SRG-fullness for dynamic IQCs

Establish the converse (SRG-fullness) for dynamic incremental IQCs: Determine whether, given multipliers (M,N) and a real symmetric 2×2 matrix Π that satisfies the frequency-domain inequality [I N(jω)]* diag(Π ⊗ I, −τ M) [I N(jω)] ≽ 0 for all ω ∈ ℝ and some τ > 0 (soft case) or the corresponding state-space LMI in Theorem 2 (hard case), the inclusion of the scaled relative graph SRG(H) (respectively, the hard scaled relative graph SRG_e(H)) in the region S(Π) is sufficient to imply that the causal system H satisfies the corresponding soft (respectively, hard) incremental IQC with multipliers (M,N).

Background

Lemma 1 shows an exact equivalence for static IQCs (SRG-fullness): a causal system satisfies a static incremental IQC with N = I and M = Π ⊗ I if and only if its (soft or hard) SRG is contained in the region S(Π). This provides a complete graphical characterization for static multipliers.

Theorems 1 and 2 extend to dynamic IQCs and show only one direction: if a system satisfies a dynamic incremental IQC with multipliers (M,N) and there exists Π satisfying the frequency-domain inequality (soft case) or the LMI (hard case), then its SRG is contained in S(Π). However, unlike the static case, it is unknown whether the converse holds for dynamic multipliers, i.e., whether SRG containment implies satisfaction of the corresponding dynamic IQC.

Resolving this would clarify whether SRG-based regions obtained via dynamic multipliers are merely overbounds or provide exact class characterizations, thereby impacting the tightness and interpretability of graphical analysis for nonlinear systems via SRGs.

References

In contrast to Lemma~\ref{lem:SRGIQC}, Theorems~\ref{th:sSRG1} and~\ref{th:hSRG1} do not establish SRG-fullness; they only show that the class of systems characterized by a dynamic IQC has its SRG contained in $\mathcal{S}(\Pi)$, but whether the converse is true remains open.

— Scaled Relative Graphs and Dynamic Integral Quadratic Constraints: Connections and Computations for Nonlinear Systems  (2604.01873 - Groot et al., 2 Apr 2026) in Section 3.2 (Overbounding hard SRGs), bullet 4