Scaled Relative Graphs and Dynamic Integral Quadratic Constraints: Connections and Computations for Nonlinear Systems

Abstract: Scaled relative graphs (SRGs) enable graphical analysis and design of nonlinear systems. In this paper, we present a systematic approach for computing both soft and hard SRGs of nonlinear systems using dynamic integral quadratic constraints (IQCs). These constraints are exploited via application of the S-procedure to compute tractable SRG overbounds. In particular, we show that the multipliers associated with the IQCs define regions in the complex plane. Soft SRG computations are formulated through frequency-domain conditions, while hard SRGs are obtained via hard factorizations of multipliers and linear matrix inequalities. The overbounds are used to derive an SRG-based feedback stability result for Lur'e-type systems, providing a new graphical interpretation of classical IQC stability results with dynamic multipliers.

• The paper establishes a novel framework linking SRGs with dynamic IQCs to compute overbounding regions for nonlinear systems.
• The paper provides LMI-based methodologies for both soft and hard SRGs, integrating static and dynamic multipliers for tighter stability bounds.
• The paper demonstrates SRG-based feedback stability tests for Lur’e systems, unifying classical and IQC-based robust control criteria.

Scaled Relative Graphs and Dynamic Integral Quadratic Constraints: Theory and Computation for Nonlinear Systems

Introduction

This paper delivers a systematic approach for the graphical analysis and robust design of nonlinear dynamical systems by introducing methods to compute overbounding regions for Scaled Relative Graphs (SRGs) utilizing dynamic Integral Quadratic Constraints (IQCs). The work addresses key open computational questions for both “soft” and “hard” SRGs in the nonlinear setting and establishes new connections between dynamic IQCs, scaled graph analysis, and classical feedback stability theorems, especially in Lur'e systems.

Scaled Relative Graphs and Their Role

SRGs generalize the graphical analysis of input-output gains and phase relationships for nonlinear and LTI systems. Unlike traditional Nyquist or Bode plots, SRGs naturally accommodate incremental and phase-type properties for wider system classes, incorporating both bounded (incrementally stable) and unbounded system behavior. Definitions are given for both soft SRGs (for systems on $\mathcal{L}_2^n$) and hard SRGs (for the extended space $\mathcal{L}_{2e}^n$). The construction translates input-output relationships into subsets of the complex plane, which for SISO LTI systems recovers Nyquist diagrams as a special case.

Integral Quadratic Constraints and System Representation

IQCs are leveraged for their expressiveness in representing input-output behavior of nonlinear systems in terms amenable to convex optimization. Both soft and hard incremental IQCs are considered, with multipliers $(M,N)$ potentially dynamic (frequency-dependent) and not restricted to static sector-type multipliers. Boundedness and causality are central, and the framework holds for a range of nonlinearity classes, including those with memory or more complex dynamic structure.

A central connection is established: for static IQCs, the set of complex $z$ associated with SRG membership aligns exactly with solutions to a pointwise quadratic matrix inequality (QMI) parameterized by $M$. This forms the geometric region $\mathcal{S}(\Pi)$.

SRG Overbounding via Dynamic IQCs

Soft SRGs

A novel approach is presented for overbounding the SRG using dynamic IQCs, with the key result formulated as a frequency-domain Linear Matrix Inequality (LMI). Given that a system satisfies a soft incremental IQC, one can search for a matrix $\Pi$ such that a parameterized matrix inequality holds for all frequencies. The set $\mathcal{S}(\Pi)$ then becomes a guaranteed overbound region for the SRG of the original system.

Hard SRGs

For the hard SRG, the inclusion of state-space realizations for dynamic multipliers enables construction of time-domain LMIs that can certify overbounding regions in $\mathcal{S}(\Pi)$ for extended (potentially unbounded) system classes. This computation requires hard factorizations of multipliers (i.e., realizable, stable LTI filter representations).

Refinement and Geometry

A constructive parameterization for $\Pi$ is adopted, covering circles, half-planes, and their complements in the complex plane. By solving for the smallest admissible SRG overbounds via LMI feasibility/optimization over the free parameters, the tightest complex region containing all admissible SRG values is found. Multiple IQCs or their convex combinations can be composed to further shrink these overbounds, particularly effective when combining static and frequency-dependent multipliers.

A worked SISO example demonstrates the utility of incorporating dynamic IQCs (not simply static sector/gain constraints): adding a Zames-Falb-type dynamic multiplier yields a strictly tighter SRG overbound than achievable with static multipliers alone.

Feedback Stability for Lur'e Systems and SRG–IQC Graphical Synthesis

A main theoretical development is an SRG-based stability theorem for Lur’e interconnections (feedback of an LTI $\mathcal{L}_{2e}^n$0 with a nonlinear $\mathcal{L}_{2e}^n$1). If the frequency-wise SRG of $\mathcal{L}_{2e}^n$2 is uniformly separated from a SRG overbound for $\mathcal{L}_{2e}^n$3 constructed via dynamic IQCs, then the closed-loop incremental gain is finite. Unlike classical IQC stability tests, this geometric test requires only the Nyquist curve of the LTI without modifications for dynamic multipliers—a substantial conceptual shift for robust nonlinear control analysis.

The approach recovers, in the static case, classical graphical stability tests (passivity, small-gain, circle criterion), thus unifying classical and IQC-based geometric conditions under the SRG framework.

Structural Limitations and Specific Cases

The paper rigorously demonstrates the structural limitations of dynamic multipliers for certain nonlinearity classes. For example, for static sector-bounded nonlinearities and Popov-type multipliers, the attainable SRG overbound coincides precisely with that generated by the static IQC alone; additional dynamic multipliers do not yield tighter bounds. This is not a limitation of the computational approach but is a consequence of intrinsic system-theoretic properties. For more restrictive classes (e.g., monotone operators, Zames-Falb multipliers), dynamic multipliers can significantly enhance overbounding regions.

Implications and Future Directions

Practically, these results enable the systematic design of robust controllers for nonlinear systems by providing tractable, certificate-based procedures for verifying incremental gain/phase margins and feedback stability using frequency-domain or state-space conditions. Theoretically, the SRG–IQC connection generalizes the reach of graphical methods in nonlinear robust control, bridging the gap between operator-theoretic IQC results and tractable graphical tests.

Potential future developments include:

• Extension to classes of non-incremental properties;
• Automated SRG construction for high-dimensional/multi-IQC systems;
• Exploration of SRG-fullness for classes characterized by dynamic IQCs;
• Incorporation into convex optimization-based robust controller synthesis pipelines.

Conclusion

The paper contributes to nonlinear control theory by formalizing overbounding techniques for scaled relative graphs using dynamic IQCs. Frequency-domain and LMI-based computations are formulated for both soft and hard SRGs, enabling tractable graphical analysis and certifiable stability for broad nonlinear dynamical classes, notably in feedback settings. The SRG–IQC connection provides both a practical computational pathway and a novel geometric perspective on robust stability in nonlinear feedback systems, revealing both the power and inherent limits of dynamic multiplier-based approaches (2604.01873).