Data-driven feature extraction for machine health monitoring has garnered significant attention, yet two key limitations remain unaddressed: lack of interpretability and the need for extensive historical fault data. To overcome these problems, an online two-dimensional optimization framework is proposed that enables interpretable time-frequency feature extraction and health index (HI) construction without requiring faulty samples for model training. Our approach introduces a convex hull-based closest point optimization model for estimating time-frequency instances and learning interpretable time-frequency features. By leveraging a small set of baseline vibration samples and recent online data, rapid fault diagnosis can be achieved based on optimized interpretable time-frequency features. This method also facilitates long-term degradation tracking by constructing and updating an HI from collected time-frequency spectrograms. Once machine faults appear, updated time-frequency features can show apparent and interpretable fault signatures for prompt fault alarming. Moreover, the proposed framework allows continuous HI updates for incipient fault detection and degradation tracking. The proposed framework is validated by using two run-to-failure datasets and ablation experiments are conducted to demonstrate its superiority.

Machine life cycle performance assessment is of great significance to use a health index to inform the time of incipient fault initiation in a normal stage and realize fault identification and fault trending in a performance degradation stage. However, most existing works consider using unexplainable model parameters and historical data to build models and infer their off-line parameters for machine life cycle performance assessment. To overcome these limitations, an online piecewise convex-optimization interpretable weight learning framework without needing any historical abnormal and faulty data is proposed in this article to generate a piecewise health index to practically implement machine life cycle performance assessment. Firstly, based on a separation criterion, the first submodel in the proposed framework is built to detect the time of incipient fault initiation. Here, the piecewise health index generated by the first submodel is continuously updated by on-line monitoring data to timely detect the occurrence of any abnormal health conditions. Secondly, once the time of incipient fault initiation is informed, online updated model weights are highly correlated with fault characteristic frequencies and informative frequency bands for immediate fault identification. Simultaneously, the second submodel integrated with monotonicity and fitness properties in the proposed framework is triggered to generate the piecewise health index to realize overall monotonic fault trending. The significance of this article is that only online monitoring data are used to continuously update interpretable model weights as fault frequencies and informative frequency bands to generate the proposed piecewise health index so as to practically realize machine life cycle performance assessment. Two run-to-failure cases are studied to show the effectiveness and superiority of the proposed framework.

Machine degradation modeling is an enabling methodology to use monitoring data to evaluate machine health conditions. Fault detection needs to confirm whether there exists an incipient fault in a machine while machine diagnostics require knowing where the fault occurs and checking a specific fault type. In this article, an anchor discrimination learning model (ADLM) for physics-informed machine degradation modeling is innovatively proposed to find a projection direction that minimizes a distance between an anchor and samples with a same label of the anchor, and simultaneously maximizes a distance between the anchor and samples with a different label of the anchor. Subsequently, the ADLM is mathematically derived and formulated as a generalized Rayleigh quotient. Instead of using hand-crafted features, this article directly inputs normal and abnormal raw square envelope spectra into the ADLM for machine degradation modeling and the responses of the ADLM, namely an optimal direction, can automatically localize informative frequency components for immediate machine fault detection and diagnostics. Unlike most data-driven methodologies, the proposed methodology is physics-informed and its outputs are capable of indicating physical fault frequencies and their relevant frequency bands for quick fault detection and diagnostics. Two experimental studies are conducted to verify the feasibility of the proposed ADLM for machine degradation modeling.

Performance degradation modeling is promising to construct an advanced health index (HI). Currently, domain knowledge including monotonicity, trendabilty, and identifiability has been widely recognized as desirable properties to evaluate the suitability of an HI. Nevertheless, a quantitative criterion to evaluate the suitability of the curvature of an HI is still lacked, which can be used to describe different machine degradation rates. In this article, a new property of an HI named “shapeness” is proposed and it expects that an HI would have a monotonic degradation rate when the HI has a monotonic curve. Subsequently, a mathematical quantitative formulation and optimization model of the shapeness is put forward to be integrated with other desirable properties, which can formulate a composite optimization degradation model. Here, the main variables of the proposed optimization degradation model are weights that are used to fuse spectra lines of vibration signals so that the sum of weighted spectra lines can be used to form a generalized HI. Besides, it is proved that the proposed optimization model is a convex optimization problem so that a global optimal solution is uniquely guaranteed. Two run-to-failure case studies are conducted and some comparisons with state-of-the-art models show that the proposed HI has better performance.

Convex hulls based maximum margin classification has been widely studied for machine fault diagnosis, while its exploration for machine degradation modeling is seldom reported. In this study, a sparse and flexible convex-hull representation for machine degradation modeling is proposed to realize degradation trajectory tracking and fault diagnosis at a same time. First, considering using vibration data as health monitoring signals, globally normal and abnormal spectral lines can be obtained based on the fast Fourier transform and they are, respectively, characterized as individually flexible convex hulls. Subsequently, a sparse and flexible convex-hull representation degradation model is constructed by simultaneously finding the closest pair of samples and its sparse regularization between normal and abnormal convex hulls. Finally, a health indicator can be developed for early fault detection and degradation trajectory tracking during a machine life cycle. Meanwhile, quick fault diagnosis can be realized by finding a difference between the optimal closest samples in a normal convex hull and an abnormal convex hull. Two experimental cases are used to show the effectiveness and superiority of the proposed model to recent existing works.

Prognostics and health management aims to use on-line sensor data to monitor and predict current and future health conditions of degraded systems and components. Nowadays, constructing composite health indices for characterizing current health conditions of a system from multiple degradation-based sensor data has attracted much attention. These kinds of “data-level” models show strong ability to provide a better degradation characterization of a degraded system than a model solely depending on data from an individual sensor. Although numerous efforts have been made to propose “data-level” fusion methodologies for process data, such as temperature, pressure, speeds, etc., little research has targeted “data-level” fusion models for nonprocess data, such as vibration and acoustic signals. In this article, a methodology for constructing a composite health index from fusion of spectrum amplitudes is proposed. Here, each spectrum amplitude can be regarded as “an individual sensor.” The goal of this article is that a composite health index generated from fusion of spectrum amplitudes can simultaneously detect incipient faults and provide a monotonically increasing trend for degradation assessment. Our proposed methodology was verified by two illustrative examples including gearbox run-to-failure vibration data and bearing run-to-failure vibration data. Results showed that our proposed methodology is better than popular sparse measures for gear and bearing health monitoring and degradation assessment.Note to Practitioners—Process data, such as temperature, pressure, speed, etc., are capable of directly showing degradation trends of degraded systems and components. “Data-level” fusion models can be directly used to fuse process data from multiple sensors to show a better-fused degradation trend than a sole trend obtained from an individual process data sensor. Being different from process data, nonprocess data, such as vibration and acoustic data, cannot be used to directly show degradation trends unless they are transformed into a health index. One of the benefits of nonprocess data is that they have been proved to be sensitive to incipient machine faults. Nevertheless, due to complicated transmission paths and multiple responses, transforming nonprocess data into a health index is still a challenging task. This article presents a methodology to fuse spectrum amplitudes to form a health index that can simultaneously detect incipient machine faults and assess monotonic machine degradation. The main idea of this article is to regard each spectrum amplitude as “an individual sensor” and the sum of weighted spectrum amplitudes as a health index. To implement the proposed methodology, it is necessary: 1) to transform temporal nonprocess data into frequency spectra by using the well-known Fourier transform; 2) to know two essential properties about detecting incipient machine faults and assessing machine degradation; and 3) to train weights based on the essential properties by any convex optimization algorithms. Once optimal weights are obtained, the health index has ability to simultaneously detect incipient machine faults and monotonically assess machine degradation.

Fault detection and degradation modeling are two main concerns in condition-based maintenance (CBM). The initial machine degradation is called first predicting time (FPT) or incipient failure time. FPT is typically assumed prior information. FPT detection aims to provide such prior information for subsequent degradation modeling in CBM. Moreover, the majority of existing methodologies regard FPT detection and degradation modeling as two separate tasks. A generic framework for integration of the FPT detection with degradation modeling is proposed in this article via fusion of spectrum amplitudes in the frequency domain to realise FPT detection and degradation modeling in a unified manner. First, a generalised health index is constructed using the sum of weighted spectrum amplitudes. Second, two properties are proposed to describe FPT detection and degradation modeling. Third, these two properties and their constraints are mathematically formulated as a quadratic programming model to find optimal weights for the fusion of spectrum amplitudes automatically. Finally, three illustrative examples are used to demonstrate the superiority of the proposed methodology over some existing commonly used sparse measures and a machine learning method in the FPT detection and degradation modeling.