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

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.