Reliability analysis for a Wiener degradation process model under changing failure thresholds

Remaining useful life (RUL) estimation based on condition monitoring data is central to condition based maintenance (CBM). In the current methods about the Wiener process based RUL estimation, the randomness of the failure threshold has not been studied thoroughly. In this work, by using the truncated normal distribution to model random failure threshold (RFT), an analytical and closed-form RUL distribution based on the current observed data was derived considering the posterior distribution of the drift parameter. Then, the Bayesian method was used to update the prior estimation of failure threshold. To solve the uncertainty of the censored in situ data of failure threshold, the expectation maximization (EM) algorithm is used to calculate the posteriori estimation of failure threshold. Numerical examples show that considering the randomness of the failure threshold and updating the prior information of RFT could improve the accuracy of real time RUL estimation.

This paper evaluates the remaining useful life (RUL) for the accelerated Wiener process with mixed random effects and measurement errors. In the discussed model, the random effects are included in both the drift and diffusion parameters of the Wiener process to model the heterogeneity among different degradation paths of products. The measurement errors are introduced to describe the inaccuracy during the measurement process. The effect of the accelerated stress level is included in the drift parameter. The expectation‐maximization (EM) algorithm is applied to obtain the estimation of the model parameters. The expressions of the probability density functions (PDFs) and cumulative distribution functions (CDFs) of the lifetime and RUL distributions of products under the normal usage stress level are derived theoretically. Finally, an example of stress relaxation data is used to illustrate the effectiveness of the developed model and method.

In this paper, a novel approach is proposed to determine target reliability index based on modeling performance degradation of concrete structures as a stochastic process. First, the Wiener process with drift is used to simulate the degradation process of relative dynamic elastic modulus of concrete, and the drift coefficient and diffusion coefficient of Wiener process are estimated according to the experimental data of degradation of relative dynamic elastic modulus. Then, based on the relations between concrete compressive strength, dynamic elastic modulus, and relative dynamic elastic modulus, the relationship between compressive strength and the amount of degradation of relative dynamic elastic modulus are deduced, and then, the statistical parameters of concrete compressive strength under different service time can be calculated. Finally, based on the established limit state equation of flexural capacity of the weak section and the statistical parameters of compressive strength at different times, the reliability index of a structure under different service life is calculated. On the other hand, according to the drift Wiener process model of relative dynamic elastic modulus degradation of concrete, the time‐varying failure probability of structural durability degradation is calculated based on the first‐passage failure probability formula of drift Wiener process. Taking a concrete plant structure as an example, the time‐varying reliability index calculated based on the limit state equation and the time‐varying failure probability obtained based on the first‐passage failure probability analysis of drift Wiener process are compared, and then the recommended values of target reliability index of structural bearing capacity are comprehensively determined.

In this work, we focus on the development of a parametric drift detection and isolation (PDDI) method for the handling of batch-to-batch parametric drifts in a batch crystallization process used to produce hen egg-white (HEW) lysozyme crystals. We consider that the batch crystallization process is controlled by an in-batch model predictive control (MPC) system and is subject to batch-to-batch parametric drifts in the solubility, growth rates, continuous-phase mass and energy balance parameters, and nucleation rate. The proposed PDDI scheme consists of two parts: preparatory stage before batch-to-batch operation and post-batch stage during batch-to-batch operation. The goal of the preparatory stage is to compute the threshold values and signatures for each parametric drift using simulations and batch process common cause variance described by noise. During batch-to-batch operation, the proposed PDDI system monitors closed-loop process residuals, which are computed by taking the difference between the time profiles of the states obtained through in-batch and postbatch measurements from the time profiles of the states obtained from the drift-free simulation with noise. While the measurements of the protein solute concentration and the temperature in the crystallizer are available in real-time, post-batch measurements are usually available for the quality of the crystal products (e.g., number of crystals, average crystal size and shape) and this key characteristic is taken into account in the PDDI method. We then compare the residuals with signatures obtained in the preparatory stage for each parametric drift for isolation of a parametric drift. The PDDI system estimates the magnitude of the parametric drift and updates the parameters of the batch process model used in the in-batch MPC system to compute a set of jacket temperatures for the production of crystals with a desired shape distribution in the next batch. The performance of the MPC with the proposed PDDI scheme is demonstrated by applying it to a multiscale simulation of a batch crystallization process with parametric drifts in the solubility and crystal growth rates. The closed-loop system simulations demonstrate that crystals with a crystal shape distribution that is closer to a desired set-point value are produced under a parametric-drift handling scheme that integrates the in-batch MPC with the proposed PDDI system compared to those under the MPC with the nominal process model.

Owing to operating condition changing, physical mutation, and sudden shocks, degradation trajectories usually exhibit multi-phase features, and the abrupt jump often appears at the changing time, which makes the traditional methods of lifetime estimation unavailable. In this paper, we mainly focus on how to estimate the lifetime of the multi-phase degradation process with abrupt jumps at the change points under the concept of the first passage time (FPT). Firstly, a multi-phase degradation model with jumps based on the Wiener process is formulated to describe the multi-phase degradation pattern. Then, we attain the lifetime’s closed-form expression for the two-phase model with fixed jump relying on the distribution of the degradation state at the change point. Furthermore, we continue to investigate the lifetime estimation of the degradation process with random effect caused by unit-to-unit variability and the multi-phase degradation process. We extend the results of the two-phase case with fixed parameters to these two cases. For better implementation, a model identification method with off-line and on-line parts based on Expectation Maximization (EM) algorithm and Bayesian rule is proposed. Finally, a numerical case study and a practical example of gyro are provided for illustration.

On the basis of the principle of degradation mechanism invariance, a Wiener degradation process with random drift parameter is used to model the data collected from the constant stress accelerated degradation test. Small‐sample statistical inference method for this model is proposed. On the basis of Fisher's method, a test statistic is proposed to test if there is unit‐to‐unit variability in the population. For reliability inference, the quantities of interest are the quantile function, the reliability function, and the mean time to failure at the designed stress level. Because it is challenging to obtain exact confidence intervals (CIs) for these quantities, a regression type of model is used to construct pivotal quantities, and we develop generalized confidence intervals (GCIs) procedure for those quantities of interest. Generalized prediction interval for future degradation value at designed stress level is also discussed. A Monte Carlo simulation study is used to demonstrate the benefits of our procedures. Through simulation comparison, it is found that the coverage proportions of the proposed GCIs are better than that of the Wald CIs and GCIs have good properties even when there are only a small number of test samples available. Finally, a real example is used to illustrate the developed procedures.

A COMPARISON OF IDENTIFICATION ALGORITHMS FOR RAPIDLY TIME-VARYING PARAMETERS ON A REAL PROCESS

Methods for predicting the remaining useful life of equipment in consideration of the random failure threshold

The two dependent competing risks are soft failure due to aging degradation and fragmentation caused by shocks and hard failure due to spring breakage caused by the same shock process. Considering the complexity of the product itself and the instability of the working environment, this study proposed a generalized reliability model for systems experiencing dependent competing failure processes (DCFPs) of degradation and random shocks, which considered two kinds of DCFP: (1) shock process could affect soft failure thresholds; (2) degradation process could affect hard failure thresholds. In case (1), we considered the effect that a cumulative number of shocks above a certain magnitude could have on the change in the soft failure threshold. In case (2), we considered not only the shock process's impact on the soft failure threshold but also the total degradation (including continuous degradation and sudden degradation caused by shock) on the hard failure threshold. The model captures the features that the shocks experienced by the system affect the degradation process, accelerating the system degradation and causing soft failures; the degradation process of the system affects the shock process, making the system more susceptible to failure from external shocks. Finally, an example using micro‐electro‐mechanical systems devices illustrates the effectiveness of the proposed approach with sensitivity analysis.

Using Motor Current Signature Analysis (MCSA) for reliability estimation of industrial robots can minimize the damage to mechanical systems and save economic costs. However, the load of the robot's reducer at the joint is changing at different stages of the mission, leading to the result that the amplitude of the motor current varies greatly throughout a single service cycle. Therefore, the method to estimate the reliability by presetting a single failure threshold is no longer applicable. In this paper, a degradation modelling method for systems with periodic and multi-stage mission is proposed, which considers different degradation rates and failure thresholds at each mission stages. Wiener process is used to model the degradation process, and the drift coefficient is considered as a random variable describing the system degradation rate. The analytical expression of reliability function for the reducer with periodic multi-stage tasks is derived, and the reliability evaluation is carried out by combining the likelihood estimation method. The effectiveness of this method is illustrated through the harmonic reducer test.

The degradation data of highly reliable products are usually analyzed by stochastic process models, such as Wiener process, gamma process and inverse Gaussian process models. If such a specific degradation model is wrongly assumed, then poor analysis results of reliability assessment would be obtained. Therefore, a class of exponential-dispersion processes, named Tweedie exponential-dispersion process (TEDP), is proposed to describe the products’ degradation paths. The TEDP model which comprises the aforementioned stochastic processes as its special cases, is more flexible and applicable for degradation modeling. Considering the nonlinear characteristics of degradation paths and the unit-to-unit variability among the product units, random-effect models are established based on the nonlinear TEDP models with random drift and dispersion parameters. To improve the mathematical tractability of these models, the variational inference, expectation maximization algorithm and differential evolution algorithm are used to estimate the unknown model parameters. Furthermore, two nonlinear TEDP models with accelerated factors and random effects are developed for accelerated degradation analysis. Finally, a simulation study and three real applications are presented to show the effectiveness and superiority of the proposed models and methods.

The development of models and methods for cure rate estimation has recently burgeoned into an important subfield of survival analysis. Much of the literature focuses on the standard mixture model. Recently, process-based models have been suggested. We focus on several models based on first passage times for Wiener processes. Whitmore and others have studied these models in a variety of contexts. Lee and Whitmore (Stat Sci 21(4):501-513, 2006) give a comprehensive review of a variety of first hitting time models and briefly discuss their potential as cure rate models. In this paper, we study the Wiener process with negative drift as a possible cure rate model but the resulting defective inverse Gaussian model is found to provide a poor fit in some cases. Several possible modifications are then suggested, which improve the defective inverse Gaussian. These modifications include: the inverse Gaussian cure rate mixture model; a mixture of two inverse Gaussian models; incorporation of heterogeneity in the drift parameter; and the addition of a second absorbing barrier to the Wiener process, representing an immunity threshold. This class of process-based models is a useful alternative to the standard model and provides an improved fit compared to the standard model when applied to many of the datasets that we have studied. Implementation of this class of models is facilitated using expectation-maximization (EM) algorithms and variants thereof, including the gradient EM algorithm. Parameter estimates for each of these EM algorithms are given and the proposed models are applied to both real and simulated data, where they perform well.

The three-axis attitude tracking manoeuvre and vibration suppression of a flexible spacecraft in the presence of external disturbances are investigated in this paper. The spacecraft consists of a rigid hub and two flexible appendages. The Euler–Bernoulli beam theory is used to model the flexible parts. The attitude dynamic equations of motion are derived using the law of conservation of angular momentum, and the flexural equations are derived. The attitude of the spacecraft is represented using the quaternion parameters. The controller is designed based on the super-twisting sliding mode control. The sliding surfaces are introduced and the global asymptotic stability of the flexible spacecraft on the sliding surfaces is assured via Lyapunov method. The control law is designed such that the sliding condition is satisfied and the system reaches the sliding surfaces in finite time. The simulation results verify the performance of the controller in the presence of bounded disturbances, sensor noises and abrupt changes in parameters.

The control problem of an unknown nonlinear dynamic system which contains the abrupt changes of parameters is concerned. Multiple models based on dynamic neural networks are used to approximate the dynamic character of unknown system. Different controllers based on these models and an effectively switching mechanism are applied to an unknown system to trace a reference trajectory. Further, we propose different switching and turning schemes for adaptive control which combine fixed and adaptive models. From the simulation, it can be shown that the multiple model adaptive control method proposed in this paper can improve the control performance greatly compared with the conventional adaptive control.

current distribution is found with Eq. (17). The cube symmetry permits easy comparison of the solutions obtained in the two cases. Figures 3, 4 show distributions of the longitudinal (v) and transverse (w) current components on the face surface for the first case (solid line), while the points denote results for the second case. The abscissa denotes distance along a line on the body face which corresponds to the v coordinate. It follows from the symmetry of the cube that on areas of the surface located identically relative to the dipole in the first and second cases, the currents should be identical. The practical coincidence of the results obtained indicates the validity of the proposed algorithm. In the calculations the cube faces were divided into elementary rectangular areas. It was found that change in the linear dimensions of the latter over the range 0.1-0.02X produces no significant change in the solution. The time required for the calculation of one discrete azimuthal harmonic for the excitation of a cube with element dimensions of the order of 0.05X comprised 2 rain on an ES-1040 computer.