[Paper Review] Low-Rank and Low-Order Decompositions for Local System Identification

Nikolai Matni, Anders Rantzer|arXiv (Cornell University)|Mar 27, 2014

Sparse and Compressive Sensing Techniques12 references15 citations

TL;DR

This paper proposes a nuclear norm minimization approach to separate local dynamics from global interconnections in large-scale distributed systems using low-rank and low-order decomposition. By exploiting the structural difference—local dynamics are low-order but full-rank, while global dynamics are high-order but low-rank—it enables accurate local system identification even when interconnection signals are hidden, achieving near-noise-level estimation error in synthetic experiments.

ABSTRACT

As distributed systems increase in size, the need for scalable algorithms becomes more and more important. We argue that in the context of system identification, an essential building block of any scalable algorithm is the ability to estimate local dynamics within a large interconnected system. We show that in what we term the "full interconnection measurement" setting, this task is easily solved using existing system identification methods. We also propose a promising heuristic for the "hidden interconnection measurement" case, in which contributions to local measurements from both local and global dynamics need to be separated. Inspired by the machine learning literature, and in particular by convex approaches to rank minimization and matrix decomposition, we exploit the fact that the transfer function of the local dynamics is low-order, but full-rank, while the transfer function of the global dynamics is high-order, but low-rank, to formulate this separation task as a nuclear norm minimization.

Motivation & Objective

• Address the challenge of scalable system identification in large-scale distributed systems where global interconnections obscure local dynamics.
• Develop a method to identify local subsystem dynamics when interconnection signals are not fully measurable, a common limitation in real-world systems.
• Leverage structural differences in system dynamics—low-order but full-rank local components versus high-order but low-rank global components—to enable separation via convex optimization.
• Provide a scalable, local algorithm that does not require communication with neighboring subsystems or prior knowledge of system structure.
• Enable accurate local model estimation even in the presence of hidden interconnection signals, with potential for integration with future global interconnection reconstruction methods.

Proposed method

• Formulate the local system identification problem as a matrix decomposition task, separating the impulse response into local (low-order, full-rank) and global (high-order, low-rank) components.
• Use nuclear norm minimization to promote low-rank structure in the global dynamics component, exploiting convex relaxation for rank minimization.
• For the hidden interconnection case, model the transfer function from local inputs and interconnection signals to local outputs as a sum of two components with distinct structural properties.
• Solve a convex optimization program (e.g., (27) and (18)) that enforces low-rank structure on the global component while preserving the low-order nature of the local component.
• Apply Frobenius norm constraints to control estimation error and ensure robustness to measurement noise.
• Use singular value decomposition (SVD) of the Hankel matrix to verify the true order of the local system and the dimension of the hidden interconnection subspace.

Experimental results

Research questions

• RQ1 Can local system dynamics be accurately identified when interconnection signals between subsystems are not directly measurable?
• RQ2 Can the structural difference between low-order/full-rank local dynamics and high-order/low-rank global dynamics be exploited to separate their contributions using convex optimization?
• RQ3 To what extent does nuclear norm minimization enable accurate local system identification in the presence of hidden interconnection signals and measurement noise?
• RQ4 Can the method correctly identify the true order of the local system and the dimension of the hidden interconnection subspace?
• RQ5 How does the performance of the method compare to classical system identification when interconnection signals are fully observed?

Key findings

• In the full interconnection measurement setting, the method achieves an estimation error of ‖Ŝⁱ − Sⁱ‖_F = 0.008, which is within the noise level relative to ‖S‖_F = 2.871.
• In the hidden interconnection case, the estimation error is ‖Ŝⁱ − Sⁱ‖_F = 0.093, which is above the noise level but still a reasonable estimate of the local dynamics.
• The top three singular values of the Hankel matrix 𝒫(H(Sⁱ)) are at least an order of magnitude larger than the remaining singular values across a broad range of parameters, indicating correct rank detection.
• The rank of each H(e^{jωₖ}) term is correctly identified as 1 for all δₕ ∈ [0, 0.15], confirming accurate identification of the hidden interconnection subspace dimension.
• Numerical experiments suggest the method is well-suited for identifying the true order of local dynamics and the dimension of the hidden interconnection subspace, enabling subsequent parametric refinement.
• The approach successfully separates local and global dynamics even when interconnection signals are not measured, demonstrating feasibility for scalable distributed system identification.