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Julius P. J. Krebbekx

Publications and source records attributed to Julius P. J. Krebbekx.

8 recordsLinked to original sources
arXiv2026

Amplitude-Dependent Bode Diagrams via Scaled Relative Graphs

Scaled Relative Graphs (SRGs) provide an intuitive graphical frequency-domain method for the analysis of Nonlinear (NL) systems, generalizing the Nyquist diagram. In this paper, we develop a method for computing $L_2$-gain bounds for Lur'e systems over bounded frequency and amplitude ranges. We do this by restricting the input space of the SRG both in frequency and energy content, and using methods from Sobolev theory. The resulting gain bounds over restricted sets of inputs are less conservative than bounds computed over the entire $L_2$, and yield three-dimensional NL generalization of the Bode diagram, plotting $L_2$-gain as function of both input frequency and energy content. In the zero-energy limit, the Linear Time-Invariant (LTI) Bode diagram is recovered, and at the infinite-energy zero-frequency limit, we recover the $L_2$-gain. The effectiveness of our method is demonstrated on an example that resembles Phase-Locked Loop dynamics.

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arXiv2026

Analysis of Non-Square Nonlinear MIMO Systems using Scaled Relative Graphs

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems. There have been recent efforts to generalize SRG analysis to Multiple-Input Multiple-Output (MIMO) systems. However, these attempts yielded only results for square systems, due to the inherent Hilbert space structure of the SRG. In this paper, we develop an SRG analysis method that accommodates non-square operators. The key element is the embedding of operators to a space of operators acting on a common Hilbert space, while restricting the input space to the original input dimension, to avoid conservatism. We generalize SRG interconnection rules to restricted input spaces and develop stability theorems to guarantee causality, well-posedness and (incremental) $L_2$-gain bounds for the overall interconnection. We show utilization of the proposed theoretical concepts on the analysis of nonlinear systems in a Linear Fractional Representation (LFR) form, which is a rather general class of systems, and the LFR is directly utilizable for control. Moreover, we provide formulas for the computation of MIMO SRGs of stable LTI operators and diagonal and non-square static nonlinear operators. Finally, we demonstrate the advantages of our embedding approach on several examples.

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arXiv2026

Scaled Graph Containment for Feedback Stability: Soft-Hard Equivalence and Conic Regions

Scaled graphs (SGs) offer a geometric framework for feedback stability analysis. This paper develops containment conditions for SGs within multiplier-defined regions, addressing both circular and conic geometries. For circular regions, we show that soft and hard SG containment are equivalent whenever the associated multiplier is positive-negative. This enables hard stability certification from soft computations alone, bypassing both the positive semidefinite storage constraint and the homotopy condition of existing methods. Numerical experiments on systems with up to 300 states demonstrate computational savings of 15-44 % for the circular containment framework. We further characterize which conic regions are hyperbolically convex, a condition our frequency-domain certificate requires, and demonstrate that such regions provide tighter SG bounds than circles whenever the operator SG is nonsymmetric.

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arXiv2025

Computing the Hard Scaled Relative Graph of LTI Systems

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems, where Linear Time-Invariant (LTI) systems are the fundamental building block. To analyze feedback loops with unstable LTI components, the hard SRG is required, since it aptly captures the input/output behavior on the extended $L_2$ space. In this paper, we develop a systematic computational method to exactly compute the hard SRG of LTI systems, which may be unstable and contain integrators. We also study its connection to the Nyquist criterion, including the multivariable case, and demonstrate our method on several examples.

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arXiv2025

Nonlinear Bandwidth and Bode Diagrams based on Scaled Relative Graphs

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of Nonlinear (NL) systems. In this paper, we restrict the SRG to particular input spaces to compute frequency-dependent incremental gain bounds for nonlinear systems. This leads to a NL generalization of the Bode diagram, where the sinusoidal, harmonic, and subharmonic inputs are considered separately. When applied to the analysis of the NL loop transfer and sensitivity, we define a notion of bandwidth for both the open-loop and closed-loop, compatible with the Linear Time-Invariant (LTI) definitions. We illustrate the power of our method on the analysis of a DC motor with a parasitic nonlinearity and verify our results in simulations.

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arXiv2025

Scaled Relative Graph Analysis of General Interconnections of SISO Nonlinear Systems

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems. However, we show that the current SRG analysis suffers from a pitfall that limits its applicability in analyzing practical nonlinear systems. We overcome this pitfall by introducing a novel reformulation of the SRG of a linear time-invariant operator and combining the SRG with the Nyquist criterion. The result is a theorem that can be used to assess stability and $L_2$-gain performance for general interconnections of nonlinear dynamic systems. We provide practical calculation results for canonical interconnections and apply our result to Lur'e systems to obtain a generalization of the celebrated circle criterion, which deals with broader class of nonlinearities, and we derive (incremental) $L_2$-gain performance bounds. We illustrate the power of the new approach on the analysis of several examples.

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arXiv2025

Reset Controller Analysis and Design for Unstable Linear Plants using Scaled Relative Graphs

In this technical communique, we develop a graphical design procedure for reset controllers for unstable LTI plants based on recent developments on Scaled Relative Graph analysis, yielding an $L_2$-gain performance bound. The stabilizing controller consists of a second order reset element in parallel with a proportional gain. The proposed method goes beyond existing approaches that are limited to stable systems only, providing a well-applicable approach to design problems in practice where the plant is unstable.

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arXiv2025

Scaled Relative Graph Analysis of Lur'e Systems and the Generalized Circle Criterion

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems. However, we show that the current SRG analysis suffers from a pitfall that limit its applicability in analyzing practical nonlinear systems. We overcome this pitfall by modifying the SRG of a linear time invariant operator, combining the SRG with the Nyquist criterion, and apply our result to Lur'e systems. We thereby obtain a generalization of the celebrated circle criterion, which deals with a broader class of nonlinearities, and provides (incremental) $L_2$-gain performance bounds.

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