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Efficient identification of linear, parameter-varying, and nonlinear systems with noise models

Alberto Bemporad, Roland Tóth

TL;DR

Addresses the challenge of identifying a broad class of dynamical systems from data by introducing a unified framework that separates deterministic dynamics $G_ ext{o}$ and the noise model $H_ ext{o}$, and parameterizes nonlinearities with artificial neural networks. The approach yields a prediction-error minimization (PEM) estimator applicable to $LTI$, $LPV$ (external and self-scheduled), and $NL$ models, with efficient training via $L$-BFGS-$B$ and automatic differentiation in $JAX$. The authors prove convergence and consistency under mild assumptions and demonstrate that training times drop from hours to seconds while achieving or surpassing state-of-the-art accuracy on benchmarks including unbalanced disk and CMG systems. This work offers a scalable, rigorous tool for data-driven system identification across a spectrum of model classes and noise structures, enabling reliable controller design and adaptation.

Abstract

We present a general system identification procedure capable of estimating of a broad spectrum of state-space dynamical models, including linear time-invariant (LTI), linear parameter-varying} (LPV), and nonlinear (NL) dynamics, along with rather general classes of noise models. Similar to the LTI case, we show that for this general class of model structures, including the NL case, the model dynamics can be separated into a deterministic process and a stochastic noise part, allowing to seamlessly tune the complexity of the combined model both in terms of nonlinearity and noise modeling. We parameterize the involved nonlinear functional relations by means of artificial neural-networks (ANNs), although alternative parametric nonlinear mappings can also be used. To estimate the resulting model structures, we optimize a prediction-error-based criterion using an efficient combination of a constrained quasi-Newton approach and automatic differentiation, achieving training times in the order of seconds compared to existing state-of-the-art ANN methods which may require hours for models of similar complexity. We formally establish the consistency guarantees for the proposed approach and demonstrate its superior estimation accuracy and computational efficiency on several benchmark LTI, LPV, and NL system identification problems.

Efficient identification of linear, parameter-varying, and nonlinear systems with noise models

TL;DR

Addresses the challenge of identifying a broad class of dynamical systems from data by introducing a unified framework that separates deterministic dynamics and the noise model , and parameterizes nonlinearities with artificial neural networks. The approach yields a prediction-error minimization (PEM) estimator applicable to , (external and self-scheduled), and models, with efficient training via -BFGS- and automatic differentiation in . The authors prove convergence and consistency under mild assumptions and demonstrate that training times drop from hours to seconds while achieving or surpassing state-of-the-art accuracy on benchmarks including unbalanced disk and CMG systems. This work offers a scalable, rigorous tool for data-driven system identification across a spectrum of model classes and noise structures, enabling reliable controller design and adaptation.

Abstract

We present a general system identification procedure capable of estimating of a broad spectrum of state-space dynamical models, including linear time-invariant (LTI), linear parameter-varying} (LPV), and nonlinear (NL) dynamics, along with rather general classes of noise models. Similar to the LTI case, we show that for this general class of model structures, including the NL case, the model dynamics can be separated into a deterministic process and a stochastic noise part, allowing to seamlessly tune the complexity of the combined model both in terms of nonlinearity and noise modeling. We parameterize the involved nonlinear functional relations by means of artificial neural-networks (ANNs), although alternative parametric nonlinear mappings can also be used. To estimate the resulting model structures, we optimize a prediction-error-based criterion using an efficient combination of a constrained quasi-Newton approach and automatic differentiation, achieving training times in the order of seconds compared to existing state-of-the-art ANN methods which may require hours for models of similar complexity. We formally establish the consistency guarantees for the proposed approach and demonstrate its superior estimation accuracy and computational efficiency on several benchmark LTI, LPV, and NL system identification problems.

Paper Structure

This paper contains 34 sections, 3 theorems, 46 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Contribution
  3. Notation
  4. Identification Setting
  5. Data-generating system
  6. Model parameterization
  7. Identification problem
  8. Model structures
  9. LTI models
  10. LPV models
  11. General nonlinear models
  12. Identification method
  13. Combined plant and noise model learning
  14. Model structure selection
  15. Initialization
  16. ...and 19 more sections

Key Result

Theorem 1

A system given by model:innovation always has a representation in the form of eq:system:proc--eq:system:noise with $n_\mathrm{x},n_\mathrm{z}\leq n_\mathrm{w}$, such that for any sample path realization of $e\in\mathbb{R}^{\mathbb{Z}_0^+}$ and $u\in\mathbb{R}^{\mathbb{Z}_0^+}$ and any initial condit

Figures (3)

  • Figure 1: Separation of the data-generating system to a deterministic and stochastic part in the considered prediction-error-minimization setting.
  • Figure 2: Considered benchmark systems, whose simulation model is used for identification.
  • Figure 3: Prediction results of the self-scheduled LPV model with LTI noise model identified for the unbalanced disk.

Theorems & Definitions (3)

  • Theorem 1: Separation of process and noise models
  • Theorem 2: Convergence
  • Theorem 3: Consistency