Prognosis of Bearing Degeneration Using Adaptive Quaternion Least Mean Biquadrate Under Framework of Hypercomplex Data

Prognostics of bearing degeneration play a crucial role in implementation of systems maintenance strategies. Although numerous classical prognostic models have been developed, these models are carried out based on channel-wise processing and therefore fail to capture the inherent nonlinear and coupling nature of multi-dimensional or multi-channel data. To address this issue, a novel prognostics method based on adaptive quaternion-valued least-mean biquadrate (AQ-LMB) algorithm is proposed for a unified processing of hypercomplex data by the virtue of quaternion algebra. The cost function of the proposed AQ-LMB algorithm is designed via the biquadrate form of system output error to adapt to the more common nonlinear and non-Gaussian data, and the update weight vector is derived through Hamilton calculus instead of traditional complex gradient calculation. First, the time series of health indicators (e.g., root-mean-square, RMS) derived from historical data are decomposed by a nonconvex sparse regularization (SR) algorithm associated with a nonconvex penalty, that is, the low frequency trend component (LFC) and high frequency noise component (HFC) are obtained. Then both LFC and HFC are respectively predicted by the AQ-LMB algorithm. The final predicted health indicators can be obtained by integrating the correspondingly predicted LFC and HFC. The separate analysis of both sub-components makes it possible to distinguish their respective contributions to the entire degeneration process, thus avoiding false deviation and improving the prediction accuracy. Finally, the effectiveness of the proposed nonconvex SR and AQ-LMB approach in improving prognostic accuracy is illustrated via three-and four-dimensional run-to-failure datasets of the rolling bearings.