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Frequency domain identification for multivariable motion control systems: Applied to a prototype wafer stage

M. van der Hulst, R. A. González, K. Classens, P. Tacx, N. Dirkx, J. van de Wijdeven, T. Oomen

TL;DR

This work targets accurate parametric identification of multivariable continuous-time systems from frequency-domain data using additive transfer-function models: $\mathbf{P}(s,\boldsymbol{\beta}) = \sum_{i=1}^K \mathbf{P}_i(s,\boldsymbol{\theta}_i)$ with $\mathbf{P}_i(s,\boldsymbol{\theta}_i) = \dfrac{1}{s^{\ell_i} A_i(s)} \mathbf{B}_i(s)$. A frequency-domain refined instrumental-variable approach is developed to estimate the additive MIMO parameters by solving a weighted LS problem for $\boldsymbol{\beta}$, and it employs pseudolinear residuals with iterative IV updates to reach a stationary point. The method is validated on a prototype wafer-stage with $n_y=13$ outputs and $n_u=4$ inputs, achieving accurate FRF reproduction across a wide frequency range while capturing 17 flexible modes, demonstrating improved model parsimony and conditioning for high-precision control. This delivers physically interpretable, well-conditioned models suitable for controller design, monitoring, and fault diagnosis in high-tech motion systems.

Abstract

Multivariable parametric models are essential for optimizing the performance of high-tech systems. The main objective of this paper is to develop an identification strategy that provides accurate parametric models for complex multivariable systems. To achieve this, an additive model structure is adopted, offering advantages over traditional black-box model structures when considering physical systems. The introduced method minimizes a weighted least-squares criterion and uses an iterative linear regression algorithm to solve the estimation problem, achieving local optimality upon convergence. Experimental validation is conducted on a prototype wafer-stage system, featuring a large number of spatially distributed actuators and sensors and exhibiting complex flexible dynamic behavior, to evaluate performance and demonstrate the effectiveness of the proposed method.

Frequency domain identification for multivariable motion control systems: Applied to a prototype wafer stage

TL;DR

This work targets accurate parametric identification of multivariable continuous-time systems from frequency-domain data using additive transfer-function models: with . A frequency-domain refined instrumental-variable approach is developed to estimate the additive MIMO parameters by solving a weighted LS problem for , and it employs pseudolinear residuals with iterative IV updates to reach a stationary point. The method is validated on a prototype wafer-stage with outputs and inputs, achieving accurate FRF reproduction across a wide frequency range while capturing 17 flexible modes, demonstrating improved model parsimony and conditioning for high-precision control. This delivers physically interpretable, well-conditioned models suitable for controller design, monitoring, and fault diagnosis in high-tech motion systems.

Abstract

Multivariable parametric models are essential for optimizing the performance of high-tech systems. The main objective of this paper is to develop an identification strategy that provides accurate parametric models for complex multivariable systems. To achieve this, an additive model structure is adopted, offering advantages over traditional black-box model structures when considering physical systems. The introduced method minimizes a weighted least-squares criterion and uses an iterative linear regression algorithm to solve the estimation problem, achieving local optimality upon convergence. Experimental validation is conducted on a prototype wafer-stage system, featuring a large number of spatially distributed actuators and sensors and exhibiting complex flexible dynamic behavior, to evaluate performance and demonstrate the effectiveness of the proposed method.

Paper Structure

This paper contains 16 sections, 1 theorem, 24 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setup and Problem formulation
  3. Experimental setup: prototype wafer-stage system
  4. Model structure
  5. Identification problem
  6. Identification strategy for additive MIMO systems
  7. Criterion for optimality
  8. Refined instrumental variables for additive systems
  9. Initialization
  10. Experimental validation
  11. Model structure
  12. Nonparametric modeling
  13. Weighting filter design
  14. Initialization and parametric identification
  15. Results
  16. ...and 1 more sections

Key Result

Lemma 1

The pseudolinear regression form of the residual eq: FREQ - residual corresponding to the $i$th submodel is expressed as with the regressor and where $\tilde{\mathbf{g}}_{f,i}(\omega_k, \boldsymbol{\beta}) = A^{-1}_i(\xi_k)\tilde{\mathbf{g}}_{i}(\omega_k, \boldsymbol{\beta})$ with $\tilde{\mathbf{g}}_i(\omega_k, \boldsymbol{\beta}) = \operatorname{vec}(\tilde{\mathbf{G}}_i(\omega_k, \boldsymbol{

Figures (4)

  • Figure 1: Experimental setup featuring a prototype wafer-stage system.
  • Figure 3: CMIF plot with $\sigma_i(\omega_k)$ the $i$th singular value of the FRF and () indicating the selected modes.
  • Figure 4: Element-wise Bode magnitude plot of the the nonparametric FRF measurement () and the estimated parametric model ().
  • Figure 5: Bode magnitude plot of the $\hat{G}_{4,13}$ entry, with the nonparametric FRF measurement (), the estimated parametric model () and residual ().

Theorems & Definitions (2)

  • Lemma 1
  • Remark 1