Identifiability and Parameter Estimation Difficulty of Structured Systems with Matrix Fraction Description Based on Finite Frequency Responses

Background

In scientific research and engineering applications, parameter identification is a core task for understanding and controlling complex systems. Whether in power systems, mechanical systems, or chemical reaction kinetics models, accurate parameter identification serves as the foundation for optimizing system behavior, reducing errors, and enhancing control performance. However, as systems become more complex, traditional parameter identification methods gradually lose effectiveness when dealing with large-scale systems, especially under highly nonlinear identification problems with massive data volumes. Therefore, new theories and computational methods need to be developed to address the practical challenges in parameter identification.

In recent years, the study of parameter estimation difficulty (sloppiness) has garnered significant attention. Sloppiness refers to situations during parameter identification where some parameters induce only minor changes in system output, making them difficult to identify from experimental data. This issue is particularly prominent in multivariable and nonlinear systems. However, existing methods often rely on the Fisher Information Matrix (FIM). While theoretically significant, these methods are computationally complex in practice and may lead to misleading experimental designs.

In this context, the study focuses on structured systems described by Matrix Fraction Description (MFD). By analyzing the identifiability and estimation difficulty of such systems from limited frequency responses, this research provides new theoretical insights and computational methods for analyzing and synthesizing large-scale systems.

Source and Author Information

The research paper titled “Identifiability and sloppiness of structured systems with a matrix fraction description using finite frequency responses” was authored by Yunxiang Ma and Tong Zhou from the Department of Automation at Tsinghua University. The paper was published in February 2025 in the journal Science China Information Sciences (Vol. 68, Issue 2, DOI: 10.1007/s11432-024-4135-9). Tong Zhou is the corresponding author, specializing in reliability analysis of complex systems, identification, and control of networked dynamic systems.

Research Workflow and Methodology

This study focuses on linear time-invariant systems with MFD to address two main issues: global identifiability of parameters and quantification of parameter estimation difficulty (sloppiness metrics). The research proceeds through several steps, systematically exploring problem formulation, assumption validation, identifiability analysis using frequency responses, and computation of sloppiness.

1. Problem Definition and Mathematical Modeling

The research begins with a mathematical description of linear time-invariant systems and their MFD: - System Model: Based on input (u(\lambda)) and output (y(\lambda)), the fractional model of the system transfer function is described as: [ D(\lambda, \theta) Y(\lambda) = N(\lambda, \theta) U(\lambda) ] Here, the numerator matrix (N(\lambda, \theta)) and denominator matrix (D(\lambda, \theta)) are affine functions of the parameters (\theta = (\theta_1, \theta_2, \dots, \theta_n)^T).

• Frequency Response Analysis: The system’s frequency response model (G(j\omega, \theta)) is treated as a critical tool for connecting input-output relationships with unknown parameter settings.

Through model formulation, the authors pointed out that global identifiability of parameters must be verified based on a finite number of frequency points. They further proposed using Singular Value Decomposition (SVD)-based methods to resolve the identifiability issues.

2. Verification of Invertibility Assumption

In parameter identification problems, it is assumed that the denominator matrix (D(\lambda, \theta)) is invertible within the parameter range. However, since the specific parameter values are unknown, straightforward numerical verification is infeasible. To address this, the paper provides a symbolic validation scheme using graph theory and matroid theory: - Core Idea: By employing rank-one decomposition of matrices, the invertibility issue is transformed into an independent matching problem on a bipartite graph. - Simplified Condition: The authors proposed that when the rank of matrix (D_i(\lambda)) is 1, the invertibility of the denominator matrix almost everywhere can be deduced by verifying the linear independence of certain known matrices and vectors.

3. Parameter Global Identifiability via Frequency Responses

The paper proposed a sufficient and necessary condition for verifying global identifiability of the system: - Theoretical Highlight: If for any two different parameter vectors (\theta) and (\tilde{\theta}), the output frequency responses (G(j\omega, \theta)) are not simultaneously equal, the parameters are globally identifiable. - Verification Method: The paper constructs a numerical matrix (\Pi(\omega_i|\omega_i, \theta)) and checks its full column rank. By recursively verifying system identifiability using limited frequency points, the method is computationally attractive, particularly for large-scale systems.

4. Quantification of Parameter Estimation Difficulty

The paper introduces metrics for absolute sloppiness and relative sloppiness: - Innovative Approach: By parameterizing solutions to a linear matrix equation and performing matrix singular value decomposition, the authors derive explicit formulas for calculating identifiability difficulty. The method is robust to slight deviations in input-output frequency responses and can provide practical measures for parameter identifiability difficulty.

• Methodological Improvement: Unlike traditional FIM-based methods, the definitions in this paper offer a clearer description of how system parameter variations influence frequency responses.

Key Research Findings

Global Identifiability of the System

The study demonstrated that global identifiability of system parameters can be established by verifying the full column rank of the numerical matrix (\Pi). Additionally, the authors proved that only a number of frequency points equal to the system order is sufficient to achieve identifiability, significantly reducing computational efforts.

Explicit Formulas for Parameter Estimation Difficulty

Compared to FIM-based approaches, the absolute and relative sloppiness metrics proposed in the paper significantly improve interpretability in parameter estimation: - Absolute Sloppiness: Quantifies the maximum ratio of parameter variations to small deviations in system frequency response. - Relative Sloppiness: Reflects the ratio of parameter variations in successive extreme cases that result in the same magnitude of frequency response deviation.

The comparison shows that the new method more effectively highlights the importance of mid-range and high-frequency responses during parameter estimation.

Importance of Frequency Point Selection

The study emphasized the critical importance of selecting the number and distribution of frequency points in reducing sloppiness. Numerical experiments revealed that mid-frequency data provides more information about system parameters than low- or high-frequency data, making it the optimal range for parameter estimation.

Conclusion and Significance

The scientific value of this study lies in providing methods for determining identifiability and quantifying parameter estimation difficulty for structured systems using finite frequency responses: 1. Theoretical Contribution: The paper is the first to offer a mathematical analysis of parameter estimation difficulty for systems with matrix fraction descriptions, extending its definitions to broader engineering applications. 2. Engineering Impact: The methods discussed have significant potential in modeling and analyzing electrical, mechanical, and neural network systems, enhancing computational efficiency in large-scale system analysis.

This research offers precise and cost-effective solutions to the parameter identification problem, holding significant implications for the design and optimization of modern complex engineering systems.