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On polynomial explicit partial estimator design for nonlinear systems with parametric uncertainties

Mazen Alamir

TL;DR

This work tackles partial estimation for nonlinear systems with parametric uncertainty by learning a data-driven map from past measurements to an observation target $z=T(x,u,p)$. It introduces a framework based on sparse multivariate polynomial identification (plars) learned from scenario-generated data, capturing the relationship from a window of measurements to $z$ while accounting for dispersion statistics $\mathcal{S}$. Validation on two nonlinear systems—the Electronic Throttle Control and the Lorentz oscillator—demonstrates that plars provides superior generalization with small data compared to ML/DL baselines, with near-perfect recovery in noiseless cases and robust performance under parameter dispersion and measurement noise. The results suggest that concise, sparse polynomial observers offer a practical path for reliable partial-state estimation in uncertain nonlinear systems, with potential extensions to rational models and more challenging problems.

Abstract

This paper investigates the idea of designing data-driven partial estimators for nonlinear systems showing parametric uncertainties using sparse multivariate polynomial relationships. A general framework is first presented and then validated on two illustrative examples with comparison to different possible Machine/Deep-Learning based alternatives. The results suggests the superiority of the proposed sparse identification scheme, at least when the learning data is small.

On polynomial explicit partial estimator design for nonlinear systems with parametric uncertainties

TL;DR

This work tackles partial estimation for nonlinear systems with parametric uncertainty by learning a data-driven map from past measurements to an observation target . It introduces a framework based on sparse multivariate polynomial identification (plars) learned from scenario-generated data, capturing the relationship from a window of measurements to while accounting for dispersion statistics . Validation on two nonlinear systems—the Electronic Throttle Control and the Lorentz oscillator—demonstrates that plars provides superior generalization with small data compared to ML/DL baselines, with near-perfect recovery in noiseless cases and robust performance under parameter dispersion and measurement noise. The results suggest that concise, sparse polynomial observers offer a practical path for reliable partial-state estimation in uncertain nonlinear systems, with potential extensions to rational models and more challenging problems.

Abstract

This paper investigates the idea of designing data-driven partial estimators for nonlinear systems showing parametric uncertainties using sparse multivariate polynomial relationships. A general framework is first presented and then validated on two illustrative examples with comparison to different possible Machine/Deep-Learning based alternatives. The results suggests the superiority of the proposed sparse identification scheme, at least when the learning data is small.

Paper Structure

This paper contains 11 sections, 17 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Definitions & notation
  3. Methodology
  4. Illustrative examples
  5. Electronic throttle controlled system
  6. The Lorentz oscillator
  7. Data generation, split and use
  8. Metrics used in the estimation error's assessment
  9. Fitting algorithms used in the comparison
  10. Results
  11. Conclusion & future works

Figures (4)

  • Figure 1: Partial estimation results (on unseen test data) for the observation target $z=x_2$ of ETC system with no parametric dispersion and different levels of noise.
  • Figure 2: Partial estimation results (on unseen test data) for the observation target $z=x_2$ of ETC system with different levels of parametric dispersion and measurement noise.
  • Figure 3: Partial estimation results (on unseen test data) for the observation target $z=x_2$ of Lorentz system with no parametric dispersion and different level of noise.
  • Figure 4: Partial estimation results (on unseen test data) for the observation target $z=x_2$ of Lorentz system with different levels of parametric dispersion and measurement noise.

Theorems & Definitions (1)

  • Definition 1: $M$-scenario