Spaces

  • The discrete-time analysis uses a Hilbert space of one-sided square-summable sequences, whose Hardy-space boundary values lie in H2.The supplied notation identifies H as a Hilbert space with an inner product and norm.Source: B. Signal Spaces · p. 2
  • Parseval’s identity expresses the sequence inner product as a frequency-domain integral and makes the sequence-to-boundary-value map an isometry onto H2.The integral uses normalized measure dθ/(2π).Source: B. Signal Spaces · p. 2
  • H2 is a proper closed subspace of L2 on the unit circle, so not every square-integrable boundary function is an admissible input.This restriction matters when selecting frequency-domain inputs for the analysis.Source: B. Signal Spaces · p. 2

C. Scaled Relative Graphs

This section introduces the operator and matrix scaled relative graphs and specifies symmetry and upper-half-plane conventions for their points.

C. Scaled Relative Graphs

  • For a linear operator T on H, the SRG is introduced as a set of complex numbers, with the definition citing prior work.The supplied passage does not include the set’s defining formula.Source: C. Scaled Relative Graphs · p. 2
  • The paper treats the set as symmetric about the real axis and takes the angle to be zero when Tv = 0.For v ≠ 0, it names z_v as the upper-half-plane SRG point generated by v.Source: C. Scaled Relative Graphs · p. 2
  • For a matrix M in C^(m×m), SRG(M) denotes the SRG of the linear map x ↦ Mx on C^m.The upper-half-plane SRG point is denoted as an element of C+.Source: C. Scaled Relative Graphs · p. 2

D. The Beltrami–Klein Map and Numerical Range

The Beltrami–Klein map sends conjugate pairs to points in the closed unit disk and real numbers to the unit circle, while mapping upper-half-plane hyperbolic geodesics to straight segments. The section also introduces a matrix transform that reduces to the scalar map and notes that numerical ranges are compact and convex.

D. The Beltrami–Klein Map and Numerical Range

III. SRG CLOSURE OF DISCRETE-TIME LTI SYSTEMS

For a Schur-stable, square discrete-time LTI system, the Beltrami–Klein image of the operator SRG closure is characterized through the frequency-response matrices. In the real SISO case, the result connects the closure to a hyperbolic convex hull of the discrete-time Nyquist locus.

III. SRG CLOSURE OF DISCRETE-TIME LTI SYSTEMS

  • The theorem applies to the causal convolution operator induced by a real-coefficient state-space realization with Schur-stable state matrix.The system is square, with m inputs and m outputs, and the realization is extended to allow complex-valued inputs.Source: §III · p. 2
  • Finite-frequency combinations are approximated by constructed inputs whose cross terms vanish as their duration increases.The proof localizes the inputs around distinct frequencies; the resulting SRG points approach the selected convex combination.Source: §III · pp. 3–4
  • Only frequencies from 0 to π are needed to construct the set whose inverse Beltrami–Klein image gives both conjugate halves of the operator SRG closure.Negative frequencies add no new points because the transformed frequency response at −θ is the transpose at θ, and transposed matrices have the same numerical range.Source: §III · p. 4
  • Theorem 1 characterizes the operator SRG closure by convexifying, in the Beltrami–Klein disk, the frequency-wise SRGs of the system.The system is assumed to be Schur stable, and each frequency-wise SRG is contained in the operator SRG closure.Source: §III · pp. 2, 4
  • For real-coefficient SISO systems, the corollary gives a hyperbolic convex-hull description in terms of the discrete-time Nyquist locus.The discrete-time Nyquist locus is the set of frequency-response values over normalized angular frequency.Source: §III · p. 4

A. SISO Filters

The examples compare SRG closures and frequency-wise SRGs for stable low- and high-pass filters, then illustrate the closure characterization on two MIMO plants. The examples show when frequency-wise SRGs need convexification and how the results relate to LMI-based SRGs.

A. SISO Filters

  • Both Schur-stable low- and high-pass filters have nearly identical SRG closures despite opposite frequency responses; their frequency-wise SRGs distinguish their direction of travel.As θ increases from 0 to π, the low-pass SRG moves from near 1 toward the origin, while the high-pass SRG moves oppositely.Source: A. SISO Filters · p. 5
  • For the Schur-stable two-input, two-output plant, the Beltrami–Klein image of the frequency-wise SRG union is convex, so the union equals the closure and LMI-based SRG.The figures show the union, its distribution over frequency, and the LMI-based SRG.Source: A. SISO Filters · p. 5, Figure 3 · p. 5
  • Plant (43) is discretized by zero-order hold (ZOH) with Ts = 0.1 s; its balanced minimal realization has 23 states and D = 0, with ρ(A) = e−0.1 ≈ 0.9048 < 1.Fig. 4 shows the resulting SRGs.Source: A. SISO Filters · p. 6, Figure 4 · p. 6
  • The frequency-wise SRG union shares the LMI-based SRG’s outer boundary but leaves a wedge near 0.6 uncovered, which multi-frequency inputs add to the closure.In the Beltrami–Klein disk, these added points are convex combinations of numerical ranges at different frequencies.Source: A. SISO Filters · p. 6
  • Table I compares computation times using 1000 frequency points for the frequency-domain method and 1000 real-axis points with MOSEK for the LMI-based method.The supplied table passage identifies the computation-time comparison but gives no timing values.Source: A. SISO Filters · p. 6, Table I · p. 6

V. CONCLUSIONS

The conclusion characterizes the closure through frequency-wise SRGs and states its SISO Nyquist interpretation. It also identifies a model-based computation route and future work on frequency-response data.

V. CONCLUSIONS

  • The SRG closure of stable, square discrete-time LTI systems is the Beltrami–Klein convexification of frequency-wise SRGs, reducing in SISO to the hyperbolic convex hull of the Nyquist locus.Source: §V · p. 6
  • With a model available, the frequency-domain computation is preferable to the LMI-based approach and uses frequency-response evaluations, numerical ranges, and planar convex hulls.Source: §V · p. 6
  • Future work will address identifying SRGs from frequency-response data.Source: §V · p. 6
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