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Interaction Identification of a Heterogeneous NDS with Quadratic-Bilinear Subsystems

Tong Zhou, Yubing Li

TL;DR

The paper addresses time-domain identification of interaction parameters in a heterogeneous NDS where each subsystem is modeled as a continuous-time QBTI system. It derives explicit time-domain decompositions into transient and steady-state components and shows that steady-state behavior relates linearly to the NDS TFMs and generalized TFMs via LFTs, enabling tangential interpolation-based estimation of subsystem interaction parameters. A two-stage estimation framework is proposed: a nonparametric stage to recover tangential conditions from multi-sine experiments, followed by a parametric stage that identifies the SIP vector through least-squares data fitting within the LFT structure. A numerical circuit example demonstrates identifiability and convergence behavior, and the results offer practical guidance on probing strategies, sampling, and weighting for accurate SIP estimation in complex QB-networked systems.

Abstract

This paper attacks time-domain identification for interaction parameters of a heterogeneous networked dynamic system (NDS), with each of its subsystems being described by a continuous-time descriptor quadratic-bilinear time-invariant (QBTI) model. The obtained results can also be applied to parameter estimations for a lumped QBTI system. No restrictions are put on the sampling rate. Explicit formulas are derived respectively for the transient and steady-state responses of the NDS, provided that the probing signal is generated by a linear time invariant (LTI) system. Some relations have been derived between the NDS steady-state response and its frequency domain input-output mappings. These relations reveal that the value of some NDS associated generalized TFMs can in principle be estimated at almost any interested point of the imaginary axis from time-domain input-output experimental data, as well as its derivatives and a right tangential interpolation along an arbitrary direction. Based on these relations, an estimation algorithm is suggested respectively for the parameters of the NDS and the values of these generalized TFMs. A numerical example is included to illustrate characteristics of the suggested estimation algorithms.

Interaction Identification of a Heterogeneous NDS with Quadratic-Bilinear Subsystems

TL;DR

The paper addresses time-domain identification of interaction parameters in a heterogeneous NDS where each subsystem is modeled as a continuous-time QBTI system. It derives explicit time-domain decompositions into transient and steady-state components and shows that steady-state behavior relates linearly to the NDS TFMs and generalized TFMs via LFTs, enabling tangential interpolation-based estimation of subsystem interaction parameters. A two-stage estimation framework is proposed: a nonparametric stage to recover tangential conditions from multi-sine experiments, followed by a parametric stage that identifies the SIP vector through least-squares data fitting within the LFT structure. A numerical circuit example demonstrates identifiability and convergence behavior, and the results offer practical guidance on probing strategies, sampling, and weighting for accurate SIP estimation in complex QB-networked systems.

Abstract

This paper attacks time-domain identification for interaction parameters of a heterogeneous networked dynamic system (NDS), with each of its subsystems being described by a continuous-time descriptor quadratic-bilinear time-invariant (QBTI) model. The obtained results can also be applied to parameter estimations for a lumped QBTI system. No restrictions are put on the sampling rate. Explicit formulas are derived respectively for the transient and steady-state responses of the NDS, provided that the probing signal is generated by a linear time invariant (LTI) system. Some relations have been derived between the NDS steady-state response and its frequency domain input-output mappings. These relations reveal that the value of some NDS associated generalized TFMs can in principle be estimated at almost any interested point of the imaginary axis from time-domain input-output experimental data, as well as its derivatives and a right tangential interpolation along an arbitrary direction. Based on these relations, an estimation algorithm is suggested respectively for the parameters of the NDS and the values of these generalized TFMs. A numerical example is included to illustrate characteristics of the suggested estimation algorithms.

Paper Structure

This paper contains 6 sections, 7 theorems, 68 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. System output decomposition
  4. Nonparametric and Parametric Estimation with a Multi-sine Probing Signal
  5. A Numerical Example
  6. Conclusions

Key Result

Lemma 1

For each admissible pair of initial conditions and input signal, the solution to the following QBTI system can be equivalently written as and for each $k \geq 2$, $x_{k}(t)$ is the solution to the following LTI system

Figures (5)

  • Figure 1: Structure of the Circuit and Input-output Properties of a Diode.
  • Figure 2: Nonparametric Estimates with $\omega_{0} = 4.5 rad/s$. $-\!\!-$: actual value and its estimates for the 1st element; $-\:-$: actual value and its estimates for the 2nd element. $\nabla$: estimate with $\sigma=0.01$; $\star$: estimate with $\sigma=0.02$; $\bf\Diamond$: estimate with $\sigma=0.03$.
  • Figure 3: Parametric Estimates with $\omega_{0} = 4.5 rad/s$. $-\!\!-$: actual value and its estimates for $V_{th,1}$; $-\:-$: actual value and its estimates for $V_{th,2}$. $\nabla$: estimate with $\widehat{\phi}_{u}(\mathbf{i}\omega_{0},\theta)$; $\star$: estimate with $\widehat{\phi}_{u}(\mathbf{i}\omega_{0},\theta)$ and $\widehat{\phi}_{u}(\mathbf{i}\omega_{0} + \mathbf{i}\omega_{0},\theta)$; $\bf\Diamond$: estimate with $\widehat{\phi}_{u}(\mathbf{i}\omega_{0},\theta)$, $\widehat{\phi}_{u}(\mathbf{i}\omega_{0} + \mathbf{i}\omega_{0},\theta)$ and $\widehat{\phi_{u}}(\mathbf{i}\omega_{0}, \mathbf{i}\omega_{0}, \theta)$.
  • Figure 4: Nonparametric Estimates with $\omega_{0} = 19.5 rad/s$. $-\!\!-$: actual value and its estimates for the 1st element; $-\:-$: actual value and its estimates for the 2nd element. $\nabla$: estimate with $\sigma=0.01$; $\star$: estimate with $\sigma=0.02$; $\bf\Diamond$: estimate with $\sigma=0.03$.
  • Figure 5: Parametric Estimates with $\omega_{0} = 19.5 rad/s$. $-\!\!-$: actual value and its estimates for $V_{th,1}$; $-\:-$: actual value and its estimates for $V_{th,2}$. $\nabla$: estimate with $\widehat{\phi}_{u}(\mathbf{i}\omega_{0},\theta)$; $\star$: estimate with $\widehat{\phi}_{u}(\mathbf{i}\omega_{0},\theta)$ and $\widehat{\phi}_{u}(\mathbf{i}\omega_{0} + \mathbf{i}\omega_{0},\theta)$; $\bf\Diamond$: estimate with $\widehat{\phi}_{u}(\mathbf{i}\omega_{0},\theta)$, $\widehat{\phi}_{u}(\mathbf{i}\omega_{0} + \mathbf{i}\omega_{0},\theta)$ and $\widehat{\phi_{u}}(\mathbf{i}\omega_{0}, \mathbf{i}\omega_{0}, \theta)$.

Theorems & Definitions (7)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Corollary 1
  • Lemma 3
  • Theorem 2
  • Theorem 3