Lifetime prognostics for deteriorating systems with time-varying random jumps

Highlights

• A degradation model with time-varying random jumps is presented.

• The approximated analytical lifetime of the proposed model is derived.

• A two-step parameters estimation method for linear model is proposed.

Abstract

In this paper, we propose a jump diffusion process with non-homogeneous compound Poisson process to model the degradation process with randomly occurring jumps, which combines two stochastic processes, i.e., traditional diffusion process to describe the continuous degradation and non-homogeneous compound Poisson process to depict random jumps with a time-varying intensity. The approximated analytical lifetime under the concept of the first passage time (FPT) is obtained by a time–space transformation technique. To identify the model parameters, we first present a general method based on Maximum Likelihood Estimation (MLE) for the proposed model, and then specifically provide a two-step approach for linear jump diffusion process via combining MLE and Expectation Conditional Maximization (ECM) algorithm. Finally, a numerical example and a study on the furnace wall are provided to illustrate the effectiveness of the proposed method.

Introduction

With the rapid development of technology, prognostics and health management (PHM) has attracted increasing attention and been applied to a large variety of industrial systems [1], [2]. Health prognostics, as an essential part of PHM, plays an increasingly important role in many engineering practice [3], [4]. Health deterioration of the systems often manifests as performance degradation, which results from physical deterioration, damage, environment erosion, running wear, etc [1], [2], [4]. In order to evaluate and predict the health state of degrading systems accurately, the general methods include two key steps: modeling the degradation paths of performance variables, and then conducting the lifetime or remaining useful life (RUL) estimation [5], [6]. According to Pecht’s research [1], he employs the techniques for modeling under the categories of physics of failure, data driven and fusion, where data-driven approach consists of the machine learning and statistics based approaches. Compared with other approaches, statistical data driven approach relies only on the available observed data and statistical models, and has advantages in mathematic properties of the estimated RUL [7]. As analyzed by Jardine et al. [8], statistical data driven approach is an effective way to achieve a well estimated lifetime and RUL. Nowadays, statistical data-driven approach has gained momentum. Particularly, Si et al. have systematically reviewed RUL estimation method based on statistical data driven approaches [7]. For example, Wiener process [9], Gamma process [10], Markov process [11], and inverse Gaussian (IG) process [12] have been widely used to model the degradation process for RUL estimation.

Due to different types of monitored degradation data, the statistical data-driven approaches could be classified into the methods based on directly observed data and the methods based on indirectly observed data [7]. Because of systems’ complexity and uncertainty, the directly observed data (e.g. fatigue crack, thickness of the furnace wall, bearing abrasion, etc.) are often unavailable [7], [8]. In this case, a widely adopted way is to measure some intermediate performance variables instead, such as the temperature sensors for the thickness of the furnace wall. As a result, the degradation process often has high dynamics, high fluctuations, and outstanding random jumps, differing from the continuous degradation case. On the other hand, in engineering practice, with the acceleration of the system’s deterioration, the stability of the system becomes worse gradually and the system is more vulnerable to the shocks. That may make some random jumps occur and their occurrence frequency increase with the aging of the system [13]. Thus, there are two impacts toward the degradation paths, i.e., time-varying random jumps and continuous degradation.

Considering the above practical desire, we propose a jump diffusion process with non-homogeneous compound Poisson process to describe such a kind of degradation process. In order to estimate the lifetime and RUL of the proposed model under the concept of the FPT and then apply such the results into practical usage, two main works have been completed as follows. Explicit forms of the lifetime and RUL under the concept of the FPT are approximately derived using the time–space transformation method in which the FPT is transferred at the first time that the diffusion process exceeds a time-varying random threshold. For parameter estimation, we first give a general method based on Maximum Likelihood Estimation (MLE). Then a two-step parameters estimation method is presented via combining MLE and expectation conditional maximization (ECM) algorithm for linear jump diffusion process, which can improve the online performance. To illustrate the applicability and rationality of our method, numerical example and a study on the practical degradation data of the furnace wall are provided.

The remainder parts are organized as follows. In Section 2, the recently related literatures are reviewed and discussed. In Section 3, the motivating examples and problem formulation are given. Section 4 includes the main result of RUL estimation. In Section 5, a new stochastic model is proposed to describe the degradation path and the framework of the ECM algorithm for unknown model parameters is presented. Two illustrative examples are presented to illustrate and demonstrate the proposed model in Section 6. This paper is concluded in Section 7.