Stability-Preserving Model Reduction of Networked Lur'e Systems

TL;DR

This work introduces a clustering-based projection method to reduce networked Lur'e systems while preserving the network's diffusive structure and absolute stability. By formulating reduced-order matrices as $\hat{A}_L = \Pi^{\dagger} A_L \Pi$, $\hat{B} = \Pi^{\dagger} B$, and $\hat{\Phi}(\cdot) = \Pi^{\dagger} \Phi(\cdot)$ with $\hat{x} = \Pi z$, the reduced model remains a network with a diagonal-plus-Laplacian form. The authors derive an explicit $\mathcal{H}_{\infty}$-based error bound that depends on a clustering-dependent quantity $\gamma_H$, and show that this bound can be minimized by an appropriate choice of the clustering matrix $\Pi$ (with $\Pi = I_N$ yielding zero error). A numerical example on a 100-node scale-free network demonstrates stability, captures inter-cluster agreement, and verifies the theoretical bound, illustrating practical viability for scalable nonlinear network reduction. The results lay groundwork for preserving key properties such as absolute stability in nonlinear network reductions and suggest future work on extending to non-scalar subsystems and other properties like synchronization and passivity.

Abstract

This paper proposes a model reduction approach for simplifying the interconnection topology of Lur'e network systems. A class of reduced-order models are generated by the projection framework based on graph clustering, which not only preserve the network structure but also ensure absolute stability. Furthermore, we provide an upper bound on the input-output approximation error between the original and reduced-order Lur'e network systems, which is expressed as a function of the characteristic matrix of graph clustering. Finally, the results are illustrated via a numerical example.

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Theorems & Definitions (7)

• Lemma 1
• proof
• Theorem 1
• proof
• Lemma 2
• Theorem 2
• Remark 1