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In signal processing, a continuous-time signal can be sampled using an impulse-train sampling technique, followed by the zero-order hold method. Impulse-train sampling involves the use of a periodic impulse train, which consists of a series of delta functions spaced at regular intervals determined by the sampling period. When a continuous-time signal is multiplied by this impulse train, it generates impulses with amplitudes corresponding to the signal's values at the sampling points.
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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
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Soft sensor for nonuniform sampling nonlinear dynamic process using irregular-time-interval latent probabilistic

Zhengxuan Zhang1, Xu Yang2, Yuri A W Shardt3

  • 1Key Laboratory of Knowledge Automation for Industrial Processes of the Ministry of Education, School of Automation and Electrical Engineering, University of Science and Technology Beijing, 100083, Beijing, China.

ISA Transactions
|November 14, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces a new deep learning model, the irregular-time-interval latent probabilistic predictability embedding supervised deep network (ILPPSDN), for industrial soft sensing. The ILPPSDN effectively handles nonlinear dynamic processes with nonuniformly sampled data, significantly improving prediction accuracy.

Keywords:
Dynamic latent variableIrregular-time intervalLatent probabilistic predictabilityNonuniform samplingSoft sensingSupervised deep network

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Area of Science:

  • Chemical Engineering
  • Data Science
  • Machine Learning

Background:

  • Dynamic latent variable (DLV) models are crucial for industrial soft sensing but struggle with nonlinear dynamics and nonuniform data.
  • Conventional DLV models are limited to linear feature extraction and exhibit poor performance with irregularly sampled data.

Purpose of the Study:

  • To propose an advanced soft sensor, the irregular-time-interval latent probabilistic predictability embedding supervised deep network (ILPPSDN), for nonlinear dynamic processes with nonuniform sampling.
  • To enhance feature predictability and model latent temporal dependencies in soft sensing applications.

Main Methods:

  • Developed an ILPPSDN incorporating a prediction regularization term in the autoencoder's decoding loss.
  • Utilized an ordinary differential equation network to parameterize the internal state derivative within a variational recurrent neural network.
  • Implemented unified training for all network components and employed pre-training and supervised fine-tuning for soft sensor development.

Main Results:

  • The ILPPSDN achieved significant reductions in root mean square error (RMSE) across various uneven sampling ratios in debutanizer and sulfur recovery units.
  • Demonstrated RMSE reductions of at least 21.1% to 26.1% in industrial case studies.
  • Ablation studies confirmed the proposed method's effectiveness, reducing RMSE by at least 5% and 6% in respective industrial cases.

Conclusions:

  • The proposed ILPPSDN offers a robust solution for soft sensing in nonlinear dynamic processes with nonuniformly sampled data.
  • This deep learning approach enhances feature predictability and temporal dependency modeling, leading to superior performance over conventional methods.
  • The ILPPSDN demonstrates practical applicability and significant improvements in industrial process monitoring and control.