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Recursive Identification of Structured Systems: An Instrumental-Variable Approach Applied to Mechanical Systems

Koen Classens, Rodrigo A. González, Tom Oomen

TL;DR

This paper tackles online identification of time-varying, interpretable dynamics by adopting a continuous-time additive model structure. It develops a recursive identification framework that combines block-coordinate descent with refined instrumental-variable iterations (additive SRIVC) to estimate time-varying submodels $G_i(p,t, heta_i)$ in both open- and closed-loop data. A consistency analysis under $oldsymbol{\alpha_k=1}$ is provided, along with practical guidance for initialization, stability, and computational efficiency; the method also accommodates marginally stable systems through first-submodel adjustments. Numerical simulations and a real-time experimental validation on a flexible beam demonstrate the ability to track shifting resonances online with parsimonious, physically interpretable parameters, enabling improved online monitoring and control for mechanical systems.

Abstract

Online system identification algorithms are widely used for monitoring, diagnostics and control by continuously adapting to time-varying dynamics. Typically, these algorithms consider a model structure that lacks parsimony and offers limited physical interpretability. The objective of this paper is to develop a real-time parameter estimation algorithm aimed at identifying time-varying dynamics within an interpretable model structure. An additive model structure is adopted for this purpose, which offers enhanced parsimony and is shown to be particularly suitable for mechanical systems. The proposed approach integrates the recursive simplified refined instrumental variable method with block-coordinate descent to minimize an exponentially-weighted output error cost function. This novel recursive identification method delivers parametric continuous-time additive models and is applicable in both open-loop and closed-loop controlled systems. Its efficacy is shown using numerical simulations and is further validated using experimental data to detect the time-varying resonance dynamics of a flexible beam system. These results demonstrate the effectiveness of the proposed approach for online and interpretable estimation for advanced monitoring and control applications.

Recursive Identification of Structured Systems: An Instrumental-Variable Approach Applied to Mechanical Systems

TL;DR

This paper tackles online identification of time-varying, interpretable dynamics by adopting a continuous-time additive model structure. It develops a recursive identification framework that combines block-coordinate descent with refined instrumental-variable iterations (additive SRIVC) to estimate time-varying submodels in both open- and closed-loop data. A consistency analysis under is provided, along with practical guidance for initialization, stability, and computational efficiency; the method also accommodates marginally stable systems through first-submodel adjustments. Numerical simulations and a real-time experimental validation on a flexible beam demonstrate the ability to track shifting resonances online with parsimonious, physically interpretable parameters, enabling improved online monitoring and control for mechanical systems.

Abstract

Online system identification algorithms are widely used for monitoring, diagnostics and control by continuously adapting to time-varying dynamics. Typically, these algorithms consider a model structure that lacks parsimony and offers limited physical interpretability. The objective of this paper is to develop a real-time parameter estimation algorithm aimed at identifying time-varying dynamics within an interpretable model structure. An additive model structure is adopted for this purpose, which offers enhanced parsimony and is shown to be particularly suitable for mechanical systems. The proposed approach integrates the recursive simplified refined instrumental variable method with block-coordinate descent to minimize an exponentially-weighted output error cost function. This novel recursive identification method delivers parametric continuous-time additive models and is applicable in both open-loop and closed-loop controlled systems. Its efficacy is shown using numerical simulations and is further validated using experimental data to detect the time-varying resonance dynamics of a flexible beam system. These results demonstrate the effectiveness of the proposed approach for online and interpretable estimation for advanced monitoring and control applications.

Paper Structure

This paper contains 21 sections, 5 theorems, 84 equations, 12 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. System setup and problem formulation
  3. System setup
  4. Problem formulation
  5. Parsimony and interpretability in continuous-time system identification
  6. Recursive estimation of continuous-time systems: A block coordinate descent method
  7. Block coordinate descent
  8. Recursive SRIVC for additive open-loop systems
  9. Consistency analysis with $\alpha_{k} = 1$
  10. Summary iterative recursive estimation algorithm
  11. Closed-loop case
  12. Marginally stable systems
  13. Practical aspects for implementation
  14. Numerical Simulation
  15. Experimental validation on an overactuated and oversensed flexible beam setup
  16. ...and 6 more sections

Key Result

Proposition 1

gonzalez2023parsimonious Consider the system in eq:system, and the model structure $G(p)=\sum_{i=1}^K B_i(p)/A_i(p)$. Opting for the model structure $G(p)=B(p)/A(p)$ during identification (with the minimal relative degree $r$ that encompasses the true system) results in a lack of parsimony for the l where $r_i=n_i-m_i$ is the relative degree of $B_i(p)/A_i(p)$. The surplus in eq:excess is the numb

Figures (12)

  • Figure 1: Block diagram for the open-loop setting studied in this paper.
  • Figure 2: Block diagram for the closed-loop setting studied in this paper.
  • Figure 3: Block diagram of the block-coordinate descent RSRIVC for open-loop systems, summarized by \ref{['alg:algorithm1']}. The section featuring SRIVC iterations is highlighted (), see \ref{['eq:srivc']} to \ref{['eq:srivcinstrument1']}, alongside the block containing the $\mathcal{A}$-iterations (), described in \ref{['subsec:Block_coordinate_descent']}.
  • Figure 4: Parameter estimates () and true parameters () related to the first submodel with decreasing denominator parameters.
  • Figure 5: Parameter estimates () and true parameters () related to the second submodel with decreasing damping coefficient $\zeta_2$, which corresponds to an increasing $a_{21}$.
  • ...and 7 more figures

Theorems & Definitions (20)

  • Proposition 1
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Theorem 1: Global convergence of Algorithm \ref{['alg:algorithm0']}
  • proof
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • ...and 10 more