Structured identification of multivariable modal systems

TL;DR

The work addresses the challenge of extracting physically interpretable, minimal‑order modal models from frequency response data for high‑dimensional MIMO mechanical systems. It introduces a two‑stage framework: (i) a frequency‑domain refined instrumental variable method to estimate an additive MIMO model, and (ii) a projection via indirect prediction error method (IPEM) to enforce the modal rank constraints and obtain the modal parameters, accommodating both general viscous damping and proportional damping. Experimental validation on a prototype wafer‑stage with 13 inputs and 4 outputs yields a 40th‑order modal model with 3 rigid‑body and 17 flexible modes, closely matching the measured FRF and providing interpretable mode shapes. The approach offers computational efficiency, robust initialization, and a principled handling of rank constraints, enabling accurate, physically meaningful models for control design, validation, and monitoring of complex industrial systems.

Abstract

Physically interpretable models are essential for next-generation industrial systems, as these representations enable effective control, support design validation, and provide a foundation for monitoring strategies. The aim of this paper is to develop a system identification framework for estimating modal models of complex multivariable mechanical systems from frequency response data. To achieve this, a two-step structured identification algorithm is presented, where an additive model is first estimated using a refined instrumental variable method and subsequently projected onto a modal form. The developed identification method provides accurate, physically-relevant, minimal-order models, for both generally-damped and proportionally damped modal systems. The effectiveness of the proposed method is demonstrated through experimental validation on a prototype wafer-stage system, which features a large number of spatially distributed actuators and sensors and exhibits complex flexible dynamics.

Figures (8)

• Figure 1: Prototype experimental wafer-stage setup. The wafer chuck (1) is suspended by gravity compensators and actively controlled in six motion degrees of freedom. The sensory equipment is mounted on the metrology frame (2), while the actuators are mounted on the force frame (3). The metrology frame is suspended from the base frame (4) through an air suspension system.
• Figure 2: Wafer chuck with force frame extracted from the machine (left). Schematic overview of the featured actuators $u_i$ and sensors $z_i$ in the out-of-plane direction (right).
• Figure 3: Outline of the estimation procedure. In Stage 1, an additive parametric model is estimated by fitting directly to the measured FRF data. A linear LS problem is first solved to obtain an initial estimate, which is subsequently refined by the RIV method via iterative minimization of a nonlinear cost function. In Stage 2, the modal model is obtained directly from the Stage 1 solution. The IPEM problem is initialized using SVDs, which subsequently projects the Stage 1 solution onto the set of models satisfying $\boldsymbol{\beta} \in \mathcal{S}_2$, yielding the final modal model estimate.
• Figure 4: CMIF plot used for model order selection and initialization of the mode frequencies, with $\sigma_i(\omega_k)$ denoting the $i$th singular value of the measured FRF dataset at frequency $k$. The frequency locations of the flexible modes are indicated by peaks in the singular-value plot, with multiplicity determined by the number of singular values peaking at each frequency.
• Figure 5: Cost-function evolution over iterations. Stage 1 ($j \leq 10$) corresponds to the () RIV algorithm, while Stage 2 ($j > 10$) applies the () IPEM algorithm.

Theorems & Definitions (7)

• Remark 1
• Remark 2
• lemma 1
• proof
• Remark 3
• Remark 4
• Theorem 1: Eckart--Young--Mirsky Horn2013MatrixEdition