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Frequency Domain Identifiability and Sloppiness of Descriptor Systems with an LFT Structure

Tong Zhou

TL;DR

The paper addresses identifiability and sloppiness of parameters in descriptor systems whose matrices depend on the parameter vector $\theta$ through a linear fractional transformation, using finite frequency responses $H(j\omega)$. It derives a rank-based necessary and sufficient condition for global identifiability with finitely many frequencies and provides a recursive algorithm to select informative frequency samples when the condition is not immediately satisfied. It also introduces absolute and relative sloppiness metrics and an ellipsoid-based approximation of the feasible parameter set, enabling quantitative assessment of parameter directions that are hard to identify from frequency data. Collectively, these results support scalable identifiability analysis and robust parameter estimation in large-scale networked dynamic systems.

Abstract

Identifiability and sloppiness are investigated in this paper for the parameters of a descriptor system based on its frequency response samples. Two metrics are suggested respectively for measuring absolute and relative sloppiness of the parameter vector at a prescribed value. In this descriptor system, system matrices are assumed to depend on its parameters through a linear fractional transformation (LFT). When an associated transfer function matrix (TFM) is of full normal row rank, a matrix rank based necessary and sufficient condition is derived for parameter identifiability with a set of finitely many frequency responses. This condition can be verified recursively which is computationally quite appealing, especially when the system is of a large scale. From this condition, an algorithm is suggested to find a set of frequencies with which the frequency responses of the system are capable to uniquely determine its parameters. An ellipsoid approximation is given for the set consisting of all the parameter values with which the associated descriptor system has a frequency response that deviates within a prescribed distance, from that corresponding to a globally identifiable parameter vector value. Explicit formulas are also derived for the suggested absolute and relative sloppiness metrics.

Frequency Domain Identifiability and Sloppiness of Descriptor Systems with an LFT Structure

TL;DR

The paper addresses identifiability and sloppiness of parameters in descriptor systems whose matrices depend on the parameter vector through a linear fractional transformation, using finite frequency responses . It derives a rank-based necessary and sufficient condition for global identifiability with finitely many frequencies and provides a recursive algorithm to select informative frequency samples when the condition is not immediately satisfied. It also introduces absolute and relative sloppiness metrics and an ellipsoid-based approximation of the feasible parameter set, enabling quantitative assessment of parameter directions that are hard to identify from frequency data. Collectively, these results support scalable identifiability analysis and robust parameter estimation in large-scale networked dynamic systems.

Abstract

Identifiability and sloppiness are investigated in this paper for the parameters of a descriptor system based on its frequency response samples. Two metrics are suggested respectively for measuring absolute and relative sloppiness of the parameter vector at a prescribed value. In this descriptor system, system matrices are assumed to depend on its parameters through a linear fractional transformation (LFT). When an associated transfer function matrix (TFM) is of full normal row rank, a matrix rank based necessary and sufficient condition is derived for parameter identifiability with a set of finitely many frequency responses. This condition can be verified recursively which is computationally quite appealing, especially when the system is of a large scale. From this condition, an algorithm is suggested to find a set of frequencies with which the frequency responses of the system are capable to uniquely determine its parameters. An ellipsoid approximation is given for the set consisting of all the parameter values with which the associated descriptor system has a frequency response that deviates within a prescribed distance, from that corresponding to a globally identifiable parameter vector value. Explicit formulas are also derived for the suggested absolute and relative sloppiness metrics.
Paper Structure (6 sections, 13 theorems, 133 equations)

This paper contains 6 sections, 13 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Parameter Identifiability
  4. Determination of Desirable Frequency Samples
  5. Existence of Sloppy Parameter Combinations
  6. Concluding Remarks

Key Result

Lemma 1

Assume that the descriptor system $\rm\bf\Sigma$ satisfies Assumptions A1) and A2). Then it is globally identifiable at a specific parameter vector value $\theta \in {\rm\bf\Theta}$, if and only if for every other element $\widetilde{\theta}$ of the set ${\rm\bf\Theta}$, the associated TFMs satisfy

Theorems & Definitions (15)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Theorem 1
  • Lemma 7
  • ...and 5 more