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Merging Parameter Estimation and Classification Using LASSO

Le Wang, Ying Wang, Yu Qiu, Mian Li, Håkan Hjalmarsson

TL;DR

The paper tackles soft sensing for wheel-force estimation across multiple operating conditions with many candidate sensors. It proposes a unified criterion that jointly minimizes data fit, inter-model distance, and sensor sparsity within MISO-FIR models, formalized as a constrained objective $\min_{\theta_1,...,\theta_K} \sum_k \|Y_k-\Phi_k \theta_k\|_2^2 + (1/2) \lambda_1 \sum_{i,j} \|\theta_i-\theta_j\|_2 + \lambda_2 \sum_{k} \sum_{j} \|\theta_k^{(j)}\|_1$, to merge conditions and prune sensors. Hyperparameters are bounded by $\lambda_{1\rm{max}}$ and $\lambda_{2\rm{max}}$, and tuned via grid search on validation data, followed by a K-means clustering post-process to finalize model groups. The approach is validated on real prototype-vehicle data, showing that BR30 and BR40 share a model while WBA40 requires a separate model, with substantial sparsity indicating irrelevant sensors and improved evaluation metrics compared to standard least-squares. The work demonstrates a practical method to reduce model count and sensor reliance in edge-deployable soft sensors for vehicle applications, with potential extensions to more scenarios and conditions.

Abstract

Soft sensing is a way to indirectly obtain information of signals for which direct sensing is difficult or prohibitively expensive. It may not \textit{a priori} be evident which sensors provide useful information about the target signal, and various operating conditions often necessitate different models. In this paper, we provide a systematic method to construct a soft sensor that can deal with these issues. We propose a single estimation criterion, where the objectives are encoded in terms of model fit, model sparsity (reducing the number of different models), and model parameter coefficient sparsity (to exclude irrelevant sensors). The proposed method is tested on real-world scenarios involving prototype vehicles, demonstrating its effectiveness.

Merging Parameter Estimation and Classification Using LASSO

TL;DR

The paper tackles soft sensing for wheel-force estimation across multiple operating conditions with many candidate sensors. It proposes a unified criterion that jointly minimizes data fit, inter-model distance, and sensor sparsity within MISO-FIR models, formalized as a constrained objective , to merge conditions and prune sensors. Hyperparameters are bounded by and , and tuned via grid search on validation data, followed by a K-means clustering post-process to finalize model groups. The approach is validated on real prototype-vehicle data, showing that BR30 and BR40 share a model while WBA40 requires a separate model, with substantial sparsity indicating irrelevant sensors and improved evaluation metrics compared to standard least-squares. The work demonstrates a practical method to reduce model count and sensor reliance in edge-deployable soft sensors for vehicle applications, with potential extensions to more scenarios and conditions.

Abstract

Soft sensing is a way to indirectly obtain information of signals for which direct sensing is difficult or prohibitively expensive. It may not \textit{a priori} be evident which sensors provide useful information about the target signal, and various operating conditions often necessitate different models. In this paper, we provide a systematic method to construct a soft sensor that can deal with these issues. We propose a single estimation criterion, where the objectives are encoded in terms of model fit, model sparsity (reducing the number of different models), and model parameter coefficient sparsity (to exclude irrelevant sensors). The proposed method is tested on real-world scenarios involving prototype vehicles, demonstrating its effectiveness.
Paper Structure (14 sections, 1 theorem, 9 equations, 11 figures, 4 tables)

This paper contains 14 sections, 1 theorem, 9 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Methodology
  4. Parameter Difference $\ell_2$-norm $\&$ LASSO Regularization
  5. Choosing the Hyperparameters
  6. Clustering
  7. Summary
  8. Experiments
  9. Experimetal Setup
  10. Experimental Results
  11. Parameter Comparison
  12. Model Classification
  13. Model Evaluation
  14. Conclusion

Key Result

Theorem 1

Let $Y_k \in \mathbb{R}^{M_k}$, $\Phi_k \in \mathbb{R}^{M_k \times n_{\theta}}$, $\theta_k \in \mathbb{R}^{n_{\theta}}, k=1,\dots,K$, where $n_{\theta}=n \times J$, $M_k$ is the size of $Y_k$. Define where Given that $\lambda_1 \geq \lambda_{1\rm{max}}$ and $\Phi_k^ {\rm{T}} \Phi_k \succ 0$, $\forall k \in {1,\dots, K}$, it holds that the solution is given by

Figures (11)

  • Figure 1: Working principle for online stage and offline stage.
  • Figure 2: The torque sensors installed on the field test vehicle.
  • Figure 3: The accelerometers (three-axis) installed on wheels.
  • Figure 4: Least-squares estimate of $\theta$ for the three used working conditions.
  • Figure 5: The resulting parameter estimates for the three used working conditions when the proposed algorithm is used.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Theorem 1