Observer-Based Virtual Sensing for Structural Health Monitoring: Experimental Validation on a Ship Hull Model

Abstract

Virtual sensing techniques, or soft sensing, aim at reconstructing physical quantities at locations where no direct measurements are available, enabling a complete representation of a system from sparse sensor information. These methods rely on the synergy between the measurement data, physics- or knowledge-based models, and statistical characterization of uncertainties. In recent years, virtual sensing has been consolidated into a broad set of approaches that include inverse finite-element formulations, data-driven methods, and control theory-based estimators. Among the latter, proportional observers (PO) and their multi-resolution extensions (MRPO) offer a practical alternative to Kalman-filters and “natural observers”. Their simple formulation, combined with frequency-band tuning, balances computational burden, noise robustness, and reconstruction accuracy. This paper presents an observer-based virtual sensing (OB-VS) framework for reconstructing strain fields in ship structures using a limited number of physical sensors. The approach builds on PO and MRPO, exploiting linear relations between the measurement vector and the estimated state-space variables. The method leverages the mechanical transfer function of the structure and a statistical description of uncertainties in the frequency domain, including measurement noise, modelling inaccuracies, and unknown excitation. A key advantage of the PO formulation is that the prediction-error covariance matrix depends quadratically on the observer gain(s), ensuring the existence of a unique global minimum. The MRPO enhances this approach by defining multiple POs, each one tailored for a specific frequency band into which the signal is decomposed via wavelet multi-resolution analysis. The methodology is applied to the experimental dataset relative to an elastic scaled ship-model campaign, performed within the “Digital Ship Structural Health Monitoring project” (dTHOR), granted by the European Defence Fund, aimed at developing a system based on innovative utilization of extensive on-board measurements, a comprehensive digital framework, and hybrid analysis and modelling. Measurements from 13 strain-gauges were utilized. Stepping forward with respect to previous 1D models, the present analysis employs a full 3D finite-element (FE) representation of the segmented hull. The observer is trained by extracting modal quantities from the FE solution (modal strains, modal mass, damping, and stiffness matrices) and constructing the corresponding transfer function. Because the FE model represents the dry hull, a correction is introduced to ensure consistency between numerical and experimental modal characteristics. The external (wave) excitation is estimated by combining acceleration, strain and rigid-body motion data. Performance is evaluated by computing the errors in signal reconstruction both in time (Time Response Assurance Criterion) and frequency (Frequency Response Assurance Criterion). Both PO and MRPO achieve a high fidelity in predicting the strains measured by sensors not involved in the virtual sensing process. An a-posteriori comparison of several sensor layouts highlights the possibility of achieving an optimal trade-off between instrumentation effort and reconstruction accuracy based on end-user needs. The OB-VS approach keeps accurate in signal tracking even under incorrect assumptions on the external excitation, thus demonstrating its robustness. Overall, this study highlights the potential of multi-resolution analysis for improving robustness to noise and spectral representation for real-time observer-based virtual sensing, enabling predictive digital-twin capabilities not only limited to on-board monitoring of ship structures.