Short communication Brownian motion with adaptive drift for remaining useful life prediction: Revisited Dong Wang ⇑ , Kwok-Leung Tsui Department of Systems Engineering and Engineering Management, City University of Hong Kong, Hong Kong, China article info Article history: Received 23 February 2017 Received in revised form 27 June 2017 Accepted 8 July 2017 Keywords: Prognostics and health management Performance degradation assessment Remaining life prediction State space modelling Filtering abstract Linear Brownian motion with constant drift is widely used in remaining useful life predic- tions because its first hitting time follows the inverse Gaussian distribution. State space modelling of linear Brownian motion was proposed to make the drift coefficient adaptive and incorporate on-line measurements into the first hitting time distribution. Here, the drift coefficient followed the Gaussian distribution, and it was iteratively estimated by using Kalman filtering once a new measurement was available. Then, to model nonlinear degradation, linear Brownian motion with adaptive drift was extended to nonlinear Brownian motion with adaptive drift. However, in previous studies, an underlying assump- tion used in the state space modelling was that in the update phase of Kalman filtering, the predicted drift coefficient at the current time exactly equalled the posterior drift coefficient estimated at the previous time, which caused a contradiction with the predicted drift coef- ficient evolution driven by an additive Gaussian process noise. In this paper, to alleviate such an underlying assumption, a new state space model is constructed. As a result, in the update phase of Kalman filtering, the predicted drift coefficient at the current time evolves from the posterior drift coefficient at the previous time. Moreover, the optimal Kalman filtering gain for iteratively estimating the posterior drift coefficient at any time is mathematically derived. A discussion that theoretically explains the main reasons why the constructed state space model can result in high remaining useful life prediction accu- racies is provided. Finally, the proposed state space model and its associated Kalman filter- ing gain are applied to battery prognostics. Ó 2017 Elsevier Ltd. All rights reserved. 1. Introduction In the research community of prognostics and health management [1], the remaining useful life (RUL) prediction [2–4] under linear Brownian motion with constant drift [5,6] has attracted much attention because its first hitting time follows the inverse Gaussian distribution given a soft failure threshold. Here, the first hitting time is defined as the time when linear Brownian motion hits the soft failure threshold for the first time. Therefore, the difference between the first hitting time and the current prediction time can be regarded as the RUL. Nevertheless, linear Brownian motion with constant drift cannot be used to model the degradation of a specific product because the drift coefficient established by a population of historical degradation data is always fixed in linear Brownian motion for degradation modelling and RUL prediction. In the pioneering work of Wang et al. [7], the authors constructed a state space model of linear Brownian motion and used Kalman filtering to http://dx.doi.org/10.1016/j.ymssp.2017.07.015 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: dongwang4-c@my.cityu.edu.hk (D. Wang), kltsui@cityu.edu.hk (K.-L. Tsui). Mechanical Systems and Signal Processing 99 (2018) 691–701 Contents lists available at ScienceDirect Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp