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Parameter Recovery from Tangential Interpolations for Systems with an LFT Structure

Tong Zhou, Yubing Li

TL;DR

The paper tackles the problem of recovering parameters $\theta$ in LFT-parameterized LTI systems from right tangential interpolations of the transfer function matrix, encoded as RTIM data. It derives a necessary and sufficient recoverability condition expressed as a vector inequality, with special cases reducing to constant-matrix FRR/FCR criteria, and proposes a practical recovery method based on a rank-constrained linear equation and a nuclear-norm relaxation implemented via an iterative algorithm. The work also establishes robustness results showing how RTIM estimation errors propagate to parameter estimates under certain rank conditions, and demonstrates the approach with a numerical SISO example where including derivatives of the transfer function improves the accuracy of natural frequency and damping recovery. Overall, the framework connects RTIM-based system identification with structured LFT models, offering actionable criteria and optimization-based procedures for reliable parameter recovery in practical settings.

Abstract

This paper investigates how to recover parameters of a linear time invariant system from values and derivatives of its transfer function matrix, along several particular directions at a prescribed set of points in the complex plane, in which system matrices depend on these parameters through a linear fractional transformation. A necessary and sufficient condition is derived for a unique determination of these system parameters, which is expressed by a vector inequality. Under some particular situations, this condition reduces to a full column rank requirement on a constant matrix. Moreover, a method is given to recover system parameters from these values and derivatives, which is expressed by a vector linear equation with some rank constraints, for which various methods exist for finding its solutions. Robustness of the suggested recovery method is also clarified. A numerical example is given to illustrate characteristics of the suggested method, as well as effectiveness of derivative information introduction in parameter recovery, in which natural frequency and damping ratio are to be recovered for a transfer function.

Parameter Recovery from Tangential Interpolations for Systems with an LFT Structure

TL;DR

The paper tackles the problem of recovering parameters in LFT-parameterized LTI systems from right tangential interpolations of the transfer function matrix, encoded as RTIM data. It derives a necessary and sufficient recoverability condition expressed as a vector inequality, with special cases reducing to constant-matrix FRR/FCR criteria, and proposes a practical recovery method based on a rank-constrained linear equation and a nuclear-norm relaxation implemented via an iterative algorithm. The work also establishes robustness results showing how RTIM estimation errors propagate to parameter estimates under certain rank conditions, and demonstrates the approach with a numerical SISO example where including derivatives of the transfer function improves the accuracy of natural frequency and damping recovery. Overall, the framework connects RTIM-based system identification with structured LFT models, offering actionable criteria and optimization-based procedures for reliable parameter recovery in practical settings.

Abstract

This paper investigates how to recover parameters of a linear time invariant system from values and derivatives of its transfer function matrix, along several particular directions at a prescribed set of points in the complex plane, in which system matrices depend on these parameters through a linear fractional transformation. A necessary and sufficient condition is derived for a unique determination of these system parameters, which is expressed by a vector inequality. Under some particular situations, this condition reduces to a full column rank requirement on a constant matrix. Moreover, a method is given to recover system parameters from these values and derivatives, which is expressed by a vector linear equation with some rank constraints, for which various methods exist for finding its solutions. Robustness of the suggested recovery method is also clarified. A numerical example is given to illustrate characteristics of the suggested method, as well as effectiveness of derivative information introduction in parameter recovery, in which natural frequency and damping ratio are to be recovered for a transfer function.

Paper Structure

This paper contains 7 sections, 11 theorems, 104 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Recoverability of the Parameter Vector
  4. Parameter Recovery from RTIM
  5. Robustness of the Parameter Recovery Procedure
  6. Numerical Example
  7. Concluding Remarks

Key Result

Lemma 1

Assume that $\Xi \in \mathbb{R}^{m_{\xi}\times m_{\xi}}$ and $\Pi \in \mathbb{R}^{m_{u}\times m_{\xi}}$ be two prescribed matrices. Moreover, assume that $\Gamma \in \mathbb{R}^{m_{y}\times m_{\xi}}$ is a prescribed RTIM. Then for each positive integer $m_{x}$ satisfying $m_{x} \geq m_{\xi}$, all t in which $G \in \mathbb{R}^{m_{\xi}\times m_{u}}$, $Z \in \mathbb{R}^{m_{\xi}\times (m_{x}-m_{\xi})

Figures (3)

  • Figure 1: Frequency-magnitude characteristics of the plant.
  • Figure 2: Typical cost function curves in parameter recovery.
  • Figure 3: Relative parameter recovery errors and their ratio.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • ...and 3 more