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

TL;DR

This paper studies the identifiability and sloppiness of descriptor systems with an LFT structure using frequency response samples, providing new metrics, conditions, and algorithms for parameter determination and system analysis.

Contribution

It introduces a rank-based identifiability condition, new sloppiness metrics, and an algorithm for selecting frequencies to uniquely identify parameters in LFT-structured descriptor systems.

Findings

Derived a necessary and sufficient rank condition for identifiability.

Proposed algorithms for frequency selection to ensure parameter identifiability.

Provided explicit formulas for absolute and relative sloppiness metrics.

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.