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Sparse Broad Learning System via Sequential Threshold Least-Squares for Nonlinear System Identification under Noise

Zijing Li

TL;DR

This work tackles robust nonlinear system identification under sensor noise by replacing the dense ridge solution in Broad Learning System (BLS) with a Sequential Threshold Least-Squares (STLS)–driven sparsity mechanism. By adding an $L_0$ sparsity penalty and performing iterative hard-thresholding and subspace least-squares projections, the proposed Sparse Broad Learning System (S-BLS) automatically prunes noise-dominated connections while preserving fast analytical updates. The approach yields substantial sparsity (often >50–70% pruning) and improved noise immunity, demonstrated on a numerical nonlinear system and a nonlinear CSTR benchmark, where S-BLS outperforms standard BLS and WBLS, especially under high noise. The results indicate that S-BLS offers a fast, robust, and interpretable alternative for industrial system identification in noisy environments, with clear gains in both accuracy and model simplicity.

Abstract

The Broad Learning System (BLS) has gained significant attention for its computational efficiency and less network parameters compared to deep learning structures. However, the standard BLS relies on the pseudoinverse solution, which minimizes the mean square error with $L_2$-norm but lacks robustness against sensor noise and outliers common in industrial environments. To address this limitation, this paper proposes a novel Sparse Broad Learning System (S-BLS) framework. Instead of the traditional ridge regression, we incorporate the Sequential Threshold Least-Squares (STLS) algorithm -- originally utilized in the sparse identification of nonlinear dynamics (SINDy) -- into the output weight learning process of BLS. By iteratively thresholding small coefficients, the proposed method promotes sparsity in the output weights, effectively filtering out noise components while maintaining modeling accuracy. This approach falls under the category of sparse regression and is particularly suitable for noisy environments. Experimental results on a numerical nonlinear system and a noisy Continuous Stirred Tank Reactor (CSTR) benchmark demonstrate that the proposed method is effective and achieves superior robustness compared to standard BLS.

Sparse Broad Learning System via Sequential Threshold Least-Squares for Nonlinear System Identification under Noise

TL;DR

This work tackles robust nonlinear system identification under sensor noise by replacing the dense ridge solution in Broad Learning System (BLS) with a Sequential Threshold Least-Squares (STLS)–driven sparsity mechanism. By adding an sparsity penalty and performing iterative hard-thresholding and subspace least-squares projections, the proposed Sparse Broad Learning System (S-BLS) automatically prunes noise-dominated connections while preserving fast analytical updates. The approach yields substantial sparsity (often >50–70% pruning) and improved noise immunity, demonstrated on a numerical nonlinear system and a nonlinear CSTR benchmark, where S-BLS outperforms standard BLS and WBLS, especially under high noise. The results indicate that S-BLS offers a fast, robust, and interpretable alternative for industrial system identification in noisy environments, with clear gains in both accuracy and model simplicity.

Abstract

The Broad Learning System (BLS) has gained significant attention for its computational efficiency and less network parameters compared to deep learning structures. However, the standard BLS relies on the pseudoinverse solution, which minimizes the mean square error with -norm but lacks robustness against sensor noise and outliers common in industrial environments. To address this limitation, this paper proposes a novel Sparse Broad Learning System (S-BLS) framework. Instead of the traditional ridge regression, we incorporate the Sequential Threshold Least-Squares (STLS) algorithm -- originally utilized in the sparse identification of nonlinear dynamics (SINDy) -- into the output weight learning process of BLS. By iteratively thresholding small coefficients, the proposed method promotes sparsity in the output weights, effectively filtering out noise components while maintaining modeling accuracy. This approach falls under the category of sparse regression and is particularly suitable for noisy environments. Experimental results on a numerical nonlinear system and a noisy Continuous Stirred Tank Reactor (CSTR) benchmark demonstrate that the proposed method is effective and achieves superior robustness compared to standard BLS.

Paper Structure

This paper contains 16 sections, 16 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Broad Learning System (BLS)
  4. Sparse Optimization Problem
  5. Proposed Method: S-BLS
  6. Model Formulation
  7. Sequential Thresholding Optimization
  8. Complexity Analysis
  9. Experiments and Analysis
  10. Experimental Setup
  11. Hyperparameter Settings
  12. Case Study 1: Numerical Nonlinear System
  13. Results and Analysis
  14. Case Study 2: CSTR Process Benchmark
  15. Results and Analysis
  16. ...and 1 more sections

Figures (2)

  • Figure 1: The architecture of the proposed Sparse Broad Learning System (S-BLS). The input $X$ is mapped to feature nodes $Z^n$, which are cascaded to enhancement nodes $H^m$. Both groups are concatenated to form the system matrix $A$. The output weights $W_{out}$ are solved sparsely using the Sequential Thresholded Least-Squares (STLS) algorithm.
  • Figure 2: Output tracking performance of the nonlinear system under noise level 0.4. S-BLS demonstrates superior smoothness and tracking accuracy compared to Standard BLS.