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Statistically Optimal Structured Additive MIMO Continuous-time System Identification

Rodrigo A. González, Maarten van der Hulst, Koen Classens, Tom Oomen

TL;DR

This work addresses the challenge of identifying structured additive MIMO systems in continuous time from time-domain data. It introduces a two-stage method: first, a refined instrumental variable approach estimates an unconstrained additive MIMO model; second, a weighted nonlinear least-squares projection enforces structure, yielding structured additive models with minimal asymptotic covariance. Theoretical results establish generic consistency and asymptotic normality, with open-loop efficiency matching the CRLB and closed-loop efficiency bounded within IV limits, supplemented by extensive simulations on a 3-mass-spring-damper system. The approach delivers both statistical optimality and practical parsimony, useful for applications requiring structured, interpretable MIMO models under open- and closed-loop data collection.

Abstract

Many applications in mechanical, acoustic, and electronic engineering require estimating complex dynamical models, often represented as additive multi-input multi-output (MIMO) transfer functions with structural constraints. This paper introduces a two-stage procedure for estimating structured additive MIMO models, where structural constraints are enforced through a weighted nonlinear least-squares projection of the parameter vector initially estimated using a recently developed refined instrumental variables algorithm. The proposed approach is shown to be consistent and asymptotically efficient in open-loop scenarios. In closed-loop settings, it remains consistent despite potential noise model misspecification and achieves minimum covariance among all instrumental variable estimators. Extensive simulations are performed to validate the theoretical findings, and to show the efficacy of the proposed approach.

Statistically Optimal Structured Additive MIMO Continuous-time System Identification

TL;DR

This work addresses the challenge of identifying structured additive MIMO systems in continuous time from time-domain data. It introduces a two-stage method: first, a refined instrumental variable approach estimates an unconstrained additive MIMO model; second, a weighted nonlinear least-squares projection enforces structure, yielding structured additive models with minimal asymptotic covariance. Theoretical results establish generic consistency and asymptotic normality, with open-loop efficiency matching the CRLB and closed-loop efficiency bounded within IV limits, supplemented by extensive simulations on a 3-mass-spring-damper system. The approach delivers both statistical optimality and practical parsimony, useful for applications requiring structured, interpretable MIMO models under open- and closed-loop data collection.

Abstract

Many applications in mechanical, acoustic, and electronic engineering require estimating complex dynamical models, often represented as additive multi-input multi-output (MIMO) transfer functions with structural constraints. This paper introduces a two-stage procedure for estimating structured additive MIMO models, where structural constraints are enforced through a weighted nonlinear least-squares projection of the parameter vector initially estimated using a recently developed refined instrumental variables algorithm. The proposed approach is shown to be consistent and asymptotically efficient in open-loop scenarios. In closed-loop settings, it remains consistent despite potential noise model misspecification and achieves minimum covariance among all instrumental variable estimators. Extensive simulations are performed to validate the theoretical findings, and to show the efficacy of the proposed approach.

Paper Structure

This paper contains 12 sections, 6 theorems, 77 equations, 5 figures.

Table of Contents

  1. Introduction
  2. System and model setup
  3. Refined instrumental variable method for MIMO additive systems
  4. Statistical analysis
  5. Generic consistency
  6. Asymptotic distribution
  7. Estimation of structured additive models
  8. Simulation results
  9. Simulation settings
  10. Monte Carlo simulation results
  11. Conclusions
  12. Technical lemmas

Key Result

Theorem 1

Consider the proposed estimator (eq: CT - estimator) for the open- and closed-loop settings, and suppose Assumptions assumption: 1 - CT - consistency to assumption: 4 - CT - consistency hold. Then, the following statements are true:

Figures (5)

  • Figure 1: Block diagrams for the open (a) and closed-loop (b) system settings.
  • Figure 2: Multivariate simulation model of a flexible motion system with nominal parameter values $m_1 = m_2 =m_3 = 1\ \text{[kg]}$, $k_1 = k_2 = k_3 = 50\ \text{[kN/m]}$ and $d_1 = 0.4955\ \text{[kN/ms]}$, $d_2 = 0.1123\ \text{[kN/ms]}$, and $d_3 =0.1367\ \text{[kN/ms]}$. For this example, the damper coefficients were chosen such that the damping ratios associated to each natural frequency is equal to $0.02$.
  • Figure 3: Frequency response of the MIMO system under study.
  • Figure 4: Open-loop setting: MSEs of each estimated parameter with respect to the sample size. The proposed structured RIV method is consistent and delivers improved MSEs for the numerator coefficients.
  • Figure 5: Closed-loop setting: MSEs of each estimated parameter with respect to the sample size. The structured RIV method for closed-loop systems is consistent, while bias is observed if the the open-loop method is used.

Theorems & Definitions (6)

  • Theorem 1: Generic consistency
  • Theorem 2: Asymptotic distribution
  • Theorem 3: As. dist. structured additive model
  • Lemma 1
  • Lemma 2
  • Lemma 3