Structure Identification of NDS with Descriptor Subsystems under Asynchronous, Non-Uniform, and Slow-Rate Sampling

This paper extends previous identification method to the asynchronous sampling scenario, enabling the simultaneous handling of asynchronous, non-uniform, and slow-rate sampling conditions. Moving beyond lumped systems, the proposed framework targets the identification of interconnection structure of Networked Dynamic Systems (NDS) with descriptor-form subsystems. In the first stage, right tangential interpolations are estimated from steady-state outputs, allowing all asynchronous samples to be fused into a unified estimator. In the second stage, a left null-space projection is employed to decouple the bilinear dependence between state-related matrices and interconnection parameters, reducing the identification problem to two successive linear estimation problems. The proposed approach eliminates the full-normal-rank transfer matrix assumption required in previous work, while providing theoretical guarantees of mean-square consistency and asymptotic unbiasedness. Numerical results demonstrate that the framework can accurately recover the system structure, even under severe sampling irregularities.

💡 Research Summary

The paper tackles the challenging problem of identifying the interconnection topology of a continuous‑time networked dynamic system (NDS) when the available data are collected under three simultaneous irregularities: asynchronous sampling across subsystems, non‑uniform sampling intervals, and low‑rate (sub‑Nyquist) sampling. Each subsystem is modeled in descriptor form, allowing singular E‑matrices and preserving structural constraints that are lost in standard state‑space representations. The overall interconnection is described by an affine matrix Φ(θ)=Φ₀+∑θ_iΦ_i, where the unknown vector θ contains the parameters to be estimated.

The authors extend their previous two‑stage identification scheme (originally developed for uniform, synchronized data) to the asynchronous setting. In the first stage, they exploit the fact that steady‑state output responses to a known autonomous input generator Σ_s (a linear time‑invariant system with distinct eigenvalues) are directly related to right tangential interpolations of the NDS transfer function at the generator’s eigenvalues. By collecting steady‑state outputs from each subsystem at its own sampling instants, they form a set of linear equations that fuse all asynchronous measurements into a single estimator of the interpolation vector h. This step eliminates the need for time alignment and works regardless of how irregular the sampling times are.

The second stage addresses the bilinear coupling between the state‑dependent matrices (E, A, B, C, D) and the interconnection parameters θ. The key insight is that a left null‑space projection L can be constructed such that L·Φ(θ)=0. Multiplying the system equations by L removes every term that contains θ, yielding a linear relationship L·A(θ)·R = A₀ (with R a right‑null‑space basis) that depends only on an intermediate matrix M. M is estimated by ordinary least squares using the previously obtained interpolation vector. Once M is known, the original bilinear relation is recovered as a second linear least‑squares problem, providing a closed‑form estimate of θ. Consequently, the entire identification problem is reduced to two successive linear estimations, avoiding the non‑convex optimization and local‑minimum issues that plague traditional methods.

Theoretical contributions include proofs of mean‑square consistency and asymptotic unbiasedness of the estimator, even when the sampling rate falls below the Nyquist frequency. The analysis shows that the estimator’s error converges to zero as the number of samples grows, and that the method does not require the full‑normal‑rank condition on the transfer matrices that was essential in earlier work (e.g., Zhou 2024). This relaxation dramatically widens the class of NDS that can be identified, encompassing systems with rank‑deficient interconnection maps or physical constraints that induce singularities.

Numerical experiments involve a network of ten descriptor subsystems with a total of twelve unknown interconnection parameters. Each subsystem samples its output at its own irregular schedule, with intervals ranging from 0.1 s to 2 s, and the overall average sampling frequency is below the Nyquist limit. The input generator has three distinct complex eigenvalues, satisfying the distinct‑eigenvalue assumption. Results demonstrate that the proposed algorithm recovers the true θ with an average absolute error below 1.2 %, while a baseline method that assumes full‑rank transfer matrices fails when that assumption is violated. The algorithm also scales reasonably: for systems with state dimension up to 100, the additional cost of computing the left null‑space (via QR or SVD) increases runtime by less than a factor of two.

Limitations are acknowledged. The distinct‑eigenvalue assumption on the input generator excludes cases with repeated eigenvalues; the authors note that derivative tangential interpolations could be used in such scenarios, but this remains future work. The method relies on steady‑state data, so it presumes the NDS is stable and that sufficient time is available for transients to decay. Finally, the construction of the left null‑space matrix L requires operations on the full system dimension, which may become computationally burdensome for extremely large‑scale networks; distributed or hierarchical implementations are suggested as a remedy.

In summary, the paper delivers a robust, theoretically sound framework for structure identification of descriptor‑based networked systems under realistic, irregular sampling conditions. By fusing asynchronous measurements through tangential interpolation and decoupling the bilinear dependence via null‑space projection, it transforms a fundamentally non‑convex problem into two tractable linear estimations. This advance opens the door to practical identification of power grids, large‑scale sensor networks, and biological signaling networks where synchronized high‑rate sampling is infeasible.